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\heading
Notes on Conformal Field Theory (incomplete)\\
by A. Beilinson, B. Feigin, B. Mazur\\
1991
\endheading


\bigskip
 
\heading 
\S 1.\ Tate's linear algebra 
\endheading 
 
\subheading{1.1\ Crossed modules and central extensions of Lie algebras}  We 
will need Lie and associative algebra versions of crossed modules: 
 
\proclaim{1.1.1\ Definition} (i) Let $L$ be a Lie algebra.  An $L$-crossed  
module is an $L$-module $L^\#$  together with a morphism   
$L^\#\ {\buildrel \partial\over\longrightarrow}\ L$ of $L$-modules.  For  
$\ell \in L$  we will denote the action of $L$ on $L^\#$  as $[\ell, \cdot\ ]$; 
so one has  $\partial[\ell, \widetilde{\ell}] = [\ell, \partial\widetilde{\ell}]$, 
$\widetilde{\ell} \in L^\#$.   
 
\noindent 
(ii) Let $R$  be an associative algebra.  An $R$-crossed module is an  
$R$-bimodule $R^\#$  together with a morphism $R^\# {\buildrel \partial\over\longrightarrow}\  
R$ of $R$-bimodules.\quad $\square$ 
\endproclaim 
 
We have canonical pairings  $\{ , \}:  Sym^2L^\# \to L$, $\langle ,\ \rangle:  
R^\# \otimes_R R^\# \to R^\#$  defined by formulas  $\{ m_1,m_2\} :=  
[\partial m_1, m_2] + [\partial m_2,m_1]$, $\langle s_1,s_2\rangle :=  
(\partial s_1)s_2 - s_1(\partial s_2)$.  These are morphisms of $L$-modules and  
$R$-bimodules respectively; one has $\partial \{ , \} = 0$, $\partial \langle , 
\rangle = 0$. 
 
Crossed modules in both versions form categories in an obvious manner.  For  
example, if  $R_1 {\buildrel f\over\longrightarrow} R_2$  is a morphism of  
associative algebras and $R_i^\#$ are $R_i$-crossed modules, then an $f$-morphism 
of crossed modules is an $f$-morphism  $f^\#: R_1 \to R_2$  of bimodules such  
that  $\partial f^\# = f\partial$.  If $R$  is an associative algebra, then  
$R$, considered as Lie algebra with commutator $ab-ba$,  will be denoted 
$R^{Lie}$.  If $R^\#$  is an $R$-crossed module, then it has also an $R^{Lie}$-crossed 
module structure  $R^{\# Lie}$  with  $[r,\widetilde{r}] =  
r\widetilde{r} - \widetilde{r}r$.  One has $\{ s_1,s_2\} =  
\langle s_1,s_2\rangle + \langle s_2,s_1\rangle$  for  $s_i \in R^\#  =  
R^\# = R^{\# Lie}$. 
 
\medpagebreak 
 
Below ``DG algebra" means ``differential graded algebra"; so ``Lie DG algebra" is the same as differential graded Lie superalgebra.   
 
\proclaim{1.1.2\ Lemma}  (i)  Let  $L$ (resp. $R$)  be a Lie  
(resp. associative) DG algebra such that  $L^i = 0$ $(R^i = 0)$ for $i > 0$.   
Then  $L^{-1} {\buildrel d\over\longrightarrow} L^0$ (resp.  
$R^{-1} {\buildrel d\over\longrightarrow} R^0$) is a Lie (resp. associative)  
algebra crossed module.  For  $m_1,m_2 \in L^{-1}$ (resp. $s_1,s_2 \in R^{-1}$) 
one has  $\{ m_1,m_2\} = d[m_1,m_2]$ (resp. $\langle s_1,s_2\rangle = d(s_1s_2)$). 
 
\noindent 
(ii)  Conversely, let  $L^\# {\buildrel \partial\over\longrightarrow} L$  
(resp. $R^\# {\buildrel \partial\over\longrightarrow} R$) be a crossed module,  
and  $i: N \subset L^\#$  (resp. $i: T \subset R^\#$)  be an $L$-submodule (resp.  
$R$-sub-bimodule) such that  $\{ L^\#, L^\#\} \subset N \subset \ker \partial$  
(resp. $\langle R^\#, R^\#\rangle \subset T \subset \ker \partial$).  Then 
$N {\buildrel i\over\longrightarrow} L^\# {\buildrel \partial\over\longrightarrow} L$ 
(resp. $T {\buildrel i\over\longrightarrow} R^\# {\buildrel\partial\over\longrightarrow}  
R$)  is a $dg$ Lie (resp. associative) $dg$ algebra placed in degrees $-2, -1, 0$.  
\quad $\square$ 
\endproclaim 
 
In other words, the lemma claims that DG algebras zero off degrees  
$-2,-1,0$  and acyclic off degrees $-1,0$  are in 1-1 correspondence with  
pairs  $(L^\# {\buildrel \partial\over\longrightarrow} L; N)$, where   
$L^\# {\buildrel \partial\over\longrightarrow} L$  is a crossed module and 
$N \subset L^\#$  is a submodule as in (ii) above.  For example, one may take 
$N =$ image of $\{ , \}$ (or image of $\langle , \rangle$  in the associative  
algebra version); we will say that the corresponding DG algebra is defined 
by our crossed module. 
 
\noindent 
1.1.3\  The simplest example of a Lie algebra crossed module is a  
central extension  $\widetilde{L} \to L$ of a Lie algebra $L$ (the bracket on 
$\widetilde{L}$  factors through an $L$-action); note that here $\{ , \} $ vanishes.   
Conversely, let  $L$  be a DG Lie algebra.  Then  $L^{-1}/dL^{-2}$,  
equipped with the bracket $[\ell_1,\ell_2] := [d\ell_1,\ell_2]^{0,-1}$ is a Lie  
algebra, and  $d: L^{-1}/dL^{-2} \to L^0$  is a morphism of Lie algebras such that 
$(H^{-1} \to L^{-1}/dL^{-2} \to d(L^{-1}))$  is a central extension of   
$dL^{-1}$ by  $H^{-1}$.  Hence if  $L^\# {\buildrel \partial\over\longrightarrow} L$ 
is an  $L$-crossed module such that $\partial$ is surjective, then  
$\ker \partial/\{ L^\#, L^\#\} \to L^\#/\{ L^\#, L^\#\} \to L$  is a central  
extension of $L$.  If  $tr: \ker \partial /\{ L^\#, L^\#\} \to {\Bbb C}$  is any  
linear functional, then it defines, by push-out, a central ${\Bbb C}$-extension   
$L^\#_{tr}$ of $L$.   
 
\noindent 
1.1.4\   The following example of a crossed module will be used below.  
Let  $L$  be a Lie algebra, and let $L_+, L_- \subset L$  be ideals.  Then we have  
an $L$-crossed module  $L_+ \oplus L_- {\buildrel \partial\over\longrightarrow} L$, 
$\partial (\ell_+, \ell_-) = \ell_+ + \ell_-$.   
We have isomorphism  $i: L_+ \cap L_-\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\  
\ker \partial$,\ $i(\ell) = (\ell, -\ell) \in L_+ \oplus L_-$.  Or we may take 
an associative algebra $R$ equipped with 2-sided ideals $R_+,R_-$, and get an  
$R$-crossed module  $R_+ \oplus R_- {\buildrel \partial\over\longrightarrow} R$. 
Note that $\{ , \}$  vanishes on  $L_+$  and  $L_-$  (and $\langle , \rangle$  
vanishes on $R_+$  and  $R_-$) and one has $\{\ell_+, \ell_-\} = i([\ell_-,\ell_+])$, 
$\langle r_+,r_-\rangle := - i(r_+r_-), \langle r_-,r_+\rangle = i(r_-r_+)$. 
 
If  $L_+ + L_- = L$, then we get a central extension  $L_+ \cap L_-/[L_+,L_-]\  
{\buildrel i\over\longrightarrow}\ \widetilde{L} \to L$ of $L$, where 
$\widetilde{L} = L_+ \oplus L_-/i([L_+,L_-])$. This central extension is equipped 
with obvious splittings $s_\pm: L_\pm \to \widetilde{L}$  such that  
$s_\pm (L_\pm)$  are ideals in $\widetilde{L}$; it is easy to see that  
$\widetilde{L}$  is universal among all central extensions of $L$ equipped with  
such splittings.  Note also that the embedding $s_+:  L_- \hookrightarrow \widetilde{L}$ 
yields an isomorphism $L_+/[L_+,L_-]\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\  
\widetilde{L}/s_-(L_-)$  and we have the Cartesian square 
$$ 
\matrix \format \c &\quad \c &\quad \c &\quad \c &\quad \c \\ 
\widetilde{L} & \longrightarrow & \widetilde{L}/s_-(L_-) & 
{\lower 1ex\hbox{$\overleftarrow{\sim}\vrule height 1ex width 0pt$}} 
& L_+/[L_+,L_-] \\ 
\big\downarrow && \big\downarrow && \big\downarrow \\ 
L & \longrightarrow & L/L_- &  
{\lower 1ex\hbox{$\overleftarrow{\sim}\vrule height 1ex width 0pt$}}   
& L_+/L_+ \cap L_-  
\endmatrix 
$$ 
and the same for $\pm$ interchanged. 
 
\noindent 
1.1.5\  Now let $tr: L_+ \cap L_-/[L_+,L_-] \to {\Bbb C}$  be any  
linear functional.  According to 1.1.3 it defines a central ${\Bbb C}$-extension  
$\widetilde{L}_{tr}$ of $L$.  One has the splittings  $s_+: L_+ \to  
\widetilde{L}_{tr}$,\ $s_-: L_- \to \widetilde{L}_{tr}$  such that  $s_\pm(L_\pm)$ 
are ideals and  $(s_+-s_-)\big\vert_{L_+\cap L_-} = tr$.  Clearly  $L_{tr}$  is  
the unique extension equipped with this data.   
 
\noindent 
1.1.6\  The above constructions are functorial with respect to  
$(L,L_\pm)$.  Hence if  $L'_\pm \subset L$  are other ideals such that  
$L_\pm \subset L'_\pm$, then we get a canonical morphism  $\widetilde{L} \to  
\widetilde{L}'$  between the corresponding central extensions of $L$.  If  
$tr: L_+ \cap L_-/[L_+,L_-] \to {\Bbb C}$  extends to $tr: L'_+ \cap L'_-/[L'_+,L'_-] 
\to {\Bbb C}$, then  $\widetilde{L}_{tr} = \widetilde{L}'_{tr}$.  In particular, 
assume that  $tr: L_+ \cap L_-/[L_+,L_-] \to {\Bbb C}$  extends to  
$tr: L_-/[L_-,L_-] \to {\Bbb C}$.  Then we may take  $L'_+ = L, L'_- = L_-$  to  
get the same extension  $\widetilde{L}_{tr}$, hence we get the splitting   
$\widetilde{s}_+: L \to \widetilde{L}_{tr}$  that extends our old   
$s_+: L_+ \to \widetilde{L}_{tr}$.  Explicitly,  $\widetilde{s}_+(\ell_+ +  
\ell_-) = s_+(\ell_+) + s_-(\ell_-) + tr \ell_-$; clearly $\widetilde{s}_+ - s_-  
= tr: L_- \to {\Bbb C}$.  In the same way, an extension of  $tr: L_+ \cap  
L_- \to {\Bbb C}$ to  $L_+$  determines the splitting  $\widetilde{s}_-:  
L \to \widetilde{L}_{tr}$  that extends the old  $s_-: L_- \to \widetilde{L}_{tr}$. 
If we have the trace functional  on the whole $L$, i.e.\  
$tr: L/[L,L] \to {\Bbb C}$, then $\widetilde{s}_+ - \widetilde{s}_- = tr:  
L \to {\Bbb C}$.   
 
\noindent 
1.1.7\  We will often use the following notation.  If  ${\frak g}$ 
is a Lie algebra, $V$ is a vector space, and  $0 \to V \to \widetilde{\frak g}  
\to {\frak g} \to 0$  is a central $V$-extension of ${\frak g}$, then for any  
$c \in {\Bbb C}$  we will denote by  $\widetilde{\frak g}_c$  a $V$-extension  
of ${\frak g}$  which is the $c$-multiple of $\widetilde{\frak g}$.  So we have a  
canonical morphism  $\widetilde{\frak g} \to \widetilde{\frak g}_c$  of central 
extensions of ${\frak g}$  that restricted to  $V$'s is multiplication by $c$.   
For example, in situation 1.1.3 one has  $(L^\#_{tr})_c = L^\#_{ctr}$. 
\vskip .15in 
 
\subheading{1.2\ Tate's vector spaces}  For subspaces $V_0,V_1$  of a vector 
space  $V$  we will write $V_0 \prec V_1$  if  $V_0/V_0 \cap V_1$  is of finite 
dimension, and  $V_0 \sim V_1$ ($V_i$  are commensurable) if  $V_0 \prec V_1$  and 
$V_1 \prec V_0$.  Clearly  $\prec$  is partial order on a set of commensurability  
classes of subspaces. 
 
\noindent 
1.2.1\  A {\it Tate's topological vector space} (or, simply,  
{\it Tate's space}) $V$ is a linearly topologized complete separated vector space  
$V$ that admits a basis $\{ V_\alpha\}$  of neighbourhoods of $0$  with  $V_\alpha$ 
mutually commensurable.  Equivalently, $V$ is the projective limit of a family of  
epimorphisms of usual vector spaces with finite dimensional kernels:  
$V = {\dsize\lim \Sb \leftarrow\\ \alpha\endSb} V/V_a$.   
 
Let  $L \subset V$  be a vector subspace.  We will say that $L$ is {\it bounded}  
if for any open  $U \subset V$  one has  $L \prec U$, and $L$ is {\it discrete}  
if for some open $U$ one has $U \cap L = 0$.  Clearly simultaneously bounded  
and discrete subspaces are just finite dimensional ones. 
 
A {\it lattice} $V_+ \subset V$  is a bounded open subspace; equivalently, this  
is a maximal (with respect to $\prec$) bounded closed subspace.  The lattices form 
 a maximal basis of neighbourhoods of $0$ that consists of mutually commensurable  
subspaces. 
 
A {\it colattice} $V_- \subset V$  is a maximal discrete subspace.  Equivalently,  
this means that for (any) lattice $V_+$  one has  $V_+ \cap V_- \sim 0$,  
$V_+ + V_- \sim V$  (or for some lattice $V_+$  one has  $V_+ \oplus V_-  
{\buildrel \sim\over\longrightarrow} V$). 
 
Tate's vector spaces form an additive category ${\Cal T}{\Cal V}$  with kernels  
and cokernels.  The category ${\Cal T}{\Cal V}$ is self-dual:  Namely, for a  
Tate's space  $V$ its {\it dual} $V^*$  is $Hom(V,{\Bbb C})$  with open subspaces  
in $V^*$  equal to orthogonal complements to bounded subspaces in $V$.  This   
$V^*$  is a Tate's space, and  $V^{**} = V$.  Note that  $V_+ \longmapsto V^\perp_+$ 
is 1-1 correspondence between lattices in $V$ and $V^*$; and the same for  
colattices. 
 
\noindent 
1.2.2\   Let  $V$  be a Tate's vector space.  One has a canonical  
${\Bbb Z}$-torsor  $Dim_V$ together with a map  $dim: \{$ Set of all lattices in 
$V \} \to Dim_V$  such that for a pair  $V_{+1},V_{+2}$  of lattices one has  
$dim V_{+1} - dim V_{+2} := dim(V_{+1}/V_{+1} \cap V_{+2}) -  
dim(V_{+2}/V_{+1} \cap V_{+2}) \in {\Bbb Z}$.  One has a natural map 
$codim: \{$ Set of all colattices in $V\} \to Dim_V$  defined by formula  
$codim V_- = dim V_+ + dim(V/V_++V_-) - dim(V_+ \cap V_-)$, where  $V_+$  is any  
lattice.  The ${\Bbb Z}$-torsor  $Dim_{V^*}$  coincides with the opposite torsor to  
$Dim_V$: one has $dim V^\perp_+ = - dim V_+$.  The group $Aut~V$ acts on  
$Dim_V$;  if $V$ is neither bounded nor discrete, then the action is  
non-trivial. 
 
\noindent 
1.2.3\  Let  $V_1,V_2$  be Tate's vector spaces.  We will say that a  
linear operator  $f \in Hom(V_1,V_2)$  is bounded if  $Im f$  is bounded, is  
discrete if $\ker f$ is open, and is finite if $Im f$  is finite dimensional. 
Denote by $Hom_+, Hom_-$ and $Hom_{00}$ respectively the corresponding spaces of 
operators; put  $Hom_0 := Hom_+ \cap Hom_-$.  Clearly  $Hom_+ + Hom_- = Hom$,  
$Hom_?$\ (where $? = +,-,0,00)$ is a 2-sided ideal in $Hom$ (i.e., if for   
$V_1 {\buildrel {f_1}\over\longrightarrow} V_2 {\buildrel {f_2}\over\longrightarrow}  
V_3$  either  $f_1$ or $f_2$  is in $Hom_?$, then  $f_2f_1$ is in $Hom_?$), and 
$Hom_-Hom_+ \subset Hom_{00}$.   
 
\demo{Remark}   Let ${\Cal T}{\Cal V}_+$, ${\Cal T}{\Cal V}_- \subset  
{\Cal T}{\Cal V}$  be full subcategories of bounded, resp. discrete, spaces.   
Then  ${\Cal T}{\Cal V}_-$  coincides with the category of usual vector spaces, 
and  $*$ identifies ${\Cal T}{\Cal V}_+$  with the dual category  
${\Cal T}{\Cal V}^{\circ}_-$; in particular these are abelian categories.  Consider  
the quotient categories ${\Cal T}{\Cal V}/+, {\Cal T}{\Cal V}/-,  
{\Cal T}{\Cal V}/0$, whose  objects  
are Tate's vector spaces, and $Hom$'s are the corresponding quotients  
$Hom/\pm := Hom/Hom_\pm, Hom/0 := Hom/Hom_0$ (clearly  ${\Cal T}{\Cal V}/\pm$  are 
just 
the quotient categories  ${\Cal T}{\Cal V}/{\Cal T}{\Cal V}_\pm$).  These quotient categories  
are abelian.  In fact, the projection  ${\Cal T}{\Cal V}/0 \to {\Cal T}{\Cal V}/+ \oplus  
{\Cal T}{\Cal V}-$  is an equivalence of categories, and embeddings ${\Cal T}{\Cal V}_\pm  
\hookrightarrow {\Cal T}{\Cal V}$  composed with projections define equivalences  
${\Cal T}{\Cal V}_+/Vect\ {\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 
0pt$}}\  
{\Cal T}{\Cal V}/-, {\Cal T}{\Cal V}_-/Vect\ {\lower 1ex\hbox{$\overrightarrow{\sim}\vrule 
height 1ex width 0pt$}}\  
{\Cal T}{\Cal V}/+$  (here  $Vect = {\Cal T}{\Cal V}_+ \cap {\Cal T}{\Cal V}_-$  is the  
category of finite dimensional vector spaces). 
\enddemo 
 
\noindent 
1.2.4\   For  $V \in {\Cal T}{\Cal V}$  consider the algebra $End V$  
equipped with 2-sided ideals  $End_\pm \supset End_0 \supset End_{00}$.  We will 
 write  ${\frak g}\ell = {\frak g} \ell V$  for $End V^{Lie} = End V$   considered 
as Lie algebra.  Since  $End^2_0 \subset End_{00}$, we have a canonical  
trace functional  $tr: {\frak g}\ell_0 \to {\Bbb C}$  which vanishes on  
$[{\frak g}\ell_+, {\frak g}\ell_-]$ 
 
According to 1.1.4, we get an $End$-crossed module  $End_+ \oplus End_- \to  
End$.  By 1.1.5, $tr$  defines a central ${\Bbb C}$-extension  
$\widetilde{{\frak g}\ell} \to {\frak g}\ell$  of ${\frak g}\ell$, together with  
canonical Lie algebra splittings  $s_\pm: {\frak g}\ell_\pm \to  
\widetilde{{\frak g}\ell}$  such that  $s_+ - s_- = tr$ on 
${\frak g}\ell_0$. 
 
\noindent 
1.2.5\  Let  $T \subset V$  be a Tate's subspace (= a closed  
subspace with induced Tate structure), and $V/T$  be the quotient.  Denote by  
$P_T\ {\buildrel i\over\hookrightarrow}\ {\frak g}\ell V$  the parabolic subalgebra  
of endomorphisms that preserve $T$; let  $\pi = (\pi_T,\pi_{V/T}): P_T \to  
{\frak g}\ell T \times {\frak g} \ell V/T$  be the obvious projection.  Let   
$\widetilde{{\frak g}\ell T \times {\frak g}\ell V/T}$  be a central  
${\Bbb C}$-extension of  ${\frak g}\ell T \times {\frak g}\ell  V/T$  which is the  
Baer sum of $\widetilde{{\frak g} \ell}T$  and  $\widetilde{{\frak g} \ell} V/T$; 
one has $\widetilde{{\frak g}\ell T \times {\frak g}\ell V/T} =$   
$\widetilde{{\frak g}\ell} T \times \widetilde{{\frak g}\ell} V/T/\{ (a_1,a_2)  
\in {\Bbb C} \times {\Bbb C}: a_1 + a_2 = 0\}$.  Clearly  
$\widetilde{{\frak g}\ell T \times {\frak g}\ell V/T}$  coincides with the 
${\Bbb C}$-extension constructed by the recipe of 1.1.4, 1.1.5 using the ideals  
${\frak g} \ell_+T \times {\frak g}\ell_+ V/T$, ${\frak g}\ell_-T \times  
{\frak g}\ell_- V/T$ and the trace functional  $tr = tr_T + tr_{V/T}$. 
 
Let  $\widetilde{P}_T = i^*\widetilde{{\frak g}\ell} V$  be the  
${\Bbb C}$-extension of  $P_T$  induced by $\widetilde{{\frak g}\ell} V$.   
Since  $P_T = P_{T+} + P_{T-}$, where  $P_{T\pm} = P_T \cap {\frak g}\ell_\pm V$, 
this ${\Bbb C}$-extension coincides with the one constructed by means of ideals 
$P_{T\pm}$  and the trace functional  $tr_V\big\vert_{P_T}$.  Note that  
$\pi(P_{T\pm}) = {\frak g}\ell_\pm T \times {\frak g}\ell_\pm V/T$  and  
$tr_V\big\vert_{P_T} = tr \circ \pi$.  By 1.1.6 this defines a canonical  
morphism  $\widetilde{\pi}: \widetilde{P}_T \to  
\widetilde{{\frak g}\ell T \times {\frak g}\ell V/T}$  of ${\Bbb C}$-extensions  
that lifts $\pi$.  In other words,  $\widetilde{P}_T$  is canonically isomorphic 
to the Baer sum of ${\Bbb C}$-extensions induced by projections  $\pi_T$, 
$\pi_{V/T}$  from  $\widetilde{{\frak g}\ell} T, \widetilde{{\frak g}\ell} V/T$.   
 
Let us consider an important particular case of this situation.  Assume that  
$T = V_+$  is a lattice.  Then we have a canonical splitting 
$s_+: {\frak g}\ell V_+ = {\frak g}\ell_+V_+ \to \widetilde{{\frak g}\ell} V_+,\  
s_-: {\frak g}\ell V/V_+ = {\frak g}\ell_-V/V_+ \to \widetilde{{\frak g}\ell} V/V_+,$ 
hence a canonical splitting $s_{V_+} = s_+\pi_{V_+} + s_-\pi_{V/V_+} :  
P_{V_+} \to \widetilde{{\frak g}\ell}_V$.  Note that  $s_{V_+}$ actually depends  
on $V_+$:  if  $V'_+$  is another lattice, then $s_{V_+} - s_{V'_+} :  
P_{V_+} \cap P_{V'_+} \to {\Bbb C}$  is given by formula  
$(s_{V_+} - s_{V'_+})(a) = tr_{V_+/V_+\cap V'_+}(a) -tr_{V'_+/V_+\cap V'_+}(a)$. 
 
Similarly, if  $T = V_-$  is a colattice, then we have the splittings  
$s_-: {\frak g}\ell V_- = {\frak g}\ell_-V_- \to \widetilde{{\frak g}\ell} V_-$, 
$s_+: {\frak g}\ell V/V_- = {\frak g} \ell_+ V/V_- \to \widetilde{{\frak g}\ell} 
V/V_-$, hence the splitting  $s_{V_-} = s_- \pi_{V_-} + s_+ \pi_{V/V_-}:  
P_{V_-} \to \widetilde{{\frak g}\ell}_V$.  On  $P_{V_-} \cap P_{V_+}$  the  
difference  $s_{V_+} - s_{V_-}: P_{V_-} \cap P_{V_+} \to {\Bbb C}$  is given by 
formula 
$$ 
(s_{V_+} - s_{V_-})(a) = tr_{V_-\cap V_+}(a) - tr_{V/V_-+V_+}(a). 
$$ 
\vskip .15in 

 The following subsection 1.3 could be omitted on first reading. 
 
\subheading{1.3\ Elliptic complexes}   Let  $(V^\cdot,d)$  be a finite complex 
of Tate's vector spaces.  We will call it elliptic, if for some (or any)  
subcomplex  $(V^\cdot_+, d) \subset (V^\cdot,d)$ formed by lattices in  $V^\cdot$ 
both  $V^\cdot_+$ and $V^\cdot/V^\cdot_+$  have finite dimensional  
cohomology spaces. 
 
Clearly, elliptic complexes have finite dimensional cohomology.   
 
\demo{Remark}  $V^\cdot$  is elliptic iff its image in the abelian category  
${\Cal T}{\Cal V}/0$  (see 3.2.2) is acyclic. 
\enddemo 
 
\noindent 
1.3.1\  Let  $(U^\cdot,d), (V^\cdot,d)$  be elliptic complexes.  
Then  $Hom = Hom(U^\cdot, V^\cdot) := \prod Hom(U^i,V^i)$  carries a bunch  
of subspaces.  First, one has the subspaces  $Hom_\pm := \prod Hom_\pm(U^i,V^i),$  
$Hom_0, Hom_{00}$  that have nothing to do with differential.  We may enlarge  
those spaces as follows.  Put  $Hom^d_\pm := \{ f \in Hom: [f,d] \in  
Hom_\pm (U^\cdot, V^{\cdot +1})\}, Hom^d_0 := Hom^d_+ \cap Hom^d_-$,  
$Hom_d := \{ f \in Hom : [f,d] = 0\}$  (= usual morphisms of complexes).  Clearly  
$Hom_\pm \subset Hom^d_\pm, Hom_0 \subset Hom^d_0$, and all $Hom^d_?$  are  
compatible with $\pm$  decomposition:  one has  $Hom^d_? = (Hom^d_? \cap Hom_+)  
+ (Hom^d_? \cap Hom_-)$.   
 
The following easy technical lemma is quite useful.  Assume that we picked  
subcomplexes $U^{'\cdot}_+ \subset U^\cdot_+ \subset U$, $V'_+ \subset  
V_+ \subset V$  formed by lattices.  Put   
$P := \{ f \in Hom(U^\cdot,V^\cdot): f(U^{'\cdot}_+) \subset V^{'\cdot}_+,  
f(U_+) \subset V_+\},$\ $P_{+d} := \{ f \in P: [f,d](U^\cdot)   
\subset V^{\cdot +1}_+\},$\ $P_{-d} := \{ f \in P: [f,d](U^{'\cdot}_+) =  
0 \},  P_{od} = P_{+d} \cap P_{-d}$. 
 
\proclaim{1.3.2\ Lemma}\ One has  $Hom^d_\pm = P_{\pm d} + Hom_{00},$\  
$Hom^d_0 = P_{0d} + Hom_{00}$. 
\endproclaim 
 
\demo{Proof}  Consider, e.g., the case of $Hom^d_+$.  One has  
$Hom^d_+ = ( P \cap Hom^d_+) + Hom_0$.  An element $f \in P \cap Hom^d_+$   
induces the linear map  $\bar{f}: U^\cdot/U^\cdot_+ \to V^\cdot/V^\cdot_+$ 
such that $\alpha = [\bar{f},d]$  is of finite rank.  One may find  $\bar{g}$ 
of finite rank such that  $[\bar{g},d] = \alpha$.  Lift $\bar{g}$  to an element  
$g \in P \cap Hom_0$; then $f-g \in P_{+d}$, and we are done.\quad $\square$ 
\enddemo 
 
Now let us define the traces.  Consider a single elliptic complex  
$(V^\cdot,d)$.  We have a bunch of Lie subalgebras in ${\frak g}\ell =  
{\frak g}\ell V^\cdot = \Pi{\frak g}\ell V^i$.  Pick subcomplexes  
$V^{'\cdot}_+ \subset V^\cdot_+ \subset V^\cdot$  formed by lattices; we get 
the corresponding parabolic subalgebra  $P \subset {\frak g}\ell$  and its standard 
subalgebras.  Define the trace functional tr:  $P_{0d} \to {\Bbb C}$ by formula 
$tr f := \Sigma (-1)^i(tr_{H^i(V/V_+)} + tr_{V^i_+/V_+^{'i}} + tr_{H^i(V'_+)})$. 
In particular, if  $V/V_+$  and  $V'_+$  are acyclic, then   
$tr = \Sigma(-1)^i tr_{V^i_+/V^{i'}_+}$.  The algebra ${\frak g}\ell_{00}$  
also carries the trace  $tr = \Sigma (-1)^i tr_{V^i}$.  Clearly on  
$P_{0d} \cap {\frak g}\ell_{00}$  these traces coincide, so, by 1.3.2, they 
define   $tr: {\frak g}\ell^d_0 \to {\Bbb C}$.   
 
\proclaim{1.3.3\ Lemma}  The trace functional  $tr: {\frak g}\ell^d_0 \to {\Bbb C}$ 
does not depend on the choice of  $V^\cdot_+, V_+^{'\cdot}$ and vanishes on  
$[{\frak g}\ell^d_0, {\frak g}\ell^d_0]$.\quad  $\square$ 
\endproclaim 
 
Let  $\widetilde{{\frak g}\ell}$    be the central extension of ${\frak g}\ell$  
by ${\Bbb C}$  which is the alternating Baer sum of $\widetilde{{\frak g}\ell} V^i$. 
Equivalently, to get $\widetilde{{\frak g}\ell}$  take the ideals  
${\frak g}\ell_\pm \subset {\frak g}\ell$  and the trace functional   
$tr = \Sigma(-1)^i tr_{V^i}$  on  ${\frak g}\ell_0$, and apply constructions 
1.1.4, 1.1.5.  We have canonical splittings  $s_\pm: {\frak g}\ell_\pm \to  
\widetilde{{\frak g}\ell}$.   
 
\proclaim{1.3.4\ Lemma}  These splittings extend to canonical splittings  
$s_\pm: {\frak g}\ell^d_\pm \to \widetilde{{\frak g}\ell}$; one has  
$s_+ - s_- = tr: {\frak g}\ell^d_0 \to {\Bbb C}$. 
\endproclaim 
 
\demo{Proof}  Consider, say, the case of $s_+$.  Let $\widetilde{{\frak g}\ell}^d_+$ 
be $\widetilde{{\frak g}\ell}$  restricted to  ${\frak g}\ell^d_+$.  Note that  
${\frak g}\ell^d_+ = {\frak g}\ell_+ + ({\frak g}\ell_- \cap {\frak g}\ell^d_+)$,  
so $\widetilde{{\frak g}\ell}^d_+$  comes from constructions 1.1.4, 1.1.5 applied 
to ${\frak g}\ell^d_+$, its ideals ${\frak g}\ell_+$  and ${\frak g}\ell_-  
\cap {\frak g}\ell^d_+$ and the trace functional $tr$.  We may even replace  
${\frak g}\ell_- \cap {\frak g}\ell^d_+$  by the larger ideal ${\frak g}\ell^d_0$ 
and, since  $tr$  extends to ${\frak g}\ell^d_0$  by 1.3.3, according to  
1.1.6 we get the desired section  $s_+: {\frak g}\ell^d_+ \to  
\widetilde{{\frak g}\ell}$.  One treats  $s_-$  in a similar way; the formula 
$s_+ - s_- = tr$ results from 1.1.6.\quad $\square$ 
 \enddemo

\subheading{1.4\ Clifford modules}  
Let $W$  be a Tate's space, and let (\ ,\ )  
be a non-degenerate symmetric form on $W$ (which is the same as symmetric  
isomorphism  $W\ {\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex  
width 0pt$}}\ W^*$). 
 
\noindent 
1.4.1\ For a lattice  $W_+ \subset W$  let  $W_+^\perp$  be the  
orthogonal complement with respect to  $(\ ,\ )$.  This is also a lattice, and  
the parity of  $dim W^\perp_+ - dim W_+ \in {\Bbb Z}$  does not depend on $W_+$ 
(and depends on $(W,(\ ,\ ))$ only).  We will say that  $W$  is even or odd  
dimensional if $dim W^\perp_+ - dim W_+$ is even or odd, respectively.   
 
\noindent 
1.4.2\  A Clifford module $M$ is a module over Clifford algebra $Cliff(W,(\ ,\ ))$   
such that $W$ acts on $M$ in a continuous way (in the discrete  
topology of $M$).  This means that for any $m \in M$  there is a lattice  $W_+$   
such that $W_+m = 0$.  Denote by  $C{\Cal M}_W$  the category of Clifford  
modules.   
 
Let  $W_+ \subset W$  be a lattice such that $(\ ,\ )|_{W_+} = 0$.  Then the  
finite-dimensional vector space  $W^\perp_+/W_+$  carries an induced non-degenerate 
form.  If  $M$  is a Clifford module, then  $M^{W_+} := \{ m \in M : W_+m = 0\}$ 
is a $W^\perp_+$-invariant subspace of $M$, hence a $Cliff(W^\perp_+/W_+, 
(\ ,\ ))$-module. 
 
\proclaim{1.4.3\ Lemma}  The functor  $C{\Cal M}_W \to C{\Cal M}_{W^\perp_+/W_+}$, 
$M \longmapsto M^{W_+}$, is an equivalence of categories.  The inverse functor is 
given by formula  $N \longmapsto Cliff(W)\ {\bigotimes_{Cliff(W^\perp_+)}} N$. 
\quad $\square$ 
\endproclaim 
 
In particular, we see that  $C{\Cal M}_W$  is a semisimple category.  There is  
1 irreducible object if  $W$  is even-dimensional, and 2 such if $W$ is odd-dimensional. 
 
Denote by  $C\ell W$ the completion ${\dsize\lim_\leftarrow}\ Cliff(W)/Cliff(W)\cdot 
W_+$, where  $W_+$  runs the set of all lattices in $W$.  It is easy to see that  
the multiplication extends to this completion by continuity, so  $C\ell W$ is an 
associative algebra.  Clearly, it acts on any Clifford module. 
 
\noindent 
1.4.4\  Let  $L_+ \subset W$  be a maximal $(\ ,\ )$-isotropic  
lattice (so either  $L^\perp_+ = L_+$  or $dim L^\perp_+/L_+ = 1$   
depending on parity of dimension of $W$).  If  $L'_+$  is another such lattice, 
put $\lambda(L_+: L'_+) := \det(L_+/L_+ \cap L'_+)$.  One has a canonical  
embedding  $i: \lambda(L_+ : L'_+) \hookrightarrow C\ell W/C\ell W \cdot L'_+$,  
given by the formula  $v_1 \wedge \cdots \wedge v_n \longmapsto \widetilde{v}_1 \cdots 
\widetilde{v}_n\ \text{mod}\ C\ell W \cdot L'_+$.  Here  $\{ v_i\}$  is a basis of  
$L_+/L_+ \cap L_+',\widetilde{v}_i$  are any liftings of  $v_i$  to elements of  
$L_+$.  For a Clifford module $M$ one has a canonical isomorphism  
$\lambda(L_+ : L_+') \otimes M^{L_+'}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\ M^{L_+}$, 
$v \otimes m \longmapsto i(v)m$. 
 
Now let  $L_- \subset L$  be a maximal isotropic colattice (so  
$codim L_- = dim L_+$  in case $dim W$  is even, or $codim L_- = dim L_+ + 1$ if 
$dim W$  is odd).  Put  $\lambda(L_+,L_-) = \det(L_+ \cap L_-)$.  For a Clifford 
module  $M$  put  $M_{L_-} := M/L_-M$.  One has a canonical isomorphism  
$\lambda(L_+,L_-) \otimes M_{L_-}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\ M^{L_+}$, 
defined by formula $v \otimes m \longmapsto v \widetilde{m}$, where  
$v \in \lambda(L_+,L_-) \subset Cliff(W)$,\ $m \in M_{L_-}$, and  
$\widetilde{m} \in M_{L_-}$, and  $\widetilde{m} \in M$  is any element such  
that  $\widetilde{m}$  mod  $L_-M = m$  and  $v\widetilde{m} \in M^{L_+}$.  If 
$M$ is irreducible, then  $dim M^{L_+} = dim M_{L_-} = 1$, and we may rewrite  
the above isomorphisms as  
$$ 
\lambda(L_+ : L'_+) = M^{L_+}/M^{L'_+},\quad \lambda(L_+, L_-) = M^{L_+}/M_{L_-}. 
$$ 
 
\noindent 
1.4.5\  The algebra  $C\ell W$  carries a natural ${\Bbb Z}/2$-grading  
such that $W$ lies in the degree 1 component.  Denote by  $C{\Cal M}_W^{{\Bbb Z}/2}$  
the corresponding category of  ${\Bbb Z}/2$-graded Clifford modules.  This is a  
semisimple category.  If  $dim W$  is odd, then it has a single irreducible  
object; if  $dim W$ is even, then there are two irreducible objects that differ  
by a shift of  ${\Bbb Z}/2$-grading.   
 
If  $dim W$  is even, then each  $M \in C{\Cal M}_W$  carries a natural  
${\Bbb Z}/2$-grading defined up to a shift.  Precisely, consider the set of all  
maximal isotropic lattices.  This breaks into two components:   lattices    
$L_+, L'_+$ lie in the same component iff  $dim L_+/L_+ \cap L'_+$  is even.   
Denote the two element set of these components by  ${\Bbb Z}/2_{{}_W}$; we will  
consider it as ${\Bbb Z}/2$-torsor.  Then any  $M \in C{\Cal M}_{{}_W}$  carries a  
canonical  ${\Bbb Z}/2_{{}_W}$-grading determined by the property that   
$M^{L_+} \subset M^\alpha$  for  $L_+ \in \alpha \in {\Bbb Z}/2_{{}_W}$. 
 
\noindent 
1.4.6\  Let  $C\ell^{Lie}W$  denote the Clifford algebra considered  
as Lie (super)algebra (with the above  ${\Bbb Z}/2$-grading; the (super)commutator 
is defined by the usual formula $[a,b] = ab - (-1)^{\alpha\beta}ba$  for  $a \in  
C\ell^{Lie}W^\alpha$,\ $b \in C\ell^{Lie}W^\beta$).  Denote by  ${\frak a}W$  the 
normalizer of  $W \subset C\ell^{Lie} W^1$  in  $C\ell^{Lie}W$.  This is a Lie 
subalgebra of  $C\ell^{Lie}W$.  As a vector space ${\frak a}W$  is the completion 
in  $C\ell W$  of the subspace of all degree $\le 2$  polynomials of elements of 
$W$.  One has  ${\frak a}W^1 = W$.  The Lie algebra $\widetilde{OW} :=  
{\frak a}W^0$  is called the spinor algebra of $W$.  The subspace  
${\Bbb C} \subset C\ell W$  coincides with center of ${\frak a}W$. One has a  
canonical isomorphism  ${\frak a}W/{\Bbb C} = OW \rtimes W$.  Here  $OW$  is the 
orthogonal Lie algebra of all (\ ,\ )-skew symmetric elements in ${\frak g}\ell W$;  
the projection  $\pi: \widetilde{OW} \to \widetilde{OW}/{\Bbb C} = OW$  is given  
by the adjoint action on $W = {\frak a}W^1$.   
 
The Lie superalgebra ${\frak a}W$  acts on any $M^\cdot \in C{\Cal M}_W^{{\Bbb Z}/2}$  
in an obvious manner.  If  $M^\cdot$  is irreducible, this action identifies  
${\frak a}W$  with the normalizer of $W$ in the Lie superalgebra  $End_{\Bbb C}M^\cdot$. 
Similarly, $\widetilde{OW}$  acts on any  $M \in C{\Cal M}_W$, and, in case $M$ 
is irreducible, $\widetilde{OW}$  coincides with the normalizer of $W$ in  
$End_{\Bbb C}M$.   
 
\noindent 
1.4.7\  Here is another construction of  $\widetilde{OW}$.  For  
$a \in {\frak g}\ell W$  denote by  ${}^ta \in {\frak g}\ell W$  the adjoint  
operator with respect to (\ ,\ ); for $a \in {\frak g}\ell_-W$ one has  
${}^ta \in {\frak g}\ell_+ W$.  Consider now the ideal ${\frak g}\ell_-W \subset 
{\frak g}\ell W$  as an $OW$-module with respect to $Ad$-action.  Then  
${\frak g}\ell_-W$  together with the surjective morphism  ${\frak g}\ell_-W\  
{\buildrel \partial\over\longrightarrow}\ OW$,\ $a \longmapsto\ a - {}^ta$, is an  
$OW$-crossed module.  The pairing   $\{ , \}: {\frak g}\ell_- W \times  
{\frak g}\ell_-W \to \ker \partial$ (see 1.1.1) is given by formula  
$\{ a_1,a_2\} = [a_1, {}^ta_2] + [a_2, {}^ta_1]$.  Clearly  $\ker \partial \subset 
{\frak g}\ell_0W$.  The usual trace $tr(1.2.4)$ vanishes on $\{ \ker \partial,  
\ker \partial\}$; put $o\ tr = 1/2 tr$.  By 1.1.3 we get a central  
${\Bbb C}$-extension  $\widetilde{OW}' = ({\frak g}\ell_-W)_{o\ tr}$ of $OW$. 
 
We define a canonical isomorphism\ $\alpha:  \widetilde{OW}'\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\  
\widetilde{OW}$  of central ${\Bbb C}$-extensions of $OW$ as follows.  One has  
a canonical identification  $W \otimes W \simeq {\frak g}\ell_{00}W$, $w_1 \otimes 
w_2$  corresponds to a linear operator  $w \longmapsto (w_2,w)w_1$.  This  
isomorphism extends by continuity to the isomorphism of completions\ 
${\dsize\lim \Sb \leftarrow\\ W_+ \endSb}$ $W \otimes (W/W_+) \simeq  
{\frak g}\ell_-W$.  Hence the map  ${\frak g}\ell_{00}W = W \otimes W \to  
Cliff(W,( , ))$, $a_1 \otimes a_2 \longmapsto  a_1a_2$, extends by continuity  
to the map  $\alpha^\#: {\frak g}\ell_-W \to C\ell W$.  Clearly  
$\alpha^\#$ maps ${\frak g}\ell_-W$  to   ${\frak a} W^0 = \widetilde{OW}$.   
For  $a_1,a_2 \in {\frak g}\ell_-W, w \in W$  one has $[\alpha^\#(a),w] =  
\partial(a)(w),$\ $[\alpha^\#(a_1), \alpha^\#(a_2)] = \alpha^\#([\partial a_1,  
a_2])$.  For $b \in \ker \partial \cap {\frak g}\ell_{00}W$  one has  
$b = 1/2(b + {}^tb) = \Sigma(w_i \otimes w'_i + w'_i \otimes w_i)$, hence  
$\alpha^\#(b) = \Sigma (w_i,w'_i) = o\ tr\ b$; by continuity this holds for any  
$b \in \ker \partial$.  This implies that  $\alpha^\#$  yields a   
map  $\alpha: {\frak g}{\ell}_- W/\ker\ tr = \widetilde{OW}' \to \widetilde{OW}$,  
which is the desired isomorphism of ${\Bbb C}$-extensions of $OW$.   
 
\noindent 
1.4.8\  Let $L_+ \subset W$ be a maximal isotropic lattice; denote  
by  $P_{L_+}O \subset OW$ the ``parabolic" subalgebra of operators that preserve  
$L_+$.  One has a canonical Lie algebra splitting $s_{L_+}: P_{L_+}O \to  
\widetilde{OW}$  defined by formula $s_{L_+}(a) = \alpha^\#(b)$, where  
$b \in {\frak g}\ell_-W$  is any operator such that $\partial(b) = a,  
b(L_+) = 0, (a-b)(W) \subset L_+$.  For any Clifford module $M$ one has  
$s_{L_+}(a)(M^{L_+}) = 0$ (and $s_{L_+}(a)$ is a unique lifting of $a$ to  
$\widetilde{OW}$  with this property). 
 
Similarly, let $L_- \subset W$  be a maximal isotropic colattice.  The  
corresponding parabolic subalgebra $P_{L_-}O \subset OW$  also has a canonical  
Lie algebra splitting  $s_{L_-}: P_{L_-}O \to \widetilde{OW}$  defined by  
formula $s_{L_-}(a) = \alpha^\#(b)$, where $b \in {\frak g}\ell_-W$  is an  
operator such that  $\partial(b)~=~a,$\ $b|_{L_-} = a|_{L_-}, b(W) \subset L_-$. 
For a Clifford module one has  $s_{L_-}(a)(M_{L_-}) = 0$ (i.e.,  
$s_{L_-}(a)(M) \subset L_-M$).   
 
According to 1.4.4 for $a \in P_{L_+}O \cap P_{L_-}O$  one has  
$(s_{L_-} - s_{L_+})(a) = tr_{L_-\cap L_+}(a) \in {\Bbb C} \subset  
\widetilde{OW}$.  If  $L'_+$  is another maximal isotropic lattice, then for  
$a \in P_{L_+}O \cap P_{L'_+}O$  one has $(s_{L_+'} - s_{L_+})(a) =  
tr_{L_+/L_+\cap L'_+}(a)$. 
 
\noindent 
1.4.9\  Let  $V$  be any Tate's vector space.  Then  $W := V \oplus  
V^*$, equipped with the form $((v,v^*), (v',v^{*'})) := v^*(v') + v^{*'}(v)$, is an  
even-dimensional space.  For any lattice  $V_+ \subset V$  and a colattice  
$V_- \subset V$  a lattice $L(V_+) = V_+ \oplus V_+^\perp \subset W$  and a  
colattice  $L(V_-) = V_- \oplus V_-^\perp \subset W$  are maximal isotropic  
ones; clearly one has a canonical isomorphisms 
$$ 
\matrix \format \l \\ 
\lambda(L(V_+) : L(V'_+)) = \det(V_+/V_+\cap V'_+)/\det (V'_+/V_+\cap V'_+)\\ 
\lambda(L(V_+), L(V_-)) = \det(V_+ \cap V_-)/\det(V/V_++V_-).			   
\endmatrix 
$$ 
 
The algebra  $C\ell W$  gets a natural ${\Bbb Z}$-grading such that the  
subspaces  $V,V^*$ $(\subset W \subset C\ell W)$  lie in degrees 1, -1,  
respectively.  Any Clifford module $M$ has a canonical $Dim_V$-grading such  
that $M^{L(V_+)}$ lies in degree $dim V_+$.   
 
The embedding $i: {\frak g}\ell V \hookrightarrow OW,\ \ell \longmapsto \ell  
\oplus (- {}^t\ell)$, lifts canonically to a morphism of ${\Bbb C}$-extensions  
$\widetilde{i}: \widetilde{{\frak g}\ell} V \longrightarrow\ \widetilde{OW}$  
constructed as follows.  For $\ell_+ \in {\frak g}\ell_+V$ choose a lattice  
$V_+ \supset Im \ell_+$.  Then  $i(\ell_+) \in P_{L(V_+)}O$.  Put  
$\widetilde{i}_+(\ell_+) = s_{L(V_+)}i(\ell_+) \in \widetilde{OW}$; by  
1.4.8 this element is independent of a choice of $V_+$.  Similarly, for  
$\ell_- \in {\frak g}\ell_-V$  choose a lattice  $V'_+ \subset \text{Ker}\  
\ell_-$; then  $i(\ell_-) \in P_{L(V'_+)}O$, and $\widetilde{i}_-(\ell_-)  
:= s_{L(V'_+)}i(\ell_-) \in \widetilde{OW}$ depends on $\ell_-$ only.  
For $\ell_0 \in {\frak g}\ell_0V$  one has  $(\widetilde{i}_- -  
\widetilde{i}_+)(\ell_0) = tr_{L(V_+)/L(V_+) \cap L(V'_+)}(i\ell_0) =  
tr \ell_0$  by 1.4.8.  According to 1.2.3 we get a canonical morphism  
$\widetilde{i}: \widetilde{{\frak g}\ell}_{-1}V \longrightarrow \widetilde{OW}$  
of ${\Bbb C}$-extensions such that  $\widetilde{i} s_\pm = \widetilde{i}_\pm:  
{\frak g}\ell_\pm V \longrightarrow \widetilde{OW}$ (here $\widetilde{{\frak g}\ell}_{-1}V 
= (\widetilde{{\frak g}\ell} V)_{-1}$, see 1.1.7). 
 
The action of  $\widetilde{{\frak g}\ell} V$ on $M$ preserves the $Dim_V$-grading.   
If $M$ is irreducible, then it is natural to denote the  
$\widetilde{{\frak g}\ell}_{-1}V$-module $M^a$,\ $a \in Dim V$, as  $\Lambda^aV$  
(``semi-infinite wedge power").  Note that  $\wedge^aV$ (as well as $M$ itself)  
is defined up to tensorization with 1-dimensional ${\Bbb C}$-vector space. 
 
\noindent 
1.4.10\  We will need a version ``with formal parameter" of the above  
constructions.  Namely, let  ${\Cal O} =  {\Bbb C}[[q]]$  be our base ring.  Consider a  
flat complete  ${\Cal O}$-module $V$ (so  ${\dsize\lim_\leftarrow} V/q^nV)$.   
A Tate structure on $V$ is given by Tate's ${\Bbb C}$-vector space structure on 
each  $V/q^nV$  such that each short exact sequence  $0 \to V/q^m V\  
{\buildrel {q^n}\over\longrightarrow}\ V/q^{m+n}V \to V/q^nV \to 0$  is strongly  
compatible with the Tate structures (i.e., $V/q^mV$ is a Tate's subspace of  
$V/q^{m+n}V$  and  $V/q^nV$  is the quotient space).  A lattice  $V_+ \subset V$  
is an ${\Cal O}$-submodule such that  $V/V_+$  is ${\Cal O}$-flat,   
$V_+ = {\dsize\lim_\leftarrow} V_+/q^nV_+$  and  $V_+/q^nV_+$  is a lattice in  
$V/q^nV$ for each $n$.  One defines a colattice  $V_- \subset V$  in a similar 
way.  For a Tate ${\Cal O}$-module $V$ one defines its dual $V^*$ in an obvious  
way; one has  $V^*/q^nV^* = (V/q^nV)^*$, $V^{**} = V$.   
 
Let  $W$ be Tate's ${\Cal O}$-module and $( , ): W \times W \to {\Cal O}$  be a  
non-degenerate symmetric form (i.e., a symmetric isomorphism  
$W\ {\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\ W^*$). 
Let  $Cliff(W)$  be the Clifford ${\Cal O}$-algebra of  $( , )$.  A Clifford  
module  $M$ is a $Cliff(W)$-module such that $M$ is flat as ${\Cal O}$-module,  
$M = {\dsize\lim_\leftarrow} M/q^nM$, and $W/q^nW$  acts on each  $M/q^nM$  in a  
continuous way (in discrete topology of $M/q^nM$).  Such $M$ carries the action of  
completed Clifford algebra 
$$ 
C\ell W = {\lim \Sb\\ \leftarrow\\ n\endSb}\ {\lim \Sb\\ \leftarrow\\ W_+^{(n)}\endSb}\  
Cliff(W)/q^nCliff(W) + Cliff(W)W_+^{(n)} 
$$  
(where $W_+^{(n)}$ is a lattice in  
$W/q^nW$).  Clearly  $M_0 ;= M/qM$  is Clifford module for  
$(W_0, ( , )_0) := (W/qW, ( , ) mod q)$; if $M'$ is another Clifford module,  
then $Hom(M,M')$ is a flat ${\Cal O}$-module and $Hom(M,M')/q Hom(M,M') =  
Hom(M_0, M'_0)$.  In particular, if $(W_0, ( , )_)$  is even-dimensional,  
then there exists a Clifford module $M$, unique up to isomorphism, such that  
$M_0$ is irreducible; one has $End M = {\Cal O}$.  All the facts 1.4.3-1.4.9   
have an obvious ${\Bbb C}[[q]]$-version. 

\newpage 
 
\pageno=13 
 
\heading 
\S2.\  Tate's residues and Virasoro-type extensions 
\endheading 
 
\subheading{2.1\ Tate's construction of the local extension}  Let  $F$  be a  
1-dimensional local field, and  ${\Cal O}_F \subset F$  be the corresponding  
local ring.  A choice of uniformization parameter  $t \in {\Cal O}_F$  
identifies  ${\Cal O}_F$  with ${\Bbb C}[[t]]$, and $F$ with ${\Bbb C}((t))$. 
Let  $E$  be an $F$-vector space of dimension $n < \infty$.  Denote by  
${\Cal D}E$ 
the algebra of $F$-differential operators acting on $E$.  A choice of a basis 
of $E$ identifies ${\Cal D}E$  with the algebra of matrix differential operators  
$a_N\partial^N_t + \cdots + a_1\partial_t + a_0$, $a_i \in Mat_n(F)$.   
 
\medpagebreak 
 
\noindent 
2.1.1\   The space $E$, considered as ${\Bbb C}$-vector space, is  
actually a Tate's vector space in a canonical way.  A basis of neighbourhoods  
of $0$ is formed by ${\Cal O}_F$-submodules of $E$ that generate $E$ as  
$F$-module.  We will denote by $End E, {\frak g}\ell_\pm E$, etc., the  
corresponding algebras of endomorphisms of $E$, considered as Tate's  
${\Bbb C}$-vector space.   
 
Clearly  ${\Cal D}E \subset End E$.  We may restrict the central extension  $\widetilde{{\frak g}\ell} E$ of ${\frak g}\ell E$ to ${\Cal D}E^{Lie}$ to get a  central extension  $0 \to {\Bbb C} \to \widetilde{{\Cal D} E} \to {\Cal D} E^{Lie} 
\to 0$  of the Lie algebra  ${\Cal D} E^{Lie}$.   
 
It is easy to compute a 2-cocycle of this extension explicitly.  Namely, let us 
choose a parameter $t \in {\Cal O}_F$  and an $F$-basis $\{ v_i\}$ in $E$.   
Put  $E_+ = {\dsize \sum_i} {\Cal O}_F v_i$, $E_- = {\dsize \sum_i} t^{-1}  
{\Bbb C}[t^{-1}] v_i$:  this is a lattice and a colattice in $E$ and $E = E_+  
\oplus E_-$.  For $\ell \in {\frak g}\ell E$  define the operator   
$\ell_+ \in {\frak g}\ell_+E$  by formula   $\ell_+|_{E_+} =  
\ell|_{E_+}, \ell_+|_{E_-} = 0$.  Clearly this map  ${\frak g}\ell E \to  
{\frak g}\ell_+E,\ \ell \longmapsto \ell_+$, lifts the canonical projection  
${\frak g}\ell E \to {\frak g}\ell E/{\frak g}\ell_-E =  
{\frak g}\ell_+E/{\frak g}\ell_0E$.  Hence by 1.1.4 it defines a section  
$\sigma: {\frak g}\ell E \to \widetilde{{\frak g}\ell} E$; the corresponding 
2-cocycle is given by formula $\ell_1,\ell_2 \longmapsto  \alpha(\ell_1,\ell_2)  
= [\sigma(\ell_1),\sigma(\ell_2)] - \sigma([\ell_1,\ell_2]) = tr([\ell_{1+},  
\ell_{2+}] - [\ell_1,\ell_2]_+)$.  Take now $\ell_1 = At^a{\frac{\partial_t^b}{b!}}$, 
$\ell_2 = A't^{a'} {\frac{\partial_t^{b'}}{b'!}}$, where $A,A' \in Ma t_n({\Bbb C})$, 
$a,a' \in {\Bbb Z}, b,b' \in {\Bbb Z}_{\ge 0}$.  Clearly  
$\alpha(\ell_1,\ell_2) = 0$ if $a - b \ne b' - a'$.  Assume that  
$a-b = b'-a'$; since $\alpha$  is skew-symmetric we may assume that   
$n = a-b \ge 0$.  Then one has  
$$ 
\alpha(\ell_1,\ell_2) = - Tr(AA') \sum^{n-1}_{i=0} {\binom i{b'}} 
{\binom {i-n}{b}}. 
$$      
 
\medpagebreak 
 
\noindent 
2.1.2\  Let  ${\Cal A} E \subset {\Cal D} E^{Lie}$  be a Lie  
subalgebra that consists of operators of order $\le 1$  with scalar symbol 
(i.e., the operators of type $a_0 + a_1\partial_t, a_0 \in End_FE, a_1 \in F$). 
Denote by ${\Cal T}_F$  the Lie algebra of vector fields on $F$.  One has a  
canonical short exact sequence of Lie algebras  $0 \to End_FE^{Lie} \to 
{\Cal A} E\ {\buildrel \sigma\over\longrightarrow}\ {\Cal T}_F \to 0$,  
$\sigma(a_0 + a_1\partial_t) = a_1\partial_t$.  Let  $\widetilde{{\Cal A}E}$ be  
the ${\Bbb C}$-extension of ${\Cal A}E$ induced from $\widetilde{{\Cal D}E}$. 
The above formulas reduce to the following ones: 
$$ 
\alpha(At^a, Bt^b) = b\delta_a^{-b} tr AB, \alpha(At^a, t^{b+1}\partial_t) =  
{\frac{a-a^2}{2}} \delta_a^{-b} tr A,\ \alpha(t^{a+1}\partial_t, t^{b+1}\partial_t)  
= {\frac{n}{6}} (a^3 - a) \delta_a^{-b}. 
$$ 
This is the Kac-Moody-Virasoro cocycle. 
 
\medpagebreak 
 
\noindent 
2.1.3\  Consider the case $E = F$.  One has an obvious embedding  
${\Cal T}_F \subset {\Cal A}F$  which defines the ${\Bbb C}$-extension  
$\widetilde{T}_F$ of ${\Cal T}_F$  with cocycle  $\alpha_{Vir}(t^{a+1}\partial_t,  
t^{b+1}\partial_t) = {\frac{1}{6}} (a^3-a)\delta_a^{-b}$.  This  
$\widetilde{\Cal T}_F$  is called (a local) Virasoro algebra.  For any  
$c \in {\Bbb C}$  consider the ${\Bbb C}$-extension $\widetilde{\Cal T}_{Fc}$ 
(see 1.1.7).  Since  ${\Cal T}_F$  is perfect, $\widetilde{\Cal T}_{Fc}$ has 
no automorphisms.  One knows that any central ${\Bbb C}$-extension of  
${\Cal T}_F$  is isomorphic (canonically) to a unique $\widetilde{\Cal T}_{Fc}$  
(one has $H^2({\Cal T}_F, {\Bbb C}) \simeq {\Bbb C})$.   
       
\medpagebreak 
 
\noindent 
2.1.4\  Now consider for $j \in {\Bbb Z}$  a 1-dimensional $F$-vector 
space  $\omega_F^{\otimes j}$ of $j$-differentials (the elements of   
$\omega_F^{\otimes j}$ are tensors  $fdt^{\otimes j}, f \in F$).  The Lie  
algebra ${\Cal T}_F$  acts canonically on  $\omega_F^{\otimes j}$  by Lie 
derivatives, i.e., we have a canonical embedding  ${\Cal T}_F \hookrightarrow  
{\Cal A}\omega_F^{\otimes j}$.  Denote by $\widetilde{\Cal T}_F^{(j)}$ the  
corresponding ${\Bbb C}$-extensions of ${\Cal T}_F$  induced from  
$\widetilde{{\Cal A}\omega}_F^{\otimes j}$.  The explicit formula for this  
action is $\varphi \partial_t(fdt^{\otimes j}) = (\varphi \partial_t(f) +  
jf\partial_t(\varphi))dt^{\otimes j}$, i.e., with respect to the basis  
$dt^{\otimes j}$  a field  $t^{a+1}\partial_t$ acts as $t^{a+1}\partial_t +  
j(a+1)t^a$.  The formulas 2.1.2 immediately show that a 2-cocycle for  
$\widetilde{\Cal T}_F^{(j)}$  coincides with $(6j^2 - 6j + 1)\alpha_{Vir}$. 
Hence $\widetilde{\Cal T}_F^{(j)}$  coincides with   
$\widetilde{\Cal T}_{F(6j^2-6j + 1)}$.   
\vskip .2in 
 
\subheading{2.2\  A geometric construction of a global extension}  Let us  
describe the above extensions in geometric language. 
 
\medpagebreak 
 
\noindent 
2.2.1\  Let  $C$  be a smooth algebraic curve (not necessary  
compact).  Denote by $\omega = \Omega^1_C$  the sheaf of 1-forms, and by  
${\Cal H} = H^1_{DR} = \Omega^1_C/d{\Cal O}_C$ the de Rham cohomology sheaf 
(in the Zariski topology of $C$).  For a vector bundle $E$  on $C$ let   
${\Cal D} = {\Cal D}E$  denote the sheaf of differential operators on $E$, and 
$E^{\circ} := \omega E^*$.  Then  $E$  is a left  ${\Cal D}$-module, $E^{\circ}$  is a right 
${\Cal D}$-module (so one has a canonical anti-isomorphism   
$t: {\Cal D}E \to {\Cal D} E^0$, see, e.g., [B]), and the pairing   
$E^0 \otimes E\ {\buildrel {\langle \ \rangle}\over\longrightarrow}\ \omega$   
quotients to the  
pairing  $E^0\ {\dsize\bigotimes_{{\Cal D} E}}\ E \to {\Cal H}$.   
 
Let  $\Delta: C \to C \times C$  be the diagonal; we will identify the sheaves 
on $C$  with ones on  $C \times C$ supported on the image of $\Delta$.  Consider the sheaf 
$E \boxtimes E^0 := p^*_1E \otimes p_2^*E^0$  on  $C \times C$.  Recall that 
one has a canonical isomorphism  $\delta: E \boxtimes E^0(\infty\Delta)/E\boxtimes E^0\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\ {\Cal D}$. 
Explicitly, for a ``kernel" $k(t_1,t_2) = e(t_1)e^0(t_2)f(t_1,t_2)$, 
$e \in E, e^0 \in E^0, f(t_1,t_2) \in {\Cal O}_{C\times C}(\infty\Delta)$, the 
corresponding differential operator $\delta(k)$  acts on sections of $E$  
according to formula $(\delta(k)\ell)(t_1) = Res_{t_2=t_1}\langle k(t_1,t_2) 
\ell(t_2)\rangle =  e(t_1)Res_{t_2=t_1} f(t_1,t_2)\langle e^0(t_2)\ell(t_2)\rangle$. 
Here  $\ell \in E, \langle e^0(t_2)\ell(t_2)\rangle \in \omega,  
\langle k(t_1,t_2)\ell(t_2)\rangle \in E \boxtimes \omega(\infty\Delta)$;  
we take the residue along the $t_2$  variable.  The right action of $\delta(k)$ 
on sections of $E^0$ is given by formula $(m\delta(k))(t_2) =  
Res_{t_1=t_2}  f(x,t_2)\langle m(t_1)e(t_1)\rangle)e^0(t_2)$.   
 
\medpagebreak 
 
\noindent 
2.2.2\ Put ${\Cal P} E_n := {\lim \Sb\\ \leftarrow \\ i\endSb} E \boxtimes  
E^0((n+1)\Delta)/E \boxtimes E^0(-i\Delta)$, ${\Cal P}E = \cup {\Cal P} E_n$, so we  
have an isomorphism  $\delta: {\Cal P}  E/{\Cal P} E_{-1}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\  
{\Cal D} E$.  Clearly  ${\Cal P} E$  is a ${\Cal D} E$-bimodule (the left and  
right actions are the obvious actions along the first, resp. the second variable),  
and  $\delta$ is a morphism of bimodules, i.e., ${\Cal P}E$ is a $DE$-crossed module 
(see 1.1).  Let  $t: {\Cal P}E \to {\Cal P}E^0$  be minus the isomorphism  
``transposition of coordinates" (here minus comes since   $E,E^0$  have ``odd"  
nature).  Then for  $k \in {\Cal P}E$  one has  ${}^t\delta(k) = \delta({}^tk)$, 
and ${}^t$  is an ``anti-isomorphism" between crossed modules.   
 
The pairing  $\langle \ \rangle: {\Cal P} E {\dsize\bigotimes_{{\Cal D} E}} 
{\Cal P} E \to {\Cal P} E_{-1}$  from 1.1, $\langle k_1,k_2\rangle =  
\delta(k_1)k_2 - k_1\delta(k_2)$, is given by formula  
$$ 
\langle k_1k_2\rangle(t_1,t_2) = (Res_{z=t_1} + Res_{z=t_2})  
\langle k_1(t_1,z)k_2(z,t_2)\rangle =  
\int_{\gamma_{t_1,t_2}}\langle k_1(t_1,z)k_2(z,t_2)\rangle. 
$$ 
Here $\langle k_1(t_1,z)k_2(z,t_2)\rangle$ is the 1-form of variable $z$ (with values in $E_{t_1} \otimes E^0_{t_2})$, and $\gamma_{t_1,t_2}$  is a loop round  $z = t_1$  and $z = t_2$.  The corresponding Lie algebra pairing  
$\{\ \}: S^2{\Cal P}E \to {\Cal P}E_{-1}$  is  $\{ k_1,k_2\} := \langle k_1,k_2\rangle  
+ \langle k_2,k_1\rangle$.  Let  $tr: {\Cal P}E_{-1} \to \omega$  be the  
composition  ${\Cal P}E_{-1} \to {\Cal P}E_{-1}/{\Cal P}E_{-2}	= E \otimes E^0  
\to \omega$.  We have 
$$ 
tr\{ k_1,k_2\} = (Res_1 - Res_2)\langle k_1(t_1,t_2)k_2(t_2,t_1)\rangle. 
$$   
Here $k_2(t_2,t_1) = {}^tk_2 \in {\Cal P}{E^0}$ is  $k_2$  with coodinates transposed, 
$\langle k_1(t_1,t_2)k_2(t_2,t_1)\rangle$  is a 2-form with poles along the  
diagonal and $Res_1, Res_2:  \Omega^2_{C\times C}(\infty\Delta) \to \omega_C$  
are residues around the diagonal along the first and second coordinates, respectively.  
Clearly,  $Res_1-Res_2$  vanishes on  $\Omega^2_{C\times C}(\Delta)$  and has 
image in exact forms.  In fact, there is a canonical map  
$\widetilde{Res}: \Omega^2_{C\times C}(\infty\Delta)/\Omega^2_{C\times C}(\Delta) 
\to {\Cal O}_C$  such that $d\widetilde{Res} = Res_1-Res_2$ (see [B Sch]  
(2.11)).  An explicit formula for $\widetilde{Res}$  is  
$$ 
\widetilde{Res}(f(t_1,t_2)(t_1-t_2)^{-i-1}dt_1 \wedge dt_2) =  
i!^{-1} \sum_{a+b=i-1} \partial^a_{t_1} \partial^b_{t_2} f(t_1,t_2)\big\vert_{t_1=t_2=t}. 
$$ 
Here  $f(t_1,t_2) \in {\Cal O}_{C\times C}$.  Hence one has $tr\{ k_1,k\} =  
d\widetilde{Res} \langle k_1,{}^tk_2\rangle$.  Note that the symmetric pairing 
${\Cal P}E \otimes {\Cal P}E \to {\Cal O}_C,\ k_1,k_2 \longmapsto \{ k_1,k_2\}^\sim  
:= \widetilde{Res} \langle k_1, {}^t k_2\rangle$  vanishes on  
${\dsize\sum_{a+b=-1}} {\Cal P} E_a \otimes {\Cal P} E_b$; in particular, it  
induces the pairing on  ${\Cal P} E_1/{\Cal P} E_{-2}$.   
 
According to 1.1.2, 1.1.3 we get a central extension  $\widetilde{DE}$  of the Lie 
algebra $DE^{Lie}$  by ${\Cal H}$  defined by a following commutative diagram: 
$$ 
\matrix \format \c &\quad \c &\quad \c &\quad \c &\quad \c &\quad \c &\quad \c &\quad \c 
&\quad \c \\ 
0 & \longrightarrow & {\Cal P} E_{-1} & \longrightarrow & {\Cal P} E &  
{\buildrel \delta\over\longrightarrow} & {\Cal D}E & \longrightarrow & 0 \\ 
&& \Big\downarrow tr && \Big\downarrow && \Bigg\Vert \\ 
&& {\buildrel {\dsize\omega}\over\downarrow}{\phantom{tr}} \\ 
0 & \longrightarrow & {\Cal H} & \longrightarrow & \widetilde{{\Cal D}E} &  
\longrightarrow & {\Cal D} E^{Lie} & \longrightarrow & 0 . 
\endmatrix 
$$ 
 
\medpagebreak 
 
\noindent 
2.2.3\  Denote by  ${\Cal A} E \subset {\Cal D} E^{Lie}$ the Lie  
subalgebra of differential operators of order  $\le 1$  with scalar symbol. 
In other words, ${\Cal A}E$  is the Lie algebra of infinitesimal symmetries of 
$(C,E)$:  the elements of  ${\Cal A}E$  are pairs $(\tau, \widetilde{\tau})$,  
where  $\tau \in {\Cal P}_C$  is a vector field, and $\widetilde{\tau}$  is an  
action of  $\tau$ on $E$  (so $\widetilde{\tau}$  is an order 1 differential  
operator with symbol equal to $\tau$).   
 
The constructions of 2.2.2 give rise to a differential graded Lie algebra  
${\Cal A}E$  defined as follows.  One has  ${\Cal A}^0E = {\Cal A}E$,\  
${\Cal A}^{-1}E$  is pre-image of  ${\Cal A}E \subset {\Cal D}E$  by the  
projection  ${\Cal P}E/\ker tr\ {\buildrel \delta\over\longrightarrow}\  
{\Cal D}_E$  (so we have short exact sequence  $0 \to \omega \to {\Cal A}^{-1}E\  
{\buildrel \delta\over\longrightarrow}\ {\Cal A}E \to 0)$, and finally  
${\Cal A}^{-2}E = {\Cal O}_C$; all the other components of ${\Cal A}^{\cdot}E$  are  
zero ones.  The differential  $d: {\Cal A}^{-2}E = {\Cal O}_C \to \omega  
\subset {\Cal A}^{-1}E$ is the de Rham differential, and ${\Cal A}^{-1}E \to {\Cal A}E$  is 
$\delta$.  The bracket components  $[\quad ]^{ij}: {\Cal A}^iE \times  
{\Cal A}^jE \to {\Cal A}^{i+j}E$  are the following.  $[\quad ]^{00}$  is  
the usual bracket  $[\quad ]^{0-1}$  comes from  ${\Cal D}^{Lie}$-action on  
${\Cal P}E$, $[\quad ]^{0,-2}$  is the action of  ${\Cal A}E$ on   ${\Cal O}_C$ 
via  $\sigma: {\Cal A}E \to {\Cal T}_C$, and $[\quad ]^{-1-1}$ is  
$\{\ ,\ \}^\sim$  defined above.  So  ${\Cal A}^\cdot E$  contains de Rham  
complex   $\Omega^\cdot_C[2]$  as an ideal, ${\Cal A}^\cdot E/\Omega^\cdot_C[2]$   
is acyclic and the central extension  $\widetilde{\Cal A}E =  
{\Cal A}^{-1}E/d{\Cal A}^{-2}E$  of  ${\Cal A}E$  by ${\Cal H}$  (see 1.13)  
coincides with restriction of $\widetilde{\Cal D}E$  to ${\Cal A}E \subset  
{\Cal D}E^{Lie}$. 
 
 
\medpagebreak 
 
\noindent 
2.2.4\  Consider the case $E = {\Cal O}_C$.  An obvious embedding  
${\Cal P}_C \hookrightarrow {\Cal A}{\Cal O}_C$  defines the central  
${\Cal H}$-extension $\widetilde{{\Cal P}}_C$  called a global Virasoro algebra. 
As in 2.1.3 for $c \in {\Bbb C}$  we will denote by  $\widetilde{{\Cal P}}_{Cc}$ 
the  ${\Cal H}$-extension of ${\Cal P}_C$ which is $c$-multiple of   
$\widetilde{{\Cal P}}_C$.   Since ${\Cal P}_C$  is perfect (see 2.5 below),  
the extensions $\widetilde{{\Cal P}}_{Cc}$  have no automorphisms.   
 
\medpagebreak 
 
\noindent 
2.2.5\  Consider for  $j \in {\Bbb Z}$  the sheaf $\omega^{\otimes j}$.  
The natural action of  ${\Cal P}_C$ on $\omega^{\otimes j}$ by Lie derivatives defines  
a canonical embedding of Lie algebras  ${\Cal P}_C  \hookrightarrow  
{\Cal A}\omega^{\otimes j}$.  Denote by  $\widetilde{\Cal P}_C^{(j)}$  the  
induced  ${\Cal H}$-extension  $\widetilde{{\Cal A}\omega}^{\otimes j}\big\vert_{{\Cal 
P}_C}$. 
Given a local coordinate  $t$, one may consider elements of  
$\widetilde{\Cal P}_C^{(j)}$ as expressions   
$$ 
\varphi^{(j)}_{(f,g)} = \left[ {\frac{f(t_1)}{(t_2-t_1)^2}} +  
j {\frac{\partial_{t_1}f(t_1)}{t_2-t_1}} + g(t_1)\right]dt_1^{\otimes j} 
dt_2^{\otimes 1-j}, 
$$ 
where  $f,g \in {\Cal O}_C$, modulo the ones of type  $\varphi_{(0,\partial_th)}$. 
The map $\widetilde{\Cal P}_C = \widetilde{\Cal P}_C^{(0)} \to  
\widetilde{\Cal P}_C^{(j)}$  defined by formula  $\varphi^{(0)}_{(f,g)} \longmapsto  
\varphi^{(j)}_{(f,(6j^2-6j+1)g)}$  is a morphism of Lie algebras, and does not  
depend on a choice of a local coordinate $t$.  Hence it defines a canonical  
isomorphism  $\widetilde{\Cal P}_{C(6j^2-6j+1)}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\  
\widetilde{\Cal P}_C^{(j)}$  of ${\Cal H}$-extensions of  $C$ (see [B Sch]).  
Unfortunately, we do not know any ``coordinate-free" explanation of this  
isomorphism.   
\vskip .2in 
 
\subheading{2.3\ Compatibility with Tate's construction}  Let  $x \in C$ be a  
point.  We may consider the constructions of 2.2 locally at $x$.  Namely, let 
${\Cal O}^\wedge_x$  be the completed local ring of $C$ at $x$,  
${\Cal O}^\wedge_{(x,x)}$  be the completed local ring of $C \times C$ at $(x,x)$,  
$F_x \supset {\Cal O}^\wedge_x$	  the local field at $x$, so if $t$ is a parameter 
at  $x$ then ${\Cal O}^\wedge_{(x,x)} = {\Bbb C}[[t_1,t_2]]$.  Denote by $R$  
the localization of ${\Cal O}^\wedge_{(x,x)}$ with respect to   
$t_1^{-1},t_2^{-1}, (t_1-t_2)^{-1}$.  Put  $\omega_{(x)} := F_x \otimes_{\Cal O}  
\omega,\ E_{(x)} := F_x \otimes_{\Cal O} E, {\Cal D}_{(x)} = {\Cal D}E_{(x)} :=  
F_x \otimes_{\Cal O} {\Cal D}E_{(x)}, {\Cal P}_{(x)} = {\Cal P}E_{(x)} =  
E \otimes_{\Cal O} R \otimes_{\Cal O} E^0$:  
these are local versions of the objects in 2.2.  We can manage all the  
constructions of 2.2 purely locally.  In particular we get the central extension 
$\widetilde{\Cal D}_{(x)}$ of  $\widetilde{\Cal D}_{(x)}^{Lie}$  by  
${\Cal H}_{(x)} = \omega_{(x)}/dF_x\  
{\buildrel {Res}\over{{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 
0pt$}}}}\   
{\Bbb C}$. 
 
\medpagebreak 
 
\noindent 
2.3.1\  By 2.1, $E_{(x)}$  is a Tate's vector space, and we have the  
embedding  $i_x: {\Cal D}_{(x)} \hookrightarrow End E_{(x)}$.  For  
$k = k(t_1,t_2) \in {\Cal P}_{(x)}$  let  $k_-,k_+ \in End E_{(x)}$ be the linear  
operator defined by formulas 
$$ 
[k_-(e)](t) = - Res_{t_2=0}\langle k(t,t_2)e(t_2)\rangle, [k_+(e)](t) =  
(Res_{t_2=t_1} + Res_{t_2=0})\langle k(t,t_2)e(t_2)\rangle. 
$$ 
Here  $e(t) \in E_{(x)}, \langle k(t,t_2)e(t_2)\rangle \in E \otimes R \otimes  
\omega$, and the residues are taken along the second variable.  According to  
2.2.1 one has  $i_x \delta(k) = k_- + k_+$.  Denote by   
$i^\#_{x\pm} : {\Cal P}_{(x)} \to End E_{(x)}$  the maps   
$i^\#_{x\pm}(k) = k_\pm$.   
 
\proclaim{2.3.2\ Lemma}  (i)  For  $k \in {\Cal P}_{(x)}$  one has  
$k_\pm \in End_\pm E_{(x)}$. 
 
\item{(ii)} The commutative diagram 
$$ 
\matrix \format \l &\quad \c &\quad \c \\ 
{\Cal P}_{(x)} & {\buildrel {i^\#_x=(i^\#_{x+},i^\#_{x-})}\over\longrightarrow}  
& End_+ E_{(x)} \oplus End_-E_{(x)} \\ 
\bigg\downarrow \delta && \bigg\downarrow \\ 
{\Cal D} & {\buildrel {i_x}\over\longrightarrow} & End\ E_{(x)} 
\endmatrix 
$$ 
is an $i_x$-morphism of crossed modules (see 1.1). 
 
\item{(iii)} For $k \in \ker \delta \subset {\Cal P}_{(x)}$  one has  
$Res_x tr(k) = tr i^\#_x(k) (= tr k_+ = - tr k_-)$. 
 
\item{(iv)}  Let us identify  $E^0_{(x)}$  with  $E^*_{(x)}$  via the pairing  
$(\ ,\ ):  E \times E^0 \to {\Bbb C}$, $(e,e^0) = Res\langle e,e^0\rangle$; 
this gives the anti-isomorphism  $t: End E_{(x)} \to End E^0_{(x)}$.  Then  
the diagram 
$$ 
\matrix \format \l &\quad \c &\quad \c \\ 
{\Cal P}E_{(x)} & {\buildrel {i^\#_+}\over\longrightarrow}  
& End_+ E_{(x)} \\ 
t\bigg\downarrow \wr && t\bigg\downarrow\wr \\ 
{\Cal P}E^0_{(x)} & {\buildrel {i^\#_-}\over\longrightarrow} & End_-E^0_{(x)} 
\endmatrix 
$$ 
commutes. 
\endproclaim 
 
 
\demo{Proof}  Assume for simplicity of notation that  $E = {\Cal O}_C$, so  
$E_{(x)} = F_x$.  The statement  $k_- \in End_-F_x$  from (i) is clear, since   
$k_-$  vanishes on the lattice  $t^N{\Cal O}^\wedge_x \subset F_x$  for  $N$ 
equal to the order of pole of  $k(t_1,t_2)$  at divisor  $t_2 = 0$.  Now the  
fact that  $k_+ \in End_+F_x$  will follow from (iv).  The statements (ii),  
(iii) are obvious.  To prove (iv) let us compute the residues integrating the  
forms along cycles.  Let  $\gamma_\pm(t)$  be the following loops in the $t_2$-complex  
plane  $t_1 = t$: 
\vskip 1.5in  
 
 
\noindent 
Then for any function  $f \in F_x$  one has  $[k_\pm(f)](t) = {\frac{1}{2\pi i}}  
\int_{\gamma_\pm(t)} k(t,t_2)f(t_2)$.   
 
Denote by  $U$  a small neighbourhood of zero in ${\Bbb C} \times {\Bbb C}$  with  
coordinate cross and diagonal removed.  One has the following 2-dimensional  
cycles  $C_\pm$ in $U$.  Fix a small real numbers  $0 < \epsilon < r \ll 1$. 
Then  $C_+ = \{(z_1,z_2) \in {\Bbb C} \times {\Bbb C}: |z_1| = {\epsilon},\  
|z_2| = r\}$, $C_- = \{ (z_1,z_2) \in {\Bbb C} \times {\Bbb C}: |z_1| = r,  
|z_2| = {\epsilon}\}$; the orientation of  $C_+$  is a standard orientation of  
$S^1 \times S^1$, and the one of $C_-$  is minus the standard orientation. 
 
The above formula for the action of  $k_\pm$  implies that for a 1-form   
$g \in F^0_x = \omega_{(x)}$  one has  $(g, k_\pm(f)) = \int_{C_\pm} g(t_1)  
k(t_1,t_2)f(t_2)$.  Since the transposition of coordinates identifies  $C_+$  with 
$C_-$, this implies that  $(g,k_+(f)) = (({}^tk)_-(g),f)$.\quad  $\square$ 
 \enddemo

\medpagebreak 
 
\noindent 
2.3.3\  Now the morphism  $i^\#_x$ 2.3.2(ii) of crossed modules  
together with compatibility    2.3.2(iii) defines the morphism of the corresponding 
${\Bbb C}$-extensions   $\widetilde{i}_x: \widetilde{\Cal D}_{(x)} \to  
\widetilde{{\frak g}\ell} E_{(x)},$ $\widetilde{i}_x(k) = s_+(k_+) + s_-(k_-)$,  
or, equivalently, the isomorphism of ${\Bbb C}$-extensions   
$\widetilde{D}_{(x)}\ {\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\  
\widetilde{{\Cal D}E}_{(x)}$ (see 2.1.1). 
       
\medpagebreak 
 
\noindent 
2.3.4\   Assume now that our curve $C$ is compact.  Let  $X =  
\{ x_i\} \subset C$ be a finite non empty set of points, and $E$  be a vector 
bundle on  $U = C\setminus X$.  Put $E_{(X)} = \Pi E_{(x_i)}, {\Cal D}_{(X)} =  
\Pi {\Cal D}_{(x_i)}$.  Denote by $\widetilde{\Cal D}_{(X)}$, a ${\Bbb C}$-extension 
of ${\Cal D}_{(X)}^{Lie}$  which is the Baer sum of ${\Bbb C}$-extensions  
${\Cal D}_{(x_i)}$, so $\widetilde{{\Cal D}}_{(X)} =  
\Pi \widetilde{{\Cal D}}_{(x_i)}/\{ (a_i) \in {\Bbb C}^X: \sum a_i = 0\}$.   
Clearly  $\widetilde{{\Cal D}}_{(X)}$  coincides with the ${\Bbb C}$-extension  
$\widetilde{\Cal D}E_{(X)}$  induced from  $\widetilde{{\frak g}\ell} E_{(X)}$  
via the embedding  ${\Cal D}_{(X)} \hookrightarrow \prod End\ E_{(x_i)}  
\hookrightarrow End\ E_{(X)}$.   
 
Put ${\Cal D}_U := H^0(U, {\Cal D} E_U)$  and consider the central extension  
$0 \to H^1_{DR}(U) \to \widetilde{{\Cal D}}_U \to {\Cal D}_U \to 0$  constructed in 2.2.2. 
One has the "localization around $x_i$"  maps ${\Cal D}_U \hookrightarrow  
\prod {\Cal D}_{(x_i)}, \widetilde{{\Cal D}}_U \to \prod  
\widetilde{{\Cal D}}_{(x_i)}$.  The composition $\widetilde{{\Cal D}}_U \to  
\prod \widetilde{{\Cal D}}_{(x_i)} \to \widetilde{{\Cal D}}_{(X)}$  vanishes 
on $H^1_{DR}(U)$  (since  ${\dsize\sum_X} Res_{x_i} = 0$).  Hence it defines a  
canonical morphism  $s_X: {\Cal D}_U^{Lie} \to \widetilde{{\Cal D}}_{(X)}$ that  
lifts the embedding  ${\Cal D}_U \hookrightarrow {\Cal D}_{(X)}$. 
 
This morphism could be constructed by pure linear algebra means.  Namely,  
consider the colattice  $E_U = H^0(U,E) \subset E_{(X)}$.  Clearly  
${\Cal D}_U^{Lie} \subset P_{E_U} \subset {\frak g}\ell E_{(X)}$, hence we have 
the splitting  $s_{E_U|{\Cal D}_U}: {\Cal D}_U^{Lie} \to \widetilde{{\Cal D}}E_{(X)}  
= \widetilde{{\Cal D}}_{(X)}$  (see 1.2.5). 
 
\proclaim{2.3.5\ Lemma}  This splitting coincides with the above $s_X$.   
\endproclaim 
 
\demo{Proof}  Let  $\partial \in {\Cal D}_U$  be a differential operator. 
Choose a section  $k \in H^0(U \times U, E \boxtimes E^0(\infty\Delta))$ such  
that $\delta(k) = \partial$.  Denote by  $k_- = (k_-^{x_i}) \in Hom(E_{(X)},E_U)$   
the morphism given by formula  $k_-(e_{x_i}) = \Sigma k_-^{x_i}(e_{x_i})$,\  
$k_-^{x_i}(e_{x_i}) = - Res_{x_i} \langle k \cdot e_{x_i}\rangle \in E_U$.   
Here  $e_{x_i} \in E_{x_i}, \langle k\cdot e_{x_i}\rangle \in H^0(U \times  
Spec F_{x_i}, E \boxtimes \omega(\infty\Delta))$  is a section obtained by convolution  
of $k$ and $e_{x_i}$  (where $e_{x_i}$  is considered as a section of  
${\Cal O}_U \boxtimes E_{(x_i)}$ independent of first variable), and $Res_{x_i}$  
is residue along the second variable at $x_i$.  Clearly  $k_-$  is a morphism of  
Tate spaces (here  $E_U$  is a discrete space). 
\enddemo 
 
Let $j = (j_{x_i}) : E_U \hookrightarrow E_{(X)}$  be the embedding.  The  
residue formula implies that for  $e \in E_U$  one has  $k_-(j(e)) = \partial(e)$. 
Hence  $j \circ k_- \in P_{E_U} \subset {\frak g}\ell E_{(X)}$, one has  
$j \circ k_- \in {\frak g}\ell_- E_{(X)}$, $\partial  - j \circ k_- \in  
{\frak g}\ell_+ E_{(X)}$, and, according to 1.2.5,  $s_{E_U}(\partial)$ coincides 
with  $s_-(j\circ k_-) + s_+(\partial - j \circ k_-)$. 
 
Now consider $j \circ k_-$  as a matrix $(j \circ k_-)^{x_i}_{x_j} \in  
Hom(E_{(x_i)}, E_{(x_j)})$.  Let  $j \circ k_-^{diag} = \Sigma(j \circ k_-)^{x_i}_{x_i}  
\in End\ E_{(X)}$  be the diagonal part of $j \circ k_-$.  According to 2.3.2, 
one has  $s_X(\partial) = s_-(j\circ k_-^{diag}) +  
s_+(\partial - j\circ k_-^{diag})$.  Hence  $s_X(\partial) - s_{E_U}(\partial)  
= tr(j \circ k_- - j \circ k_-^{diag})$.  This is a trace of a matrix in  
${\frak g}\ell_0E_{(X)}$  with zero diagonal component which is zero, q.e.d. 
\quad $\square$ 
 
\medpagebreak 
 
\noindent 
2.3.6\  We will often use the morphism $s_X$  for appropriate  
subalgebras of ${\Cal D}_U^{Lie}$, say, for  ${\Cal A} E_U$.   
\vskip .2in 
 
\subheading{2.4\ Spinors and theta-characteristics}  Let  $W$  be a vector  
bundle on our curve  $C$  equipped with a symmetric non-degenerate pairing 
$(\ ,\ ): W \times W \to \omega$. 
       
\medpagebreak 
 
\noindent 
2.4.1\  One may consider  $(\ ,\ )$  as an isomorphism  
$W \simeq W^0$, hence we have the involution  ${}^t: {\Cal D}W \to {\Cal D}W$ 
such that ${}^t(\partial_1\partial_2) = {}^t\partial_2 {}^t \partial_1$, and  
${}^t$  acts on degree $n$ symbols as multiplication by $(-1)^n$.  Denote by 
${\Cal O}{\Cal D}W$ the anti-invariants of ${}^t$; this is a Lie subalgebra of  
${\Cal D}W^{Lie}$. 
 
The isomorphism  $W \simeq W^0$  also defines an involution  
${}^t: {\Cal P}W \to {\Cal P}W$  (see 2.2.2) such that  ${}^t\delta = \delta^t$. 
Let  ${\Cal O}{\Cal P}W$  be the anti-invariants of  ${}^t$  in ${\Cal P}W$; put 
$o\delta = \delta\big\vert_{{\Cal O}{\Cal P}W}$.  The action of  
${\Cal D}W$  on ${\Cal P}W$  defines the ${\Cal O}{\Cal D}W$-action on ${\Cal O}{\Cal 
P}W$, 
and $o\delta: {\Cal O}{\Cal P}W \to {\Cal O}{\Cal D}W$  is an ${\Cal O}{\Cal D}W$-crossed 
module.  The trace $o tr$ which is  $- {\frac{1}{2}}$  of the composition  
$\ker o\delta \to W \otimes W^0\ {\buildrel {(\ ,\ )}\over\longrightarrow}\  
\omega \longrightarrow {\Cal H}$  defines by 1.1.3, a canonical central  
${\Cal H}$-extension $\widetilde{{\Cal O}{\Cal D}W}$  of ${\Cal O}{\Cal D}W$.    
In ${\Cal O}{\Cal D}W$  we have a Lie subalgebra ${\Cal O}{\Cal A}W =  
{\Cal A}W \cap {\Cal O}{\Cal D}W$ of  
infinitesimal symmetries of $(C,W,(\ ,\ ))$:  this is an extension of ${\Cal P}_C$  
by an orthogonal Lie algebra ${\Cal O}W \subset End\ W$.  Denote by  
$\widetilde{{\Cal O}{\Cal A}W}$  the central extension  
$\widetilde{{\Cal O}{\Cal D}W}\big\vert_{{\Cal O}{\Cal A}W}$.  Note that if  
$rk W = 1$, i.e., if  $W = \omega^{\otimes 1/2}$ is a theta-characteristic, then  
${\Cal O}\omega^{\otimes 1/2} = 0$, hence ${\Cal O}{\Cal A}\omega^{\otimes 1/2} =  
{\Cal T}_C$.  The formula from 2.2.5 applied to  $j = 1/2$ gives a canonical  
isomorphism $\widetilde{{\Cal O}{\Cal A}\omega^{1/2}} = \widetilde{\Cal T}_{C-1/2}$. 
       
\medpagebreak 
 
\noindent 
2.4.2\  If  $E$  is any vector bundle, and  $W = E \oplus E^0$  with 
obvious $(\ ,\ )$, then the Lie algebras embedding  $j: {\Cal D} E \to  
{\Cal O}{\Cal D}W$, $\partial \longmapsto (\partial, - {}^t\partial)$, lifts to a  
morphism of crossed modules  $j^\#: {\Cal P}E \to {\Cal O}{\Cal P}W$,  
$k \longmapsto (k, - {}^tk)$.  For  $k \in \ker \delta$  one has  
$o tr(j^\#k) = -tr k$.  So we get a canonical morphism  $\widetilde{j}:  
\widetilde{{\Cal D}E}_{-1} \to {\Cal O}{\Cal D}W$  of  ${\Cal H}$-extensions  
(see 1.1.7 for -1 index). 
       
 
\medpagebreak 
 
\noindent 
2.4.3\  Let us consider a local version of the above construction. 
Now our curve is a punctured disc  $Spec F_x$, so one has the identification  
$Res_x: {\Cal H}(F_x)\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\  
{\Bbb C}$.  The Tate ${\Bbb C}$-vector space  $W_{(x)}$  carries a non-degenerate 
symmetric form $(\ ,\ )_\bullet$  defined by formula $(w_1,w_2)_\bullet =  
Res_x(w_1,w_2)$.  The action of ${\Cal D}W_{(x)}$  on $W_{(x)}$  gives the  
embedding  $oi_X: {\Cal O}{\Cal D}W_{(x)} \hookrightarrow {\Cal O}W_{(x)}$. 
It lifts to an $oi_x$-morphism $oi_x^\#: {\Cal O}{\Cal P}W_{(x)} \longrightarrow 
{\frak g}\ell_-W_{(x)}$ of crossed modules (for the latter crossed module see 
1.4.7), $oi_x^\#(k) = k_-$, according to 2.3.2 (i),(ii),(iv).  For $k \in \ker \delta$ 
one has  $o tr(k) = {\frac{1}{2}} tr k_- = o tr(k_-)$ by 2.3.2 (iii), 1.4.7.  Hence 
$oi_x^\#$ defines a canonical morphism of ${\Bbb C}$-extensions $\widetilde{oi_x}: 
\widetilde{{\Cal O}{\Cal D}W_{(x)}} \hookrightarrow \widetilde{{\Cal O}W_{(x)}}$. 
 
\medpagebreak 
 
\noindent 
2.4.4\  Assume we are in a situation 2.3.4, i.e., we have a compact  
curve $C$, a finite set of points $X \subset C$, and our bundle $(W, (\ ,\ ))$   
on  $U = C\setminus X$.  We get a Tate vector space  $W_{(X)} = \prod W_{(x_i)}$  
with the form $(\ ,\ )_{(X)} = \sum (\ ,\ )_{(x_i)}$, a central ${\Bbb C}$-extension 
$\widetilde{{\Cal O}{\Cal D}W_{(X)}} \subset \widetilde{{\Cal O}W_{(X)}}$ of  
${\Cal O}{\Cal D}W_{(X)} = \prod {\Cal O}{\Cal D}W_{(x_i)} \subset {\Cal O}W_{(X)}$. 
Just as in 2.3.4 a localization at $X$ morphism ${\Cal O}{\Cal D}W_U :=  
H^0(U, {\Cal O}{\Cal D}W) \longrightarrow {\Cal O}{\Cal D}W_{(X)}$  lifts  
canonically to a morphism  $s_X: {\Cal O}{\Cal D}W_U \longrightarrow  
\widetilde{{\Cal O}{\Cal D}W_{(X)}}$; as in 2.3.5 this $s_X$  coincides with the  
lifting  $s_{W_U}\big\vert_{{\Cal O}{\Cal D}W_U}$ from 1.4.8.  Certainly  $s_X$  
extends in an obvious manner to a morphism of Lie superalgebras  
${\Cal O}{\Cal D}W_U \ltimes W_U \to {\frak a}W_{(X)}$  (here $W_U$  has odd  
degree, for  ${\frak a} W_{(X)}$, see 1.4.6).   
 
\medpagebreak 
 
\noindent 
2.4.5\  By Serre's duality  $W_U$  is a maximal isotropic  
colattice in  $W_{(X)}$.   
\vskip .2in 
 
\subheading{2.5\ Simplicity of Lie algebra of vector fields}  The following lemma 
will be of use: 
 
\proclaim{2.5.1\ Lemma}  Let  $C$  be a smooth curve.  Then the Lie algebra  
$T = H^0(C, {\Cal T}_C)$  of vector fields on $C$ is simple. 
\endproclaim 
 
\demo{Proof}  The case of compact $C$ is clear, so we will assume that $C$ is  
affine.  Let  $I \subset T$ be a non-zero ideal; we have to show that  
$I = T$.  Let $\tau \in I$  be a non-zero vector field.  Note that if  
$g \in {\Cal O}(C)$  is a function such that  $g\tau \in I$  and  $f \in  
{\Cal O}(C)$  is any function, then  $\tau(f)g\tau = {\frac{1}{2}}([g\tau,  
f\tau] + [\tau, fg\tau]) \in I$.  Let $A_{\tau} \subset {\Cal O}(C)$  be the subalgebra 
of functions generated by all functions  $\tau(f), f \in {\Cal O}(C)$.  The  
previous remark implies (by induction) that  $A_\tau\tau \subset I$.  One may  
describe  $A_\tau$ explicitly, namely  $A_\tau$  consists precisely of those  
$f \in {\Cal O}(C)$  that take equal values at zeros of $\tau$  and  
$ord_x(f - f(x)) \ge ord_x(\tau)$  for any  $x \in C$; this condition  
is non empty only for $x =$ zero of $\tau$.  (To see this, consider the  
morphism  $\pi: C \to C' = Spec A_\tau$.  Clearly  $A_\tau$  is a curve.  An  
easy local analysis at points at $\infty$ of $C$ shows that $\pi$ is finite. 
If  $x,y \in C, x \ne y$, are not zeros of $\tau$, then a finite jet at $x,y$  
of the functions  $\tau(f), f \in {\Cal O}(C)$, could be arbitrary ones, hence  
$\pi$  is isomorphism on the complement of zeros of  $\tau$.  An easy local  
analysis at zeros of $\tau$ finishes the proof).  In particular, any function  
that vanishes at zeros of $\tau$  with large order of zero lies in $A_\tau$.  Hence  
$I$ contains any vector field that vanishes at zeros of $\tau$ with sufficiently  
large order of zero (namely, twice that of $\tau$).  A trivial local analysis  
at zeros of $\tau$ (take brackets of elements of $I$  with vector fields  
non-vanishing at zeros of $\tau$) shows that  $I = T$.\quad $\square$ 
\enddemo 
 
\proclaim{2.5.2\ Corollary}  If  $C$  is an affine curve, then  $T$  has  
no non-trivial finite dimensional representations. \quad $\square$ 
\endproclaim
 
\newpage 
 
\pageno=23 
 
 
\heading \S 3.\ Localization of representations 
\endheading 
 
\subheading{3.1\ Harish-Chandra modules}  Recall some definitions. 
 
\medpagebreak 
 
\noindent 
3.1.1\ Let  $K$  be a pro-algebraic group.  A $K$-module $M$ is a comodule  
over the co-algebra ${\Cal O}(K)$.  Equivalently, $M$ is a vector space with  
an algebraic $K({\Bbb C})$-action.  Here ``algebraic" means that $M$ is a union  
of finite dimensional  $K({\Bbb C})$-invariant subspaces $M_\alpha$  such that 
$K({\Bbb C})$  acts on  $M_\alpha$  via an algebraic action of a factor group  
$K/K_\alpha$  of finite type.  Any $K$-module is a Lie $K$-module in a natural  
way. 
 
\medpagebreak 
 
\noindent 
3.1.2\ A Harish-Chandra pair $({\frak g}, K)$  consists of a Lie algebra  
${\frak g}$ and a pro-algebraic group $K$ together with an ``adjoint" action  
$Ad$ of $K({\Bbb C})$  on ${\frak g}$  and a Lie algebra embedding  
$i: Lie K \hookrightarrow {\frak g}$  that satisfy the compatibilities: 
 
\item{(i)} The embedding $i$ commutes with adjoint actions of $K$. 
 
\item{(ii)}  The action  $Ad$  is ``pro-algebraic":\  for any normal  
subgroup  $K' \subset K$  such that  $K/K'$  has finite type the action of  
$K({\Bbb C})$ on ${\frak g}/i(Lie K')$  is algebraic. 
 
\item{(iii)} The    $ad \circ i$-action of Lie $K$ on ${\frak g}$  coincides with 
the differential of the $Ad$-action. 
	      
\medpagebreak 
 
\noindent 
3.1.3\ 	Let  $({\frak g}, K)$  be a Harish-Chandra pair.  A $({\frak g}, K)$-module, 
or a Harish-Chandra module, is a ${\Bbb C}$-vector space equipped with  
${\frak g}$- and  $K$-module structures such that  
 
\item{(i)}  For $k \in  K, h \in {\frak g}, m \in M$  one has   
$Ad_k(h)m = khk^{-1}(m)$. 
 
\item{(ii)} The two Lie $K$-actions on $M$ (the one that comes from  
${\frak g}$-action via $i$, and the differential of $K$-action) coincide. 
 
We denote by $({\frak g},K)$-mod  the category of $({\frak g},K)$-modules. 
 
\medpagebreak 
 
\noindent 
3.1.4\ Let $T$ be any $K$-torsor.  Denote $({\frak g},K)_T = ({\frak g}_T,K_T)$  
the $T$-twist of $({\frak g},K)$ with respect to adjoint action; this is a  
Harish-Chandra pair.  If  $M$ is a $({\frak g},K)$-module, then the $T$-twist  
$M_T$  is a $({\frak g}_T, K_T)$-module, and  $M \longmapsto M_T$  is equivalence 
of categories $({\frak g},K)$-mod  
${\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
({\frak g}_T, K_T)$-mod. 
 
\medpagebreak 
 
\noindent 
3.1.5\ The following version of the above definitions is quite convenient. 
 
A pro-algebraic groupoid ${\Cal V}$  is a groupoid such that for any object $X$  
the group  $Aut X$  carries a pro-algebraic structure and for any   
$f: X \to Y$  the map  $Ad_f: Aut Y\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
Aut X$  preserves the pro-algebraic structures (the objects of ${\Cal V}$  form  
a usual set with no algebraic structure).  A ${\Cal V}$-module is a functor  
$M: {\Cal V} \to  Vect_{\Bbb C}$  such that for any  $X \in {\Cal V}$  the  
$Aut X$-action on $M_X$  is algebraic. 
 
A Harish-Chandra groupoid $({\frak g}, {\Cal V})$  is a pro-algebraic groupoid  
${\Cal V}$  together with a functor  $X \longmapsto ({\frak g}_X, K_X)$  from 
${\Cal V}$  to the category of Harish-Chandra pairs equipped with a canonical  
identification of ``group part" $K_X$  of the functor with  $Aut X$; we assume  
that for  $g \in Aut X = K_X$  the ``functorial" action of $g$  on  
${\frak g}_X$  coincides with the $Ad$-action from 3.1.3. 
 
One defines a representation of our Harish-Chandra groupoid (or simply a  
$({\frak g}, {\Cal V})$-module) in the obvious manner.  For any  $X \in {\Cal V}$   
one has a canonical ``fiber" functor $({\frak g}, {\Cal V})$-mod $\to  
({\frak g}_X, K_X)$-mod, $M \longmapsto M_X$.  If  ${\Cal V}$  is connected,  
this functor is an equivalence of categories.  Note that if  $T$  is a $K_X$-torsor, 
and  $X_T \in {\Cal V}$  is $T$-twist of $X$ (i.e., $X_T$  is an object of  
${\Cal V}$  equipped with isomorphism of $K_X$-torsors\  
$T\ {\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
\text{Hom}(X, X_T)$, then one has a canonical isomorphism\  
$({\frak g}_{X_T}, K_{X_T}) = ({\frak g}_X,K_X)_T$, $M_{X_T} = (M_X)_T$ (see 3.1.4). 
 
\medpagebreak 
 
\noindent 
3.1.6\ We will need to consider the above objects depending on parameters. 
 
Let  $S$  be a scheme, and $K$ be a pro-algebraic group.  A $K$-torsor on $S$ is a  
projective limit of $K/K'$-torsors in the \'etale topology of $S$; here  $K' \subset  
K$  is any normal subgroup such that $K/K'$  has finite type.   
 
Let  ${\Cal V}$  be a pro-algebraic groupoid.  An $S$-object $Y_S$  of ${\Cal V}$ 
is a rule that assigns to each object  $X \in {\Cal V}$  on $Aut X$-torsor  
$Y_S(X)= \underline{Hom}(X,Y_S)$  on $S$ together with canonical identifications  
of  $Aut X$-torsors $Y_S(X) = Y_S(X')_{Hom(X,X')}$ (= the twist of $Y_S(X')$ by 
$Aut X'$-torsor $Hom(X,X')$) for each $X,X' \in {\Cal V}$; these identifications  
should satisfy an obvious compatibility condition for three objects  
$X, X', X'' \in {\Cal V}$.  In other words, $Y_S$ is a functor from ${\Cal V}$ 
to schemes over $S$ such that the $Aut X$-action defines on $Y_S(X)$  the  
structure of $Aut X$-torsor, and for any connected component $S'$ of $S$ the  
objects $X$ for which $Y_{S'}(X) = Y_S(X)_{S'}$  is non-empty are isomorphic. 
If $M$ is a ${\Cal V}$-module, then an $S$-object $Y_S$ of ${\Cal V}$  defines  
a locally free ${\Cal O}_S$-module  $M_{Y_S}$ on $S$.  If  $Y_S(X)$  for  
$X \in {\Cal V}$  is non-empty then  $M_{Y_S}$  coincides with $Y_S(X)$-twist of  
$M_X \otimes {\Cal O}_S$. 
 
Let now $({\frak g}, {\Cal V})$  be a Harish-Chandra groupoid, and  $Y_S$  be an  
$S$-object of ${\Cal V}$  (considered as pro-algebraic groupoid).  We get a sheaf 
${\frak g}_{Y_S}$  of ${\Cal O}_S$-Lie algebras; ${\frak g}_{Y_S}$  is a  
projective limit of locally free ${\Cal O}_S$-modules.  For any   
$({\frak g},{\Cal V})$-module $M$ the  ${\Cal O}_S$-module $M_{Y_S}$  is a  
${\frak g}_{Y_S}$-module. 
\vskip .2in 
 
\subheading{3.2\ Lie algebroids}  Let $S$  be a scheme. 
 
\medpagebreak 
 
\noindent 
3.2.1\ A Lie algebroid on $S$ (which is an infinitesimal version of Lie  
groupoid) is a sheaf ${\Cal A}$  of Lie algebras on $S$ together with an  
${\Cal O}_S$-module structure on ${\Cal A}$  and an ${\Cal O}_S$-linear map  
$\sigma: {\Cal A} \to {\Cal T}_S$  such that $\sigma$ is a morphism of Lie algebras, 
and the formula  $[a,fb] = \sigma(a)(f)b + f[a,b]$  holds for $a,b \in {\Cal A}, 
f \in {\Cal O}_S$.  Clearly  ${\Cal A}_{(0)} = \ker \sigma$ is a sheaf of 
${\Cal O}_S$-Lie algebras.  In the case when $S$ is smooth we will say that ${\Cal A}$ 
is transitive if $\sigma$ is surjective. 
 
The Lie algebroids form a category $Lie(S)$  with final object ${\Cal T}_S$.   
This category has products:  for ${\Cal A}, {\Cal B} \in Lie(S)$  we have  
${\Cal A} \times {\Cal B} = {\Cal A} \times_T {\Cal B}$  in the obvious notations. 
The categories  $Lie(S)$  form a fibered category over the category of schemes. 
For a morphism  $f: S' \to S$  of schemes and ${\Cal A} \in Lie(S)$  the inverse  
image  $f^*{\Cal A} \in Lie(S')$  is defined by the formula $f^*{\Cal A} =  
{\Cal T}_{S'} \times f^*({\Cal A})$.  Here  $f^*({\Cal A}), f^*({\Cal T}_S)$   
are inverse images in the categories of ${\Cal O}$-modules, and the fibered product  
is   $f^*({\Cal T}_S)$  taken with respect to projections  ${\Cal T}_{S'}\  
{\buildrel {df}\over\longrightarrow}\ f^*({\Cal T}_S)  
{\buildrel {f^*(\sigma)}\over\longleftarrow}\ f^*({\Cal A})$. 
 
\medpagebreak 
 
\noindent 
3.2.2\ Let  ${\Cal A}$  be a Lie algebroid.  An ${\Cal A}$-module is a sheaf  
${\Cal F}$  of ${\Cal A}$-modules on $S$ together with an ${\Cal O}_S$-module  
structure such that for  $a \in {\Cal A}$, $f \in {\Cal O}_S$, $m \in {\Cal F}$  
one has  $a(fm) = \sigma(a)(f)m + f(am)$.  We will also call such a structure  
an action of ${\Cal A}$  on ${\Cal O}_S$-module ${\Cal F}$.  If  
${\Cal A}, {\Cal B}$  are Lie algebroids,  ${\Cal F}$ is an ${\Cal A}$-module,  
$G$  is a ${\Cal B}$-module, then  ${\Cal F} \otimes_{{\Cal O}_S}  G$  
is  ${\Cal A} \times {\Cal B}$-module:  for $(a,b) \in {\Cal A} \times {\Cal B}$, 
$m \in {\Cal F}, n \in G$  one has  $(a,b)(m \otimes n) = (am) \otimes n +  
m \otimes (bn)$.   
	      
\medpagebreak 
 
\noindent 
3.2.3\ Let ${\Cal A}$  be a Lie algebroid, and ${\frak g}$  an ${\Cal O}_S$-Lie  algebra equipped with an ${\Cal A}$-action.  An ${\Cal A}$-morphism  
$\psi: {\Cal A}_{(0)} \to {\frak g}$  is a morphism of  ${\Cal O}_S$-Lie  
algebras that commutes with ${\Cal A}$-action (here the ${\Cal A}$-action on  
${\Cal A}_{(0)}$  is adjoint one).  Note that if  $\varphi: {\Cal A} \to  
{\Cal B}$  is a morphism of Lie algebroids, then ${\Cal A}$ acts on  
${\Cal B}_{(0)}$ by $ad \circ \varphi$, and $\varphi_{(0)}: {\Cal A}_{(0)}  
\to {\Cal B}_{(0)}$  is an ${\Cal A}$-morphism. Conversely, for an ${\Cal A}$-morphism 
$\psi: {\Cal A}_{(0)}\to {\frak g}$  let  ${\Cal A}_\psi$  be the quotient  
of the semi-direct product  ${\Cal A} \ltimes {\frak g}$  by the ideal  
${\Cal A}_{(0)} \hookrightarrow {\Cal A} \ltimes {\frak g}, a \longmapsto  
(a, - \psi(a))$.  Then  ${\Cal A}_\psi$  is a Lie algebroid, ${\Cal A}_{\psi_{(0)}}  
= {\frak g}$, and we have a canonical morphism  $\psi: {\Cal A} \to {\Cal A}_\psi$ 
with $\psi_{(0)} = \text{old}\  \psi$.  These constructions are mutually inverse:  if 
${\frak g} = {\Cal B}_{(0)}$, $\varphi: {\Cal A} \to {\Cal B}$  is a morphism of 
Lie algebroids, and  $\psi = {\varphi}_{(0)}$, then we have a canonical morphism   
$i: {\Cal A}_\psi \to {\Cal B}$   which is an isomorphism if  ${\Cal A}$  is  
transitive. 
 
\medpagebreak 
 
\noindent 
3.2.4\ Let ${\Cal A}$  be a Lie algebroid.  A central extension of  ${\Cal A}$  by 
${\Cal O}_S$  is a Lie algebroid   $\widetilde{\Cal A}$  together with a   
surjective morphism  $\pi: \widetilde{\Cal A} \to {\Cal A}$  and a central  
element  $1 \in \ker \pi$  such that the map  ${\Cal O}_S\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
\ker \pi$, $f \longmapsto f \cdot 1$, is isomorphism.  Note that the adjoint action of  $\widetilde{\Cal A}$  on $\widetilde{\Cal A}_{(0)}$  quotients to an  ${\Cal A}$-action.  We will call a central extension ${\Cal L}$  of  
${\Cal T}_S$  by ${\Cal O}_S$  an invertible Lie algebroid (so ${\Cal L}_{(0)} =  {\Cal O}_S$). 
 
\medpagebreak 
 
\demo{3.2.5\ Remarks} (i)\ Let $B$ be any Lie algebroid, and let  
$tr: {\Cal B}_{(0)} \to {\Cal O}_S$  be a ${\Cal B}$-morphism (we will call such  
$tr$  a trace on ${\Cal B}$).  If  ${\Cal B}$  is transitive, then  ${\Cal B}_{tr}$  
is an invertible algebroid. 
 
\noindent 
(ii)\ Let  $\widetilde{\Cal A}\ {\buildrel\pi\over\longrightarrow}\ {\Cal A}$  be  
a central  extension of  ${\Cal A}$  by ${\Cal O}_S$, and  $\gamma: {\Cal A}_{(0)}  
\to \widetilde{\Cal A}$  be  an ${\Cal O}$-linear section of $\pi$  such that  
$\gamma$   commutes with adjoint action of ${\Cal A}$.  Then  
$\gamma({\Cal A}_{(0)})$  is ideal in $\widetilde{\Cal A}$, and   
$\widetilde{\Cal A}/\gamma({\Cal A}_{(0)})$  is invertible algebroid.   
\enddemo 
 
\medpagebreak 
 
\noindent 
3.2.6\  The invertible Lie algebroids form a category  ${\Cal P} Lie(S)$  
which is a Picard category, and, more generally,  a ``${\Bbb C}$-vector space"  
in categories.  This means that for  $\alpha,\beta, \in {\Bbb C}$,\ ${\Cal A},  
{\Cal B} \in {\Cal P} Lie(S)$  we may form the linear combination\  
$C = \alpha {\Cal A} + \beta {\Cal B} \in {\Cal P} Lie(S)$:  by definition  
$C = ({\Cal A} \times {\Cal B})_{tr_{\alpha,\beta}}$, where  
$tr_{\alpha,\beta}(f,g) = \alpha f + \beta g$.  For ${\Cal A} \in {\Cal P} Lie(S)$ 
we have  $Aut {\Cal A} = \Omega_S^{1 c\ell}$:  for a closed 1 form $\omega$ the  
corresponding automorphism of ${\Cal A}$  is $a \longmapsto a +  
\langle\omega\sigma(a)\rangle\cdot 1.$  
The trivial invertible algebroid is ${\Cal T}_{SO} = {\Cal T}_S \ltimes  
{\Cal O}_S$  (where  $O: {\Cal T}_{S(0)} = 0 \to {\Cal O}_S$  is the trivial trace 
map).  The      locally trivial invertible Lie algebroids form a full  
${\Bbb C}$-linear subcategory canonically equivalent to the one of  
$\Omega^{1c\ell}$-torsors. 
 
\medpagebreak 
 
\noindent 
3.2.7\  For  ${\Cal A} \in {\Cal P} Lie(S)$  define  ${\Cal D}_{\Cal A}$  to be  
the sheaf of associative ${\Bbb C}$-algebras on $S$  together with a morphism  
of ${\Bbb C}$ Lie algebras  $i: {\Cal A} \to {\Cal D}_{\Cal A}$  such that   
$i\big\vert_{{\Cal O}_S}$  is a morphism of associative algebras (in particular, 
$i(1)$  is 1 in ${\Cal D}_{\Cal A}$) and one has  $i(f)i(a) = i(fa)$  for  
$f \in {\Cal O}_S$, $a \in {\Cal A}$, and universal with respect to these data. 
For example, if  ${\Cal A}$  is trivial, then  ${\Cal D}_{\Cal A}$  is the usual 
algebra of differential operators on $S$.  For arbitrary ${\Cal A}$  this is a  
twisted differential operators ring, see, e.g. Appendix to [BK] for details. 
Clearly a  ${\Cal D}_{\Cal A}$-module ${\Cal F}$  is the same as an ${\Cal A}$-module 
such that  $1 \in {\Cal A}$  acts on ${\Cal F}$  as the identity operator.  Since  
${\Cal D}_{\Cal A}$  carries an obvious filtration with  $gr {\Cal D}_{\Cal A} =  
Sym{\Cal T}_S$,  for a coherent  ${\Cal D}_{\Cal A}$-module ${\Cal F}$   
we have  
its singular support  $SS{\Cal F}$  which is a closed conical subset in the cotangent 
bundle of $S$.  A  ${\Cal D}_{\Cal A}$-module ${\Cal F}$  is called lisse if  
$SS{\Cal F} = (0)$:  this condition is equivalent to the fact that  ${\Cal F}$ 
is a vector bundle (as ${\Cal O}_S$-module).   
 
\medpagebreak 
 
\noindent 
3.2.8\  The standard example of a Lie algebroid is the current (or Atiyah) algebra 
${\Cal A}(E)$  of a vector bundle  $E$.  This is the Lie algebra of infinitesimal  
symmetries of $E$.  The sections of  ${\Cal A}(E)$  are pairs  $(\sigma(\tau), 
\tau)$, where  $\sigma(\tau) \in {\Cal T}_S$   and  $\tau$  is an action of  
$\sigma(\tau)$  on $E$, or, equivalently, a first order differential operator 
 on $E$ with symbol  $\sigma(\tau) \cdot id_E$.  Clearly ${\Cal A}(E)$  is  
transitive and ${\Cal A}(E)_{(0)} = {\frak g}\ell(E)$.  If $L$ is a line bundle, 
then ${\Cal A}(L)$  is invertible algebroid; one has  ${\Cal A}(L_1 \otimes L_2)  
= {\Cal A}(L_1) + {\Cal A}(L_2)$, i.e., ${\Cal A}: {\Cal P}ic(S) \to {\Cal P} Lie(S)$ 
is a morphism of Picard categories.  The ring  ${\Cal D}_{{\Cal A}(L)}$   
coincides with the algebra  ${\Cal D}_L$  of differential operators on $L$.  If  $E$ is any vector bundle, then  tr: ${\frak g}\ell(E) \to {\Cal O}_S$  is a trace on 
${\Cal A}(E)$, and ${\Cal A}(E)_{\text{tr}} = {\Cal A}(\det E)$:  this canonical 
isomorphism comes from a natural action of ${\Cal A}(E)$ on det $E$ given  
explicitly by the Leibnitz rule $a(e_1 \wedge \ldots \wedge e_n) = ae_1 \wedge e_2  
\wedge \ldots \wedge e_n + \cdots + e_1 \wedge \ldots \wedge ae_n$.   
\vskip .2in 
 
\subheading{3.3\ Localization of $({\frak g},K)$-modules}  Below we will explain  
the general pattern how to transform  
representations to ${\Cal D}$-modules.  We will start with some  
notations. 
 
\medpagebreak 
 
\noindent 
3.3.1\ Let  $(\widetilde{\frak g}, {\Cal V})$  be a Harish-Chandra groupoid.  We 
will say that it is centered if for any  $X \in {\Cal V}$  there is a fixed  
central element  $1 \in \widetilde{\frak g}_X$, $1 \notin Lie Aut X$, that depends  
on $X$ in a natural way.  Put  ${\frak g}_X = \widetilde{\frak g}_C/{\Bbb C} 1$, 
so  $\widetilde{\frak g}_X$  is a central ${\Bbb C}$-extension of  ${\frak g}_X$. 
 
 
Our $(\widetilde{\frak g}, {\Cal V})$  defines several Harish-Chandra groupoids  
with the same underlying proalgebraic groupoid ${\Cal V}$.  Namely, we have the  
groupoid $({\frak g}, {\Cal V})$ that corresponds to  ${\frak g}_X$; 
for any  $c \in {\Bbb C}$  we have the centered groupoid $(\widetilde{\frak g}_c, {\Cal V})$ 
with $\widetilde{\frak g}_{cX}$  equal to $c$-multiple of the central extension 
$\widetilde{\frak g}_X$  of ${\frak g}_X$.  Denote by  $(\widetilde{\frak g}, {\Cal V})_c$-mod 
the category of $(\widetilde{\frak g}_c, {\Cal V})$-modules on which  
$1 \in {\Bbb C} \subset \widetilde{\frak g}_c$  acts as identity. 
 
\medpagebreak 
 
\noindent 
3.3.2\ Let  $S$  be a smooth scheme, $K$ be a proalgebraic group and  $Y_S$  be 
a $K$-torsor over $S$.  Denote by  ${\Cal A} Y_S$  the Lie algebroid of  
infinitesimal symmetries of  $(S, Y_S)$.  Its sections are pairs  
$(\tau, \tau_{Y_S})$, where  $\tau \in \tau_{Y_S}$  and  ${\Cal T}_{Y_S}$  is a  
lifting of  $\tau$ to $Y_{S}$  that commutes with $K$-action.  Clearly  
${\Cal A} Y_{S(0)} = Lie K_{Y_S}$ ($= Y_S$-twist of Lie $K \widehat{\otimes}  
{\Cal O}_{S}$  with respect to the adjoint action of $K$); ${\Cal A}Y_S$  is a  
transitive groupoid.  If  $({\frak g}, K)$  is a Harish-Chandra pair, then we  
have the ${\Cal O}_S$-Lie algebra  ${\frak g}_{Y_S}$ ($= Y_S$-twist of  
${\frak g} \widehat{\otimes} {\Cal O}_S$  with respect to the adjoint action). 
The Lie    algebroid  ${\Cal A} Y_S$  acts on  ${\frak g}_{Y_S}$  in an obvious 
manner, and the canonical embedding  $i: {\Cal A} Y_{S(0)} = Lie K_{Y_S}\  
\hookrightarrow\ {\frak g}_{Y_S}$  is an ${\Cal A}Y_S$-morphism.  According to  
3.2.3 we get the transitive Lie algebroid  ${\Cal A}{\frak g}_{Y_S} =  
{\Cal A} Y_{Si}$  with  ${\Cal A}{\frak g}_{Y_S(0)} = {\frak g}_{Y_S}$.  If  
$M$ is a $({\frak g}, K)$-module, then  $M_{Y_S}$ (= $Y_S$-twist of   
$M \otimes {\Cal O}_S$)  is an ${\Cal A}{\frak g}_{Y_S}$-module. 
 
\medpagebreak 
 
Now let  $({\frak g}, {\Cal V})$  be a Harish-Chandra groupoid, and let $Y_S$  be an  
$S$-object of ${\Cal V}$.  The above construction defines a transitive Lie  
algebroid  ${\Cal A}{\frak g}_{Y_S}$  on $S$ with ${\Cal A}{\frak g}_{Y_S(0)} =  
{\frak g}_{Y_S}$.  If  $M$ is a $({\frak g}, {\Cal V})$-module, then  $M_{Y_S}$   
is an ${\Cal A}{\frak g}_{Y_S}$-module in a natural way.  Note that if   
$(\widetilde{\frak g}, {\Cal V})$  is a centered groupoid, then   
${\Cal A} \widetilde{\frak g}_{Y_S}$  is a central  ${\Cal O}_S$-extension of  
${\Cal A}{\frak g}_{Y_S}$. 
 
\medpagebreak 
 
\proclaim{3.3.3\ Definition}  Let  $S$  be a smooth scheme and   
$(\widetilde{\frak g}, {\Cal V})$  be a centered Harish-Chandra groupoid.  An  
$S$-localization data  $\psi$  for $(\widetilde{\frak g}, {\Cal V})$  is a  
collection  $(Y_S, N, \varphi, \widetilde{\varphi}_{(0)})$ where 
 
\item{(i)}  $Y_S$  is an $S$-object of ${\Cal V}$. 
 
\item{(ii)}  $N$ is a transitive Lie algebroid on $S$. 
 
\item{(iii)} $\varphi: N \to {\Cal A} \frak{g}_{Y_S}$  is a morphism of Lie  
algebroids. 
 
\item{(iv)}  $\widetilde{\varphi}_{(0)}: N_{(0)} \to \widetilde{\frak g}_{Y_S}$   
is a lifting of  $\varphi_{(0)}$  such that for $n \in N, m \in N_{(0)}$  one 
has $\widetilde{\varphi}_{(0)}([n,m]) = [\varphi(n), \varphi_{(0)}(m)]$.\hfill $\square$ 
\endproclaim 
 
\medpagebreak 
 
\noindent 
3.3.4\ A localization data $\psi$ defines an invertible Lie algebroid  ${\Cal A}_\psi$ 
on $S$ as follows.  Consider a fiber product  ${\Cal A}\widetilde{\frak g}_{Y_S}N  
= {\Cal A}\widetilde{\frak g}_{Y_S} \times_{{\Cal A}{\frak g}_{Y_S}} N$:  this is  
a central  ${\Cal O}_S$-extension of $N$.  This central extension splits over  
$N_{(0)}$  by means of the section $s: N_{(0)} \to {\Cal A}{\frak g}_{Y_S}N_{(0)}$,  
$s(m) = (\widetilde{\varphi}_{(0)}(m),m)$.  Put  ${\Cal A}_\psi :=  
{\Cal A}\widetilde{\frak g}_{Y_S} N/s(N_{(0)})$.  Let  $D_\psi = D_{{\Cal A}_\psi}$ 
be the corresponding algebra of twisted differential operators.   
 
\medpagebreak 
 
\noindent 
3.3.5\ Let  $M \in (\widetilde{\frak g}, {\Cal V})_1$-mod  be a Harish-Chandra  
module such that $1$ acts as  id$_M$.  Then  $M_{Y_S}$  is an  
${\Cal A}\widetilde{\frak g}_{Y_S}N$-module (via the projection  
${\Cal A}\widetilde{\frak g}_{Y_S}N \to {\Cal A}\widetilde{\frak g}_{Y_S})$,  
and  $\Delta_\psi M = M_{Y_S}/s(N_{(0)})M_{Y_S}$  is ${\Cal A}_\psi$-module on  
which  $1 \in {\Cal A}_\psi$  acts as identity.  Hence  $\Delta_\psi M$  is a  
$D_\psi$-module.  Clearly  $\Delta_\psi: (\widetilde{\frak g}, {\Cal V})_1$-mod 
$\to D_\psi$-mod is a right exact functor; we call it the $S$-localization functor that  
corresponds to $\psi$.  Note that for a point  $s \in S$  we have a Lie algebra  
map  $N_{(0)s} \to \widetilde{\frak g}_{Y_S}$  (where  $N_{(0)s} =  
N_{(0)}/m_sN_{(0)}$), hence the fiber  $\Delta_\psi(M)/m_s\Delta_\psi(M)$   
coincides with coinvariants  $M_{Y_s}/N_{(0)s} M_{Y_S}$. 
 
\medpagebreak 
 
\noindent 
3.3.6  The above constructions are functorial with respect to morphisms of  
localization data.  Precisely, let  $(\widetilde{\frak g}', {\Cal V}')$  be  
another centered Harish-Chandra groupoid, and  $r:   
(\widetilde{\frak g}, {\Cal V}) \to (\widetilde{\frak g}', {\Cal V}')$  is a  
morphism of centered groupoids.  One defines an $r$-morphism of $S$-localization 
data  $r^\#: \psi \to \psi'$  in an obvious manner.  Such   $r^\#$  defines the 
isomorphisms  $r^\#_{\Cal A}: {\Cal A}_\psi\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
{\Cal A}_{\psi'}$, $r^\#_D: D_\psi\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
D_{\psi'}$.  For  $M \in (\widetilde{\frak g}, {\Cal V})_1$-mod,  
$M \in (\widetilde{\frak g}', {\Cal V}')_1$-mod  and an $r$-morphism   
$\ell: M \to M'$  we have  $r^\#_D$-morphism  $r^\#_\Delta:   
\Delta_\psi(M) \to \Delta_{\psi'}(M')$.   
 
One has also functoriality with respect to base change.  If  $f: S' \to S$  is 
a morphism of smooth schemes, and  $\psi$  is an $S$-localization data for  
$(\widetilde{\frak g}, {\Cal V})$, then one gets an $S'$-localization data   
$f^*\psi$ for  $(\widetilde{\frak g}, {\Cal V})$.  One has  ${\Cal A}_{f^*\psi}  
= f^*{\Cal A}_\psi$, and for  $M \in (\widetilde{\frak g}, {\Cal V})_1$-mod one  
has a natural isomorphism  $f^*\Delta_\psi(M) = \Delta_{f^*\psi}(M)$  of  
$D_{f^*\psi}$-modules. 
 
\medpagebreak 
 
\noindent 
3.3.7\ An $S$-localization data  $\psi$  for  $(\widetilde{\frak g}, {\Cal V})$  
defines in an obvious way for each  $c \in {\Bbb C}$ an $S$-localization data  
$\psi_c$  for  $(\widetilde{\frak g}_c, {\Cal V})$.  One has  
${\Cal A}_{\psi_c} = c {\Cal A}_\psi$  (see 3.2.6). 
\vskip .2in 
 

 
 
\subheading{3.4\ Localization along moduli of curves}  This section collects  
some basic examples of the above localization constructions. 
 
\medpagebreak 
 
\noindent 
3.4.1\ Let us describe a centered Harish-Chandra groupoid  $(\widetilde{\Cal T}, 
{\Cal V})$  called the Virasoro groupoid.  The underlying connected proalgebraic  
groupoid  ${\Cal V}$  is the groupoid of one-dimensional local fields with residue field equal $\Bbb C$ (the  
morphisms are isomorphisms of the local fields).  Precisely, let  $F \in {\Cal V}$  be a local  
field,  ${\Cal O}_F \subset F$  the corresponding local ring, and  
$m_F \subset {\Cal O}_F$   the maximal ideal.  A choice of uniformizing  
parameter $t$  identifies  $F$  with  ${\Bbb C}((t))$ and  ${\Cal O}_F$  with 
${\Bbb C}[[t]]$.   
The   group  $Aut F = Aut {\Cal O}_F$  is the projective limit of groups  
$Aut {\Cal O}_F/m^n_F = Aut F/Aut_nF$.  These groups are obviously algebraic  
groups, our $Aut F$  is a proalgebraic group, and ${\Cal V}$  is a  
proalgebraic groupoid.  Note that  $Aut F/Aut_1F = {\Bbb C}^*$, and   
$Aut_iF/Aut_{i+1}F$  is isomorphic to ${\Bbb C}$  for  $i \ge 1$; in particular 
$Aut_1F$  is the pro-unipotent radical of  $Aut F$.  Explicitly, $Aut {\Bbb C}((t))$  
coincides with the group of power series  $a_1t + a_2t^2 + \cdots, a_1 \ne 0$, 
with multiplication law equal to composition of series. 
 
Now for  $F \in {\Cal V}$  let  ${\Cal T}_F$  be the Lie algebra of vector  
fields on $F$  and  $\widetilde{\Cal T}_F$  be the Virasoro ${\Bbb C}$-extension  
of ${\Cal T}_F$  defined in 2.1.3.  The Lie algebra  ${\Cal T}_F$  carries a  
canonical filtration   ${\Cal T}_{iF}$;  for  $F = {\Bbb C}((t))$  one has  
${\Cal T}_{iF} = t^{i+1}{\Bbb C}[[t]]\partial_t$.  The subalgebra ${\Cal T}_{-1F}$ 
preserves the lattice ${\Cal O}_F \subset F$, hence we have a canonical splitting  
$s_{{\Cal O}_F}: {\Cal T}_{-1 F} \to \widetilde{\Cal T}_F$.  Clearly   
$Lie Aut F = {\Cal T}_{0F}$, and the embedding  $s_{{\Cal O}_F}: Lie Aut F  
\hookrightarrow \widetilde{\Cal T}_F$  together with the natural $Aut F$-action on  
$\widetilde{\Cal T}_F$  define the Harish-Chandra pair $(\widetilde{\Cal T}_F,  
Aut F)$.  This    defines our centered Virasoro groupoid  $(\widetilde{\Cal T}, 
{\Cal V})$.   
 
\medpagebreak 
 
\noindent 
3.4.2\ Let  $S$  be a scheme.  It is easy to see that an $S$-object  $Y_S$  of  
${\Cal V}$  is the same as a ``family of formal discs" over $S$ or, equivalently,  
a formal  ${\Cal O}_S$-algebra ${\Cal O}_Y$  locally isomorphic to ${\Cal O}_S[[t]]$. 
The corresponding Lie algebroid  ${\Cal A}Y_S$  consists of pairs  
$(\tau, \tau_{Y_S})$  where  $\tau \in {\Cal T}_S$  and  $\tau_{Y_S} \in  
Der {\Cal O}_{Y_S}$  is a $\tau$-derivation of  ${\Cal O}_{Y_S}$. 
	      
\medpagebreak 
 
\noindent 
3.4.3\ Now let  $\pi: C_S \to S$  be a family of smooth projective curves and  
$a: S \to C_S$  be a section of $\pi$.  These define an $S$-localization data  
$\psi = \psi(C_S,a)$  for $(\widetilde{\Cal T}, {\Cal V})$  as follows.  Our  
$Y_S$  is the formal completion of  $C_S$  along  $a(S)$, and $N$ is the Lie algebroid  
of pairs  $(\tau, \tau_U)$  where  $\tau \in {\Cal T}_S$  and  $\tau_U$  is a   
lifting of  $\tau$  to  $U = C_S\setminus a(S)$.   
Clearly  ${\Cal A}{\Cal T}_{Y_S}$  is the Lie algebroid 
of pairs  $(\tau, \tau_{Y_{S\setminus (a)}})$, where  $\tau \in {\Cal T}_S$ 
and $\tau_{Y_{S\setminus (a)}}$  is a lifting of  $\tau$  to a meromorphic 
vector field on  $Y_S$  with possible pole at  $a(S)$.  Our  $\varphi: N \to  
{\Cal A}{\Cal T}_{Y_S}$  is just the restriction of a vector field  $\tau_U$  on  $Y_S\setminus \{ a\} =$\  punctured neighbourhood of  $a$.  Now the lifting  
$\widetilde{\varphi}_{(0)}: N_{(0)} = \pi_* {\Cal T}_{U/S} \to  
\widetilde{\Cal T}_{Y_S}$  is the restriction to ${\Cal T}_{U/S} \subset D_{U/S}$ of the morphism   
$s_a: \pi_* {\Cal D}_{U/S} \to \widetilde{D}_{(a)}$  (here  $D = D_{{\Cal O}_{C/S}})$ 
defined in 2.3.4 (more precisely, in  
2.3.4 we considered the case of a single curve, $S =$ point; the generalization  
to families is immediate).  These  $(Y_S, N, \varphi, \widetilde{\varphi}_{(0)})$ 
is our localization data  $\psi(C_S,a)$.  According to 3.3.4, 3.3.5, 3.3.7 for any 
$c \in {\Bbb C}$  we have the localization functor  $\Delta_{\psi_c(C_S,a)}:  
(\widetilde{\Cal T}, {\Cal V})_c$-mod  $\to {\Cal D}_{\psi_c(C_S,a)}$-mod. 
 
\medpagebreak 
 
\noindent 
3.4.4\ Here is an explicit description of  ${\Cal A}_{\psi(C_S,a)}$  and  
$\Delta_{\psi(C_S,a)}$.  Choose (locally on $S$) a formal parameter $t$ at $a$,  
so ${\Cal O}_{Y_S} = {\Cal O}_S[[t]]$.  Consider the space  $B$  of triples  
$(\tau, \tau_U, \widetilde{\tau}_U^v)$, where  $\tau \in {\Cal T}_S$, 
$\tau_U$  is a lifting of $\tau$ to $U$, and $\widetilde{\tau} _U^v:  
S \to \widetilde{\Cal T}_{{\Bbb C}((t))}$  is a lifting of a vertical component 
of $\tau_U$, $\tau_U^v = \tau_U(t) \partial_t: S \to {\Cal T}_{{\Bbb C}((t))}$. 
This $B$ is a Lie algebroid on $S$  in an obvious manner.  We have a canonical 
 morphism  ${\Cal T}_{U/S} \to B_{(0)}$,\ $\nu \longmapsto (o, \nu, s_a(\nu))$,  
see 2.3.4.  One has  ${\Cal A}_{\psi(C_S,a)} = {\Cal B}/{\Cal T}_{U/S}$.  Now 
let  $M$  be a $(\widetilde{\Cal T}, {\Cal V})_c$-module.  One has   
$M_{Y_S} = M_{{\Bbb C}((t))} \otimes {\Cal O}_S$.  The algebroid ${\Cal B}$  acts 
on  $M_{Y_S}$  by formula  $(\tau, \tau_U, \widetilde{\tau}_U^v)(m \otimes f)  
= m \otimes \tau(f) + \widetilde{\tau}_U^v(m \otimes f)$.  One has  
$\Delta_{\psi(C_S,a)}(M) = M_{Y_S}/{\Cal T}_{U/S} M_{Y_S}$. 
 
\medpagebreak 
 
\subheading{3.4.5\ Variant}  For any non empty finite set $A$ we may consider the 
centered groupoid  $(\widetilde{\Cal T}^A, {\Cal V}^A)$.  Here  ${\Cal V}^A$  is  
the $A$-th power of ${\Cal V}$ and  $\widetilde{\Cal T}^A_{\{ F_a\}}$  is the  
Baer sum of ${\Bbb C}$-extension  $\widetilde{\Cal T}_{F_a}$, $a \in A$ (so  
$\widetilde{\Cal T}^A_{\{ F_a\}}$  is a ${\Bbb C}$-extension of   
${\dsize\prod_{a\in A}} {\Cal T}_{F_a}$).  A family  $\pi: C_S \to S$  of curves  
together with a disjoint set  $A$ of sections  (where  ``disjoint" means that for 
$a_i \ne a_j \in A$  and any  $s \in S$  one has  $a_i(s) \ne a_j(s) \in C_S$)  
defines an $S$-localization data  $\psi(C_S,A)$ for  $(\widetilde{\Cal T}^A,  
{\Cal V}^A)$  in a way similar to 3.4.2.  For example, the corresponding Lie 
algebroid $N$ consists of pairs  $(\tau, \tau_U)$, where  $\tau \in {\Cal T}_S$  
and $\tau_U$  is a lifting of  $\tau$  to  $U = C_S\setminus {\dsize\coprod_{a\in A}} 
a_i(S)$. 
 
\medpagebreak 
 
\demo{3.4.6\ Remark}  Let  $B \subset A$  be a non-empty subset.  The groupoids 
$(\widetilde{\Cal T}^B, {\Cal V}^B)$  and  $(\widetilde{\Cal T}^A, {\Cal V}^A)$  
are related by an obvious correspondence  $(\widetilde{\Cal T}^B, {\Cal V}^B)\  
{\buildrel {\pi_B}\over\longleftarrow}\ (\widetilde{\Cal T}^{A,B}, {\Cal V}^A)\  
{\buildrel {i_A}\over\longrightarrow}\ (\widetilde{\Cal T}^A, {\Cal V}^A)$, where 
$\widetilde{\Cal T}^{A,B}_{\{ F_a\}} = \widetilde{\Cal T}^B_{{\{ F_b\}_{b\in B}}}  
\times {\dsize\prod_{a\in A\setminus B}} {\Cal T}_{-1 F_a}\ \hookrightarrow\  
\widetilde{\Cal T}^A_{\{ F_a\}}$.  Any family of curves  $\pi: C_S \to S$ 
and a set  $A$  of disjoint sections defines an $S$-localization data  
$\psi(C_S,A,B)$  for  $(\widetilde{\Cal T}^{A,B}, {\Cal V}^A)$  in an obvious  
manner 	together with corresponding morphisms $\psi(C_S,B)$ $ 
{\buildrel {\pi^\#_B}\over\longleftarrow}\ \psi(C_C, A,B)\  
{\buildrel {i_A}\over\longrightarrow}\ \psi(C_S,A)$.  These define the corresponding  
isomorphisms  $D_{\psi_c(C_S,B)}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
D_{\psi_c(C_S,A,B)}\ {\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
D_{\psi_c(C_S,A)}$.  For  $M_B \in ({\Cal T}^B, {\Cal V}^B)_c$-mod, $M_A \in  
({\Cal T}^A, {\Cal V}^A)$-mod a morphism  $f: M_B \to M_A$  is, by definition,  
an  $i_A$-morphism from  $M_B$, considered as $(\widetilde{\Cal T}^{A,B}, 
{\Cal V}^A)$-module via  $\pi_B$, to  $M_A$.  Since  
$\Delta_{\psi_c(C_S,B)} M_B = \Delta_{\psi_c(C_S,A,B)} M_B$, such an $f$  defines a morphism  $\Delta(f): \Delta_{\psi_c(C_S,B)} M_B \to  
\Delta_{\psi_c(C_S,A)} M_A$.  For example, if  $M_A =  
Ind^{\widetilde{\Cal T}^A}_{\widetilde{\Cal T}^{A,B}} (M_B)$ and  $f$  is the  
canonical embedding, then  $\Delta(f)$  is isomorphism. 
\enddemo 
 
\medpagebreak 
 
Note that the above canonical identification  $D_{\psi_c(C_S,A)} =  
D_{\psi_c(C_S,B)}$  for   $B \subset A$  actually provides a canonical algebra  
$D_{\psi_c(C_S)}$  that depends on  $C_S$  only together with canonical  
isomorphisms $D_{\psi_c(C_S)} = D_{\psi_c(C_S,A)}$  for any set  $A$ of disjoint  
sections.  To construct  $D_{\psi_c(C_S)}$  we may assume, working locally in 
\'etale topology of $S$, that  $C_S$  has many sections.  To construct  
$D_{\psi_c(C_S)}$  it suffices to define for any two sets  $A,A'$  of disjoint  
sections a canonical isomorphism  $D_{\psi_c(C_S,A)} = D_{\psi_c(C_S,A')}$.   
Choose a non-empty set $B$ of sections such that both  $A \sqcup B, A' \sqcup B$ 
are sets of disjoint sections.  Our isomorphism is  $D_{\psi_c(C_S,A)} =  
D_{\psi_c(C_S,A\sqcup B)}  = D_{\psi_c(C_S,B)} = D_{\psi_c(C_S,A'\sqcup B)} =  
D_{\psi_c(C_S,A')}$.  One verifies easily that this does not depends on the choice  
of $B$.  We will compute  $D_{\psi_c(C_S)}$  explicitly in 3.5.6. 
 
\medpagebreak 
 
\subheading{3.4.7\ Variant}  Often the Virasoro modules are integrable only with  
respect to the subgroup  Aut$_1F$  (see 3.4.1).  To localize them one needs to consider 
the groupoid  $(\widetilde{\Cal T}, {\Cal V}_1)$.  The objects of  ${\Cal V}_1$ 
are pairs  $(F,\nu)$, where  $F$  is a local field and  $\nu \in m_F/m^2_F$,  
$\nu \ne 0$, is a 1-jet of a parameter.  One has  Aut$(F,\nu) = \text{Aut}_1F$. 
The Lie algebra  $\widetilde{\Cal T}_{(F,\nu)}$  is $\widetilde{\Cal T}_F$.   
If  $\pi: C_S \to S$  is a family of curves, $a: S \to C_S$   a section, and 
$\nu \in a^*\Omega^1_{C_S/S}$   a 1-jet of parameters at $a$, then we get an  
$S$-localization data  $\psi(C_S, a, \nu)$  for  $(\widetilde{\Cal T}, {\Cal V}_1)$. We may also consider many points, as in 3.4.5. 
 
\medpagebreak 
 
We have a ``forgetting of $\nu$" morphism  $r: (\widetilde{\Cal T}, {\Cal V}_1)  
\to (\widetilde{\Cal T}, {\Cal V})$  and a corresponding $r$-morphism of  
localization data  $\psi_c(C_S, a, \nu) \to \psi_c(C_s,a)$.  This defines a  
canonical isomorphism  $r_D:  D_{\psi_c(C_S,a, \nu)}\  
{\lower 1ex\hbox{$\overleftarrow{\sim}\vrule height 1ex width 0pt$}}\   
D_{\psi_c(C_S,a)}$\  and for any  $M \in ({\Cal T}, {\Cal V})_c$-mod the  
$r_D$-isomorphism  $r_M: \Delta_{\psi_c(C_S,a, \nu)} M\  
{\lower 1ex\hbox{$\overleftarrow{\sim}\vrule height 1ex width 0pt$}}\   
\Delta_{\psi_c(C_S,a)} M$. 
 
\medpagebreak 
 
\noindent 
3.4.7.1\ Let  $C$  be a fixed curve,  $a \in C$, and $\nu$ a 1-jet of parameter  at $a$.  Consider a constant  ${\Bbb C}^*$-family  $C_{{\Bbb C}^*} = C \times  
{\Bbb C}^*$  with constant point $a$, and put  $\nu^\vee(u) = u\nu$  for   
$u \in {\Bbb C}^*$.  We get the corresponding  ${\Bbb C}^*$-localization data  
$\psi = \psi(C_{{\Bbb C}^*}, a, \nu^\vee)$.  One has  $D_\psi =  
D_{\psi(C_{{\Bbb C}^*}, a, \nu^\vee)} = D_{\psi(C_{{\Bbb C}^*}, a)} =  
D_{{\Bbb C}^*}$ -- the usual ring of differential operators.  In particular, we 
have  $\lambda\partial_\lambda \in D_{\psi_c}$.  Let us compute the action of  
$u\partial_u$  on  $\Delta_{\psi_c}(M)$  for $M \in ({\Cal T}, {\Cal V}_1)_c$-mod.  
Choose a parameter  $t_a$  at $a$  on  $C$  such that  $dt(a) = \nu$.  Then  
$t_{au} = ut$  is a ${\Bbb C}^*$-family of parameters which identifies our  
${\Cal O}_{Y_{{\Bbb C}^*}}$  with  ${\Cal O}_{{\Bbb C}^*}[[t]]$.  We have  
$M_{Y_{{\Bbb C}^*}} = M_{{\Bbb C}((t))} \otimes {\Cal O}_{{\Bbb C}^*}$, and  
$\Delta_{\psi_c}(M)$  is a quotient of	$M_{Y_{{\Bbb C}^*}}$.  For  
$m \in  M_{{\Bbb C}((t))}$  denote by  $\overline{m}$  its image in   
$\Delta_{\psi_c}(M)$.  Put  $L_0 = s_{{\Bbb C}[[t]]}(t\partial_t) \in  
\widetilde{\Cal T}_{{\Bbb C}((t))}$.  One has  $u\partial_u(\overline{m}) =  
\overline{L_0m}$.  In particular, if  $M$  is a higher weight module (see  
7.3.1), then  $\Delta_\psi M$  is smooth along  ${\Bbb C}^*$  with monodromy 
equal to the action of  $T = \text{exp}(2\pi i L_0)$  (see 7.3.2). 
 
\medpagebreak 
 
\noindent 
3.4.8\ Now consider the case ``vector symmetries".  Our ``Virasoro-Kac-Moody" 
centered Harish-Chandra groupoid  $(\widetilde{\Cal A}, {\Cal V}{\Cal V})$  defined as follows. The objects of  ${\Cal V}{\Cal V}$  are pairs $(F,E_{\Cal O})$ where  $F$  is a local field,  and $E_{\Cal O}$  is a free ${\Cal O}_F$-module of finite rank; we put  $E_F = F \otimes E_{\Cal O}$.  The morphisms are defined in an obvious manner.  Clearly  Aut$(F, E_{\Cal O})$  is extension of  Aut $F$ by GL$(E_{\Cal O}) = \text{Aut}_{{\Cal O}_F}(E_{\Cal O})$; this is a proalgebraic  
group.  We put  $\widetilde{\Cal A}(F, E_{\Cal O}) = \widetilde{{\Cal A}E}_F$,  
see 2.1.2. The canonical embedding  $s_{E_{\Cal O}}$:  Lie Aut $(F,E_{\Cal O})  
\to \widetilde{{\Cal A}E}_F$  defines the Harish-Chandra pair   
$(\widetilde{{\Cal A}E}_F, \text{Aut}(F,E_{\Cal O}))$.  This defines our  
centered groupoid  $(\widetilde{\Cal A}, {\Cal V}{\Cal V})$.   
 
Let  $S$  be a scheme.  An $S$-object of ${\Cal V}{\Cal V}$  is a pair  
$(Y_S, E_{Y_S})$, where  $Y_S$  is an $S$-object of   ${\Cal V}$ (see 3.4.2)  
and  $E_{Y_S}$  is a locally free ${\Cal O}_{Y_S}$-module of finite rank. 
 
Assume that  $S$  is smooth.  Let  $\pi: C_S \to S$  be a famly of smooth  
projective curves,  $a: S \to C_S$  a section, and let $E$  be a vector bundle 
on  $C_S$.  These define an $S$-localization data  $\psi(C_S, E, a)$.  Namely,  
the corresponding  $S$-object of  ${\Cal V}{\Cal V}$  is the completion of  $C_S,E$ 
 along $a$.  The Lie algebroid $N$ consists of triples  $(\tau, \tau_U,  
\tau_{E_U})$, where  $\tau \in {\Cal T}_S$,  $\tau_U$  is a lifting of  $\tau$ 
to  $U = C_S \setminus a(S)$, and  $\tau_{E_U}$  is an action of $\tau_{E_U}$ on 
$E_U$.  The morphisms  $\varphi, \widetilde{\varphi}_{(0)}$, appear precisely  
as in 3.4.3 from 2.3.4. 
 
As above, this localization data gives rise to a localization functor.  The  
versions  3.4.5-3.4.7 are immediate. 
 
\medpagebreak 
 
3.4.9\ Let us consider now the spinor or ``fermionic" version.  The corresponding 
centered Harish-Chandra groupoid  $(\widetilde{{\Cal O}{\Cal A}}, {\Cal O}{\Cal V})$ 
is defined as follows. Its objects are triples   $Q = (F, W_{\Cal O}, (\ ,\ ))$, where  $F$  is a local field, $W$  is a free ${\Cal O}_F$-module of finite rank,  and (\ ,\ ):  $W_{\Cal O} \times W_{\Cal O} \to \omega_{{\Cal O}_F}$  is a  
symmetric bilinear form with values in 1-forms of  ${\Cal O}_F$.  We assume that (\ ,\ )  is maximally non-degenerate, i.e., the cokernel of the corresponding map 
$W_{\Cal O} \to W_{\Cal O}^0 = \text{Hom}_{{\Cal O}_F}(W_{\Cal O}, 
\omega_{{\Cal O}_F})$  is either trivial (such $Q$ is called even) or a 1-dimensional 
${\Bbb C}$-vector space (such  $Q$  is called odd).  The morphisms in   
${\Cal O}{\Cal V}$  are the obvious ones.  For  $Q$  as above, put  $W_F =  
F \otimes W_{\Cal O}$; our (\ ,\ ) extends  to non-degenerate form $(\ ,\ ):  
W_F \times W_F \to \omega_F$.  Note that our condition means that $W_{\Cal O}$   
is a maximal isotropic lattice in  $W_F$.  We may consider $W_F$  as a Tate's  
${\Bbb C}$-vector space with form $(\ ,\ )_\bullet = \text{Res}(\ ,\ )$ (see 2.4.3);  
then  $W_{\Cal O}$  is also a maximal isotropic $(\ ,\ )_\bullet$-lattice so  
$Q$ is even iff $W_F$  is even-dimensional, see 1.4.1.  We put   
$\widetilde{{\Cal O}{\Cal A}}(Q) = \widetilde{{\Cal O}{\Cal A}W}_F$  (see 2.4.1). 
The Lie algebra  Lie Aut $Q \subset {\Cal O} {\Cal A} W_F$  preserves  $W_{\Cal O}$, 
hence we have a canonical embedding  $s_{W_{\Cal O}}: \text{Lie\ Aut}\ Q  
\hookrightarrow \widetilde{{\Cal O}{\Cal A}}(Q)$.  This defines the Harish-Chandra 
pair  $(\widetilde{{\Cal O}{\Cal A}}(Q), \text{Aut}\ Q)$, and we get the groupoid 
$(\widetilde{{\Cal O}{\Cal A}}, {\Cal O}{\Cal V})$. 
 
\medpagebreak 
 
\demo{Remark}  Clearly  $Q$  is even (resp. odd) iff  $(W_F, (\ ,\ )_\bullet)$   
is even (resp. odd) dimensional, see 1.4.1.  The two objects of $Q$ are isomorphic 
iff the  $W$'s have the same rank and parity.   
\enddemo 
 
\medpagebreak 
 
Now let  $S$  be a smooth scheme.  Let  $\pi: C_S \to S$  be a family of smooth 
projective curves,  $a: S \to C_S$   a section,  $W$   a vector bundle on  
$C_S$, and $(\ ,\ ): W \times W \to \omega_{C_S/S}$  a symmetric bilinear  
pairing.  Assume that the cokernel of the corresponding map  $W \to W^0 =  
\text{Hom}(W, \omega_{C_S/S})$  is either trivial or supported on $a(S)$ and is  an ${\Cal O}_S$-module of rank 1.  This collection  $(C_S, a, W, (\ ,\ ))$  
defines an $S$-localization data $\psi$  for  $(\widetilde{{\Cal O}{\Cal A}},  
{\Cal O}{\Cal V})$  in a way similar to 3.4.3, 3.4.8.  Namely, the formal  
completion of $W$ along $a$ defines an $S$-object of ${\Cal O}{\Cal V}$.  The  
Lie algebroid $N$ consists of triples  $(\tau, \tau_U, \tau_{W_U})$, where 
$\tau \in {\Cal T}_S$, $\tau_U \in {\Cal T}_U$  is a lifting of $\tau$  to  
$U = C_S\setminus a(S)$, and  $\tau_{W_U}$  is an action of  $\tau_U$  on  
$W_U$  that preserves  $(\ ,\ )$.  The corresponding map $\varphi$  is obvious,  
and  $\widetilde{\varphi}_{(0)}$  comes from 2.4.4.   
 
\medpagebreak 
 
One has immediate variants of this construction for the case of several points  
and points with 1-jet of a parameter (see 3.4.6, 3.4.7). 
 
\medpagebreak 
 
\noindent 
3.4.10\ Note that we have a canonical morphism  $r: (\widetilde{\Cal A}^._{-1},  {\Cal V}{\Cal V}) \to (\widetilde{{\Cal O}{\Cal A}}, {\Cal O}{\Cal V})$  of  
centered Harish-Chandra groupoids.  It assigns to  $(F, E_{\Cal O}) \in  
{\Cal V}{\Cal V}$  the triple  $(F, E_{\Cal O} \oplus E^0_{\Cal O}, (\ ,\ ))$ 
where  (\  ,\ )  is the obvious pairing.  The morphism $\widetilde{\Cal A}E_F  
\to \widetilde{{\Cal O}{\Cal A}}(E_F \oplus E^0_F)$  was defined in 2.4.2.  
Now for a scheme $S$ and a collection  $(C_S, a, E)$  from 3.4.8  we have  $(C_S, a, E \oplus E^0, (\ ,\ ))$  from 3.4.9.  We have an obvious  
$r$-morphism of corresponding localization data  $r^\#: \psi_c(C_S, a,E)  
\to \psi_{-c}(C_S, a, E \oplus E^0,(\ ,\ ))$  (see 2.4), hence the isomorphism 
$r_D: D_{\psi_c(C_S, a,E)}\ 
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
D_{\psi_c(C_S, a,E \oplus E^0,(\ ,\ ))}$. 
\vskip .2in 


\subheading{3.5\ Fermions and determinant bundles}\ In this section the rings of 
twisted differential operators  $D_\psi$  that appeared in 3.4 will be  
canonically identified with the rings  ${\Cal D}_L$  for some natural line  
bundles $L$  (see 3.2.8).  Equivalently, we will construct a $D_\psi$-module $L$ 
which is a line bundle (as ${\Cal O}$-module).  This will be done by means of 
Clifford modules. 
 
\medpagebreak 
 
\noindent 
3.5.1\ Let us start with the situation in 3.4.9.  For $Q = (F, W_{\Cal O},(\ ,\ )) \in  
{\Cal O}{\Cal V}$  denote by  $M_Q$  the Clifford module (for Clifford algebra  
$C\ell(Q) = C\ell$ $(W_F. (\ ,\ )_\bullet )$, see 1.4)  generated by a single  
fixed vector $v$ with the only relation  $W_{\Cal O}v = 0$.  If $Q$ is even, then  
$M_Q$  is irreducible; if $Q$ is odd, then $M_Q$  is the sum of two non-isomorphic  
irreducible modules.  Note that  $M_Q$  carries a canonical  Aut $Q$-action (the 
only one) that leaves $v$ invariant.  By 2.4.3  $M_Q$  is an 
$\widetilde{{\Cal O}{\Cal A}} W_F = \widetilde{{\Cal O}{\Cal A}}_Q$-module.  Clearly  
these actions are compatible, hence  $M_Q$  is an $(\widetilde{{\Cal O}{\Cal A}}_Q,  
\text{Aut}\ Q)$-module.  This way we get the $(\widetilde{{\Cal O}{\Cal A}}, 
{\Cal O}{\Cal V})$-module $M$. 
 
Let $S$  be a smooth scheme, and  $(C_{S}, a, W,(\ ,\ ))$   the geometric  
data from 3.4.9 that defines the corresponding  $S$-localization data $\psi$  
for  $({\Cal O}{\Cal V}, \widetilde{{\Cal O}{\Cal A}})$.  Let  $Q_S =  
(F_S, W_{{\Cal O}_{F_S}}, (\ ,\ ))$  be the corresponding $S$-object of  
${\Cal O}{\Cal V}$ (= the completion of our data along $a$), and  $M_{Q_S}$   
be the corresponding  ${\Cal O}_S$-module with $\widetilde{{\Cal O}{\Cal A}}_{Q_S}$-
action. 
Certainly, $M_{Q_S}$  is a Clifford module for the ${\Cal O}_S$-Clifford  
algebra  $C\ell(W_{F_S}, (\ ,\ )_\bullet)$  generated by the section  $v$ 
with the only relation  $W_{{\Cal O}_{F_S}} v = 0$.  Note that  
$\pi_*W_U = \pi\big\vert_{U^*} (W\big\vert_U)$  is an $S$-family of maximal 
isotropic colattices in  $W_{F_S}$  (see 2.4.5).  Put  $L_\psi =  
M_{Q_S}/\pi_*W_UM_{Q_S}$.  This is a line bundle on  $S$  if  $Q_{S}$  is even  
(which means that $(\ ,\ ): W \times W \to W_{C_S/S}$  is non-degenerate).  
If  $Q_S$  is odd, then  $L_\psi$  is a two-dimensional vector bundle which  
splits canonically as a sum of two line bundles on the 2-sheeted covering of $S$  
that corresponds to a choice of  $\gamma \in  
W^\perp_{{\Cal O}_{F_S}}\big/ W_{{\Cal O}_{F_S}}$  with $(\gamma,\gamma)_\bullet  
= 1$. 
 
\medpagebreak 
 
 
\proclaim{3.5.2\ Lemma}  $L_\psi$  is naturally a $D_\psi$-module:  it is a  
$D_\psi$-module quotient of  $\Delta_\psi M$ . 
\endproclaim 
	      
\medpagebreak 
 
\demo{Proof} Consider the action of Lie algebroid  ${\Cal A}  
\widetilde{{\Cal O}{\Cal A}}_{Q_S} N$  (see 3.3.4) on  $M_{Q_S}$.  Since for 
$(a,n) \in {\Cal A}\widetilde{{\Cal O}{\Cal A}}_{Q_S} N = {\Cal A} \widetilde{{\Cal O}{\Cal 
A}}_{Q_S}  
{\lower 1.5ex\hbox{$\buildrel{\times}\over{{{\Cal A}{\Cal O}{\Cal A}_{Q_S}}}$}}  
\pi_* {\Cal O}{\Cal A}W_U$  and   
$w \in \pi_* W_U$  one has  $[(a,n),w] = n(w)$  (as operators on  $M_{Q_S}$),  
we see that this action "quotients down" to  $L_\psi$.  It remains to show that  $L_\psi$  is actually an $A_\psi$-module.  We need to prove that the  
${\Cal O}_S$-Lie subalgebra  $s(N_{(0)}) \subset \widetilde{{\Cal A}{\Cal O}{\Cal 
A}}_{Q_S}N$ 
acts trivially on  $L_\psi$.  Note that  $s(N_{(0)}) = \pi_* {\Cal O}{\Cal A}W_{U/S}$ 
acts on  $L_\psi$ ${\Cal O}_S$-linearly, hence it suffices to consider the case  when $S $ is a point.  Then  $N_{(0)} = {\Cal O}{\Cal A} W_U$  is an extension of ${\Cal T}_U$ 
by the orthogonal Lie algebra ${\Cal O} W_U$.  Since both ${\Cal O} W_U$ and  
${\Cal T}_U$  are perfect ${\Bbb C}$-Lie algebras, we see that  $N_{(0)}$  is  
perfect, hence every 1-dimensional representation of  $N_{(0)}$  is trivial.  
Since  $L_\psi$  is either 1-dimensional or a sum of two 1-dimensional  
$N_{(0)}$-invariant subspaces, we are done.\hfill $\square$ 
\enddemo 
 
\medpagebreak 
 
Actually we have proven that  $L_\psi$  is a quotient of the $D_\psi$-module  
$\Delta_\psi(M)$.  Certainly, 3.5.2 implies 
 
\proclaim{3.5.3\ Proposition}  One has a canonical    isomorphism of twisted  
differential operators algebras  $D_\psi = D_{L_\psi}$  if  $Q_S$  is even, and   
$D_{\psi_2} = D_{\det L_\psi}$  if  $Q_S$  is odd. \hfill $\square$ 
\endproclaim 
 
\medpagebreak 
 
\demo{3.5.4\ Remarks}  (i)\ According to 1.4.4 the fibers  $L_{\psi_s}$,  
$s \in S$, are canonically identified with  det $H^0(C_s,W_s)$  if  $Q_S$  is   
even, i.e., if     (\ ,\ ) is non degenerate (if  $Q_S$  is odd, one has 
det $L_{\psi_s} = \det^{\otimes 2} H^0(C_s, W_s))$.  Hence the automorphism  
- id$_W$  of our data acts on  $L_\psi$  as $\pm 1$  depending on whether  
dim $H^0(C_s, W_s)$  is even or odd.  This proves the theorem of Mumford that  
the parity of dim does not jump. 
 
\noindent 
(ii)\ Of course we may consider the situation with several points  
$a_1,\ldots ,a_n \in C$.  By a reason similar to 3.4.6 one may see that the  
corresponding line bundle  $L_\psi$ actually does not depend on these points;  
certainly, we may delete only ``even" points where (\ ,\ ) is non-degenerate. 
\hfill $\square$ 
\enddemo
 
\medpagebreak 
 
Now let us consider the situation 3.4.8 of vector symmetries.  By 3.4.10 we have  
a canonical isomorphism $D_{\psi_c(C_S,a,E)} = D_{\psi_{-c}(C_S,a,E \oplus E^0, 
(\ ,\ ))}$.  By 3.5.4(i) the fibers of the line bundle $L_\psi =  
L_{\psi}(C_S,a,E \oplus E^0, 
(\ ,\ ))$  coincide with det $H^0(C_s,E) \otimes \det H^0(C_s,E^0_s) =  
\det H^0(C_s,E)/\det H^1(C_s,E) = \det R\Gamma(C_s,E)$.  It is easy to see that  
$L_\psi = \det R\pi_*E =$ the determinant line bundle of $E$ (about determinant 
line bundles, see e.g. [KM]).  By 3.5.4 (ii) and a version of     3.4.6 for vector  
symmetries we may delete a point $a$ above.  Hence 
 
\proclaim{3.5.5\ Corollary}  One has a canonical isomorphism   
$D_{\psi_c(C_S,E)} = D_{\det^{\otimes -c}R\pi_*E}$.\hfill $\square$ 
\endproclaim 
 
Consider finally the pure Virasoro situation.  We have an obvious embedding 
of Harish-Chandra groupoids  $r: ({\Cal V}, \widetilde{\Cal T}) \to  
({\Cal V}{\Cal V}, \widetilde{\Cal A})$, $F \longmapsto (F, {\Cal O}_F)$,  
$\widetilde{\Cal T} \hookrightarrow \widetilde{{\Cal A}F}$  (see 2.1.3).  If 
$C_S$  is an $S$-family of curves, $a$ is an $S$-point of $C_S$, we have an  
obvious $r$-morphism of localization data  $\psi_{(C_S,a)} \longrightarrow 
\psi_{(C_S,a,{\Cal O}_{C_S})}$  which identifies $D_{\psi_c(C_S,a)}$  with  
$D_{\psi_c(C_S,a,{\Cal O}_{C_S})}$.  Now 3.5.5 implies 
 
\proclaim{3.5.6\ Corollary}  One has a canonical isomorphism   
$D_{\psi_c(C_S)} = D_{\det^{\otimes -c}R\pi_*{\Cal O}_{C_S}}$.\hfill $\square$ 
\endproclaim 
\vskip .2in 
 
\subheading{3.6\ Quadratic degeneration}\  In this section we will describe the  
determinant bundle of a family of curves that degenerates quadratically.  Below 
$S = \text{Spec}\ {\Bbb C}[[q]]$  is a formal disc, $0 \in S$  is the 
special point  $q = 0$, $\eta = \text{Spec}\ {\Bbb C}((q))$  is the generic point. 
 
\medpagebreak 
 
\proclaim{3.6.1\ Lemma}\ There is a canonical 1-1 correspondence between the  
following data (i) and (ii):       
 
\item{(i)}\  A proper $S$-family of curves, $C_S$ such that  $C_\eta$  is smooth and  $C_0$   
has exactly one singular point $a$ which is quadratic, together with formal  
coordinates  $t_1,t_2$  at $a$ such that  $q = t_1t_2$. 
 
\item{(ii)} A proper smooth $S$-family of curves  $C^\vee_S$  together with two  
disjoint points  $a_1, a_2 \in C_S(S)$  and formal coordinates  $t_i$  at  $a_i$. 
\endproclaim 
 
\demo{Proof}  Here is a construction of mutually inverse maps.  Note that,  
according to Grothendieck, we may replace any proper $S$-curve  $B_S$  by the  
corresponding formal scheme  $\widehat{B}_S =$ the completion of  $B_S$   
along  $B_0$.   
 
\noindent 
(i) $\longmapsto$ (ii).  Let  $C_S, t_1,t_2$  be a (i)-data.  The corresponding  
$C^\vee_S, a_i,t_i$  are the following ones.  One has $C^\vee_0 =$ normalization  
of $C_0$, so $t_i$  define formal coordinates at points  $a_1(0), a_2(0) \in  
C^\vee_S$.  To define $C^\vee_S$ as a formal scheme, we have to construct the  
corresponding sheaf  $\widehat{\Cal O}_{C^\vee_S}$  of functions on  
$C^\vee_0$.  We demand that on  $U = C^\vee_S \setminus \{ a_1,a_2\} =  
C_0 \setminus \{ a\}$  our  $\widehat{\Cal O}_{C^\vee_S}$ coincides with  
$\widehat{\Cal O}_{C_S}$.   Note 
that any function  $\varphi \in  \widehat{\Cal O}_{C_S}(V)$, where  $V \subset U$, 
has Laurent series expansions  $\varphi_i(t_i,q) \in {\Bbb C}((t_i))[[q]]$  at  
$a_i(0)$.  We say that  $\varphi$  is regular at  $a_i(0)$  if $\varphi_i(t_i,q)  
\in {\Bbb C}[[t_i,q]]$.  This defines $\widehat{\Cal O}_{C^\vee_S}$.  The points 
$a_i$  are defined by equations $t_i = 0$.   
 
\noindent 
(ii) $\longmapsto$ (i).  Let  $C^\vee_S, a_i,t_i$ be (ii)-data.  The zero fiber  $C_0$  of our curve  $C_S$  is $C^\vee_0$  with points  $a_1,a_2$  glued  
together.  The sheaf $\widehat{\Cal O}_{C_S}$  coincides with  
$\widehat{\Cal O}_{C^\vee_S}$  on  $U = C_0\setminus \{ 0\} =  
C^\vee_0 \setminus \{ a_1,a_2\}$.  For a Zariski open $V \subset U$  a  
function  $\varphi \in \widehat{\Cal O}_{C_S}(V)$ is regular at $a$ if the  
$t_i$-Laurent series expansions  $\varphi_i \in {\Bbb C}((t_i))[[q]]$  of  
$\varphi$ at $a_i$  lie in ${\Bbb C}[[t_1,t_2]] \subset {\Bbb C}((t_i))[[q]]$ 
and $\varphi_1 = \varphi_2 \in {\Bbb C}[[t_1,t_2]]$.  Here the embedding  
${\Bbb C}[[t_1,t_2]] \hookrightarrow  {\Bbb C}((t_1))[[q]]$ is   
$t_1 \longmapsto t_1,t_2 \longmapsto q/t_1$, and the one  
${\Bbb C}[[t_1,t_2]] \hookrightarrow  {\Bbb C}((t_2))[[q]]$  is  
$t_1 \longmapsto q/t_2,\ t_2 \longmapsto t_2$.  This defines  
$\widehat{\Cal O}_{C_S}$. \hfill $\square$ 
\enddemo 
 
\medpagebreak 
 
Below we will say that a vector bundle $E$ on a scheme $X$ is {\it stratified}  
at $x \in X$  if we are given an isomorphism  $E \simeq A \otimes_{\Bbb C}  
{\Cal O}_X$  on a formal neighbourhood of $x$ (here $A$ is a vector space;  $A = E_x$).   
 
\medpagebreak 
 
\proclaim{3.6.2\ Lemma}  Let  $C_S$  and  $C^\vee_S$  be the $S$-curves from 
3.6.1.  There is natural 1-1 correspondence between the data  
 
(i)  A vector bundle $E$ on  $C_S$  together with a stratification of $E$ at $a$. 
 
(ii) A vector bundle $E^\vee$  on  $C^\vee_S$  together with a stratifications  
of $E^\vee$  at $a_1,a_2$ and an isomorphism of fibers   
$E^\vee_{a_1} \simeq E^\vee_{a_2}$. \hfill $\square$ 
\endproclaim 
 
\proclaim{3.6.3\ Proposition}\ Let $(C_S,E), (C^\vee_S,E^\vee)$  be the related  objects from 3.6.1, 3.6.2.  Then there is a canonical stratification of the line bundle ${\Cal L} = \det R\pi_*E/\det R\pi^\vee_* E^\vee$  on $S$. 
\endproclaim 
 
\medpagebreak 
 
\demo{Remark}  Here ``stratification" = ``stratification at 0" = (isomorphism 
${\Cal L} \simeq {\Cal L}_0 \otimes {\Cal O}_S$).  Note that ${\Cal L}_0 =  
\det R\Gamma(C_0,E_0)/\det R\Gamma(C^\vee_0, E^\vee_0)$  is naturally  
isomorphic to  $\det^{-1} E_a$, so 3.6.3 is canonical isomorphism   
$\det R\pi_*^\vee(C^\vee, E^\vee) = \det E_a \det R\pi_*(C,E)$. 
\enddemo 
 
\medpagebreak 
 
\demo{Proof.\ Construction}  Let us compute our determinant bundles.  Below we  
will use notations from the proof of 3.6.1.  Put  $A = E_a = E^\vee_{a_1} =  
E^\vee_{a_2}$.  Our data identifies the formal completion  $E_{\widehat{a}}$  
of $E$  at $a$ with $A \otimes {\Bbb C}[[t_1,t_2]]$, and the formal completion  
of  $E^\vee_{\widehat{a}_i}$  of  $E^\vee$  at  $a_i$  with   
$A \otimes {\Bbb C}[[t_i, q]]$.  The restrictions of $E$ and $E^\vee$  to the  
formal scheme  $\widehat{U} = (U, \widehat{\Cal O}_U)$  coincide;  put  
$P =   H^0(U, E\vert_{\widehat{U}}) = {\dsize\lim_\leftarrow} H^0(U, E/q^nE)$.   
Also put  $V = A \otimes \{ {\Bbb C}((t_1))[[q]] \oplus {\Bbb C}((t_2))[[q]]\}$, 
$V_{+0} = A \otimes \{ {\Bbb C}[[t_1,q]] \oplus {\Bbb C}[[t_2,q]]\}$, 
$V_{+1} = A \otimes \{ {\Bbb C}[[t_1,t_2]]$.   We may compute  $R\pi_*E$, 
$R\pi^\vee_*E^\vee$  by means of ``adelic" complexes for our formal schemes.   
Namely,  $R\pi^\vee_*E^\vee = \text{Cone}(P \oplus V_{+0} \to V)[-1]$,  
$R\pi_*E = \text{Cone}(P \oplus V_{+1} \to V)[-1]$; here the map  $P \to V$   
is minus the Laurent series expansion map, the map  $V_{+1} \to V$  is given by  
formula  $a \otimes t^m_1t^n_2 \longmapsto a \otimes \{ q^nt_1^{m-n} +  
q^m t_2^{n-m}\}$  (see the proof of 3.6.1), and  $V_{+0} \to V$  is the obvious 
embedding.  
\enddemo 
 
Note that  $V$  is a flat complete ${\Bbb C}[[q]]$-module with the obvious Tate  
structure (see 1.4.10), $V_{+0}, V_{+1}$  are lattices in $V$ and $P$ is a  
colattice in $V$.  So to compute our determinants we may use Clifford modules. 
Namely, take  $W = V \oplus V^*$  with the standard form (\ ,\ ); let $M$  be the  
corresponding Clifford module such that $M_0 = M/qM$  is an irreducible Clifford  module for  $(W_0,  (\ ,\ )_0)$.  Then  $L(P) = P \oplus P^\perp$,  
$L(V_{i+}) = V_{i+} \oplus V^\perp_{i+}$  are maximal isotropic colattice and  
lattices respectively.  A  ${\Bbb C}[[q]]$-version of 1.4.9 shows that coinvariants  
$M_{L(P)}$  and invariants  $M^{L(V_{i+})}$  are free ${\Bbb C}[[q]]$-modules  
of rank one, and there are canonical isomorphisms 
$$ 
\det R \pi^\vee_* E^\vee  = M^{L(V_{0+})}/M_{L(P)},  
\det R \pi_* E  = M^{L(V_{1+})}/M_{L(P)} . 
$$ 
Hence  $\det R \pi_* E/\det R\pi_*^\vee E^\vee  = M^{L(V_{1+})}/M^{L(V_{0+})}$.   
In this description of the ratio of determinants all the ``global" data that may  
vary (encoded in $P$) disappeared; we've got the standard ``local" expression  
for it. 
 
It remains to fix an isomorphism  $\gamma : M^{L(V_{0+})} \to M^{L(V_{1+})}  
\otimes \det A$; the desired stratification of the ratio of determinants then  
will be  $\gamma(v)/v$  for a non-zero generator $v$  
(clearly it does not depend on $M$).   Let  $a_1,\ldots ,a_\ell$  be a basis of  
$A$.  Consider the vectors  $e^k_{\alpha 1} = a_\alpha \otimes t^k_1, e^k_{\alpha 2}  
= a_\alpha \otimes t_2^k$,\ $k \in {\Bbb Z},\ \alpha = 1,\ldots ,\ell$.  This 
is a basis (in an obvious sense) of $V$; denote by   $e^{k*}_{\alpha i} \in V^*$ 
the dual basis.  The vectors  $\{ e^k_{\alpha i}\}, k \ge 0$, form a basis of  
$V_{0+}$, and the vectors  $f^k_{\alpha 1} := e^k_{\alpha 1} + q^k e^{-k}_{\alpha 2}$, 
$f^k_{\alpha 2} := q^k e^{-k}_{\alpha 1} +  e^{k}_{\alpha 2}, e^0_{\alpha 1} + 
e^0_{\alpha 2},$\ $k \ge 1$, form a basis of  $V_{1+}$.  In a slightly informal 
way our  $\gamma$  could be defined as follows.  A generator of  $M^{L(V_{0+})}$ is an infinite wedge product  ${\dsize\bigwedge \Sb\\ {k\ge 0}\\ \alpha,i\endSb}  e^k_{\alpha i}$, a  
generator of  $M^{L(V_{1+})} \otimes \det A$  is   
${\dsize\bigwedge \Sb\\ k \ge 1 \\ \alpha, i \endSb} f^k_{\alpha i} \wedge  
{\dsize\bigwedge_\alpha} (e^0_{\alpha 1}  + e^0_{\alpha 2}) \otimes  
{\dsize\bigwedge_\alpha} a_\alpha$, and $\gamma$  just identifies these generators. 
To be precise, consider the elements   $\gamma_n =  
{\dsize\prod \Sb 1\le k\le n \\ \alpha\endSb}  
(f^k_{\alpha 1}f^k_{\alpha 2} e^{k*}_{\alpha 2} e^{k*}_{\alpha 1}) \in  
\text{Cliff}(W)$.  These $\gamma_n$  do not depend on a choice of basis  
$\{ a_\alpha\}$  in $A$, and it is easy to see that  $\gamma_\infty = {\dsize\lim_n} \gamma_n  
\in C\ell W$  is correctly defined.  Let  $V_{0++} \subset V_{0+},  
V_{1++} \subset V_{1+}$  be sublattices with bases  $\{ e^k_{\alpha i}\},  
k \ge 1$, and  $\{ f^k_{\alpha i}\}, k \ge 1$, respectively.  It is easy to see 
that  $\gamma_\infty (M^{L(V_{0++})}) = M^{L(V_{1++})}$  (more precisely,   
$\gamma_n(M^{L(V_{0+})}) \equiv M^{L(V_{1+})} \text{mod} q^{n+1}M)$.  Since  
$M^{L(V_{0+})} = {\dsize\bigwedge_{\alpha,i}} e^0_{\alpha i} \cdot  
M^{L(V_{0++})}, M^{L(V_{1+})} =  
{\dsize\bigwedge_\alpha} (e^0_{\alpha 1} + e^0_{\alpha 2}) \cdot M^{L(V_{1++})}$,  
we have  ${\dsize\bigwedge_\alpha} (e^{0*}_1 - e^{0*}_2) \cdot  
\gamma_\infty M^{L(V_{0+})} = M^{L(V_{1+})}.$  Put  
${\dsize\bigwedge_\alpha} (e^{0*}_1 - e^{0*}_2) \cdot  
\gamma_\infty \otimes {\dsize\bigwedge_\alpha} a_\alpha \in C\ell W \otimes  
\det A$.  This $\gamma$ does not depend on a choice of basis $\{ a_\alpha\}$  of  
$A$, and the desired  $M^{L(V_{0+})}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
M^{L(V_{1+})} \otimes \det A$  is just multiplication by $\gamma$.  
\hfill $\square$ 
 
\medpagebreak 
  
\noindent 
3.6.4\ Let  $C^\vee$  be a curve,  $a_1,   a_2 \in C^\vee $, $\ a_1 \ne a_2$,   
a pair of points, and $t_i$  a formal parameter at  $a_i$.  Consider the constant $S$-family  $C^\vee_S := C^\vee \times S$; let $a_i \in C^\vee_S(S), t_i$  be the  
``constant" points and parameters.  According to 3.6.1 these define an  
$S$-curve  $C_S$  with quadratic singularities along zero fiber and smooth  
generic fiber.  Consider the trivial vector bundles  ${\Cal O}_{C_S},  
{\Cal O}_{C^\vee_S}$; they correspond to each other by 3.6.2 correspondence. 
Note that  $\det R\pi^\vee_* {\Cal O}_{C^\vee_S} = \det R\Gamma(C^\vee,{\Cal O}_{C^\vee})  
\otimes {\Cal O}_S$  is obviously stratified, hence 3.6.3 defines the  
stratification of  $\det R\pi_* {\Cal O}_{C_S}$  which is a natural generator 
$\gamma$  of the ${\Bbb C}[[q]]$-module  $\det^{-1}R\Gamma(C^\vee, {\Cal O}_{C^\vee})  
\otimes_{{\Bbb C}[[q]]} \det R\pi_* {\Cal O}_{C_S}$.  Let us compute $\gamma$ in a couple 
of simple situations. 
 
\medpagebreak 
 
\noindent 
3.6.5\ Assume that $C^\vee$  is a disjoint union of two copies of   
${\Bbb P}^1$'s, $C^\vee = {\Bbb P}^1_1 \coprod {\Bbb P}^1_2$, $a_1 \in {\Bbb P}^1_1, 
a_2 \in {\Bbb P}^1_2$  are ``zero" points, $t_i$  are standard parameters at  
$a_i$.  Then the $S$-curve $C_S$  is the compactification of the family of affine  
curves  ${\Bbb A}^2 \to S$, $q = t_2t_2$.  This is a genus 0 curve, hence  
$R\pi_* {\Cal O}_{C_S} = {\Cal O}_S$, so we have a canonical trivialization  
$\alpha$ of $\det R\pi_* {\Cal O}_{C_S}$  of ``global" origin.  In fact, it  
coincides with our $\gamma$.  To see this, note that (in the notations of  
proof of 3.6.3) in our case $P$ is colattice with basis  
$\{ e^k_1,e^k_2\}, k \le 0$, so one has  $P \oplus V_{1++} = V =  
P \oplus V_{0++}$.  The operator $(e^0_1 + e^0_2)\cdot$  identifies   
$M^{L(V_{1++})}$  with  $M^{L(V_{1+})}$, hence  $\det R \pi_* {\Cal O}_{C_S} =  
M^{L(V_{1++})}/M_{L(P)}$.  The ``global" trivialization $\alpha$ comes from  
the isomorphism $M^{L(V_{1++})}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
M_{L(P)}$, $m \longmapsto m\ \text{mod}\ L(P)M$.  The trivialization $\gamma$ 
comes from composition $M^{L(V_{1++})}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
M^{L(V_{0++})}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
M_{L(P)}$  where the first arrow is inverse to multiplication by $\gamma_\infty$  
and the second one is projection  $m \longmapsto m\ \text{mod}\ L(P)M$.  Since 
$f^k_i = e^k_i\ \text{mod}\ P$  for $k \ge 1$, the formula for  $\gamma_\infty$  
shows that this composition coincides with projection   
$M^{L(V_{1++})}\ \to\ M_{L(P)}$, hence $\alpha = \gamma$.   
 
\medpagebreak 
 
\noindent 
3.6.6\ Assume now that  $C^\vee = {\Bbb P}^1, a_1 = 0, a_2 = \infty$  and  
$t_1,t_2$  are standard parameters $t$ and  $t^{-1}$  respectively.  Then the  
curve  $C_S$  coincides with the standard Tate elliptic curve (see, e.g.,  
[DR]), $q$  is a standard parameter on moduli space of elliptic curves at  
$\infty$.  The Tate curve carries a canonical relative 1-form $\nu$  (that  
corresponds to the standard invariant form on $G_m$  via Tate's uniformization). One has  $R^0\pi_* {\Cal O}_{C_S} = {\Cal O}_S$, $R^1\pi_* {\Cal O}_{C_S} =  
(R^0 \pi_* \omega_{C_S})^*$  by Serre duality (here  $\omega_{C_S}$  is  
relative dualizing sheaf), hence $\det R \pi_* {\Cal O}_{C_S} =  
R^0 \pi_* \omega_{C_S}$ and $\nu$  is a canonical   trivialization of  
$\det R \pi_* {\Cal O}_{C_S}$.  Let us calculate $\gamma$.  The colattice $P$  
has basis  $\{ e^k_1 + e^k_2\}, k \in {\Bbb Z}$.  One has ${\Cal O}_S =  
R^0 \pi_* {\Cal O}_{C_S} = {\Cal O}_S(e^0_1 + e^0_2) = P \cap V_{1+}$,  
$R^1 \pi_* {\Cal O}_{C_S} = V/P + V_{1+} = V/P+V_{1++}$.  The relative  
differential  $\nu$  in local coordinates  $t_i$  is   
${\frac{dt_1}{t_1}} = - {\frac{dt_2}{t_2}}$, and the Serre duality morphism is the  
sum of local residues at  $a_i$.  Hence the functional   
$\nu \in (R^1 \pi_* {\Cal O}_{C_S})^* = (V/P + V_{1+})^* \subset V^*$  is  
$e_1^{0*} - e_2^{0*}$.  As above, multiplication by  $e^0_1 + e^0_2$   
identifies  $M^{L(V_{1++})}$  with  $M^{L(V_1)}$, hence  
$\det R \pi_* {\Cal O}_{C_S} = M^{L(V_{1++})}/M_{L(P)}$.  The trivialization  
$\nu$  comes from the isomorphism  $M^{L(V_{1++})} \to M_{L(P)}$,  
$m \longmapsto (e^0_1m)$ mod $L(P)M$.  The trivialization  $\gamma$  comes  
from composition  $M^{L(V_{1++})}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
M^{L(V_{0++})}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
M_{L(P)}$  where the first arrow is inverse to multiplication by $\gamma_\infty$ 
isomorphism  $M^{L(V_{0++})}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
M^{L(V_{1++})}$\   and the second arrow is  $m \longmapsto (e^0_1m)$ mod  
$L(P)M$.  Since  $f^k_1 = (1 - q^k) e^k_1$ mod $P$, $f^k_2 = (1 - q^k)e^k_2$  
mod $P$  we see that  $\gamma = [{\dsize \prod_{k\ge 1}} (1 - q^k)^2]\nu$, or,  
in terms of Dedekind's $\eta$-function  $\eta(q) = q^{1/24} {\dsize\prod_{k\ge 1}} 
(1 - q^k)$, one has  
$$ 
\gamma = q^{-1/12} \eta(q)^2 \nu . 
$$ 
One may reformulate this as follows.  Recall that the line bundle  
$\lambda = \det R \pi_* {\Cal O}_C = \pi_* \omega_C$  on moduli space of  
elliptic curves carries a canonical global integrable connection  $\nabla$  such  
that the discriminant  $\Delta$  is a global horizontal section of  
$\lambda^{\otimes 12}$  (with respect to the corresponding connection on 
$\lambda^{\otimes 12}$). We see that  
our $\gamma$  is a horizontal section of a connection  $\nabla +  
{\frac{1}{12}} {\frac{dq}{q}}$.   
 

 
\newpage 
 
\pageno=41 
 
\heading 
\S4.\ Fusion Categories 
\endheading 
 
\subheading{4.1\ Recollections from symplectic linear algebra}  Let  $V$  be a  
symplectic ${\Bbb R}$-vector space of dimension $2g$ with symplectic form  
$\langle\ ,\ \rangle$.       To $(V, \langle\ ,\ \rangle)$  there corresponds  
a canonical transitive groupoid  ${\Cal T}_V$.  In 1.1-1.3 below we give three  
different constructions of  ${\Cal T}_V$.  Assume first that  $V \ne 0$.   
 
\medpagebreak 
 
\noindent 
4.1.1\ Let  $H = H_V$  be the Siegel upper half plane of $V$.  A point of $H$ is a  
complex Lagrangian subspace  $L \subset V_{{\Bbb C}} := V \otimes {\Bbb C}$  such  
that  $i\langle x, \overline{x}\rangle > 0$  for $x \ne 0 \in L$.  Equivalently, 
one may consider a point of $H$ as a complex structure  $\ell$  on $V$  such that 
the form  $\langle \cdot, i_\ell \cdot\rangle$  is symmetric and positive definite; 
here  $i_\ell \in \text{End}\ V$  is multiplication by  $i \in {\Bbb C}$  with  
respect to $\ell$   (the 1-1 correspondence  $\ell \longleftrightarrow L$  is  
$\ell \longmapsto L_\ell :=\ \text{the}\ i$-eigenspace of  $i_\ell$,  
$L \longmapsto \ell_L  :=$  the complex structure that comes from the isomorphism   
$V {\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
V_{\Bbb C}/L)$.  The space $H$ is a complex variety, and the $L$'s form a rank  
$g$ holomorphic bundle ${\Cal L}$ on $H$.  Put  $\lambda := \det {\Cal L} :$   
this is a  
holomorphic line bundle on $H$.  Denote by  $\widetilde{H}$  the space of  
$\lambda^{\otimes 2}\setminus \{$ zero section $\}$; the projection  
$\widetilde{H} \to H$  is a ${\Bbb C}^*$-fibration.  Let  ${\Cal H}$  be the space 
of  $C^\infty$-sections  $H \to \widetilde{H}$.  One has obvious maps 
$$ 
{\Cal H} \longleftarrow {\Cal H} \times H \longrightarrow \widetilde{H},\quad 
\varphi \leftarrow (\varphi, h) \mapsto \varphi(h).  \tag 4.1.1.1 
$$ 
 
Since $H$ is contractible, these are homotopy equivalences.  Note that for any  
$a \in \widetilde{H}$  the map  $i_a: S^1 \hookrightarrow \widetilde{H}$,  
$i_a(e^{i\theta}) := e^{i\theta}a$, is a homotopy equivalence which defines a  
canonical identification  
$$ 
\pi_1(\widetilde{H}, a) = {\Bbb Z} .  \tag 4.1.1.2 
$$ 
 
For a topological space $X$ let  ${\Cal T}(X)$  be the fundamental groupoid of 
$X$:  its objects are points of  $X$, and its morphisms are homotopy classes of paths. 
Put  ${\Cal T}'_V := {\Cal T}(\widetilde{H})$.     
 
\medpagebreak 
 
\noindent 
4.1.2\ Denote by  $\Lambda = \Lambda_V$  the grassmannian of real non-oriented  
Lagrangian subspaces of $V$; the planes form a canonical Lagrangian sub-bundle  
${\Cal L}_{\Bbb R}$ of $V_\Lambda := V \times \Lambda$.  Put $\lambda_{\Bbb R} := \det 
{\Cal L}_{\Bbb R}$:  this is 
a real line sub-bundle of  $\Lambda^gV_\Lambda$.  Let  $\Lambda'$  be the space  
$\lambda_{\Bbb R}\setminus \{\text{zero\ section}\}/\pm 1$:  the map 
$x \longmapsto  x^2$  identifies  $\Lambda'$  with the ``positive ray" of   
$\lambda^{\otimes 2}_{\Bbb R}$.  The obvious projection  $\Lambda' \longrightarrow  
\Lambda$  is an ${\Bbb R}^*_+   $-torsor, hence a homotopy equivalence.  One has a  
canonical map  
$$ 
v: \Lambda' \longrightarrow {\Cal H}  \tag 4.1.2.1 
$$ 
defined by the formula  $v(x^2) (h) = \lambda^2$, where  $\lambda \in \det L_h  
\subset \wedge^gV_{\Bbb C}$  is the unique vector such that  vol$(x \wedge 
\lambda) = 1$  (here vol$ = {\frac{\langle\ ,\ \rangle}{g!}} \in \Lambda^{2g}V^*$   
is the canonical volume).  The map  $v$  induces an isomorphism of fundamental groupoids. 
Put  ${\Cal T}''_V: = {\Cal T}(\Lambda)$.  According to (1.1.1), 1.2.1) we have 
a canonical equivalence of groupoids 
$$ 
\alpha : {\Cal T}''_V {\buildrel \sim\over\longrightarrow}\ {\Cal T}'_V.  \tag 4.1.2.2 
$$ 
 
\noindent 
4.1.3\ Here is the third construction of   ${\Cal T}_V$.  For 3 Lagrangian planes  
one defines, according to Kashiwara, their Maslov index  $\tau(L_1,L_2,L_3)$  as  
the signature of the quadratic form $B$ on $L_1 \oplus L_2 \oplus L_3$  given by  
the formula 
$B(x_1,x_2,x_3) = \langle x_1,x_2\rangle + \langle x_2,x_3\rangle + \langle x_3,  
x_1\rangle$  (see [LV] (\quad )).  Let  ${\Cal T}_V'''$  be the following groupoid. 
Its set of objects is $\Lambda$.  For  $L_1,L_2 \in \Lambda$ we put   
Hom$_{{\Cal T}'''_V}(L_1,L_2) = {\Bbb Z}$, and   the composition of morphisms  
$L_1 {\buildrel n\over\longrightarrow} L_2 {\buildrel m\over\longrightarrow} L_3$  
is given by the formula  $m \circ n := m + n + \tau(L_1,L_2,L_3)$.  Since $\tau$  
satisfies a cocycle formula [LV] (\quad ), the composition is associative. 
 
Let us define a canonical isomorphism 
$$ 
\beta: {\Cal T}'''_V {\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
{\Cal T}''_V   \tag 4.1.3.1 
$$ 
which is the identity on objects.  To construct $\beta$ we need to choose for each  
pair  $L_1,L_2 \in \Lambda$  a canonical path  $\gamma_{L_1,L_2} \in  
\text{Hom}_{{\Cal T}''_V}(L_2,L_1)$  such that   
$$ 
\gamma_{L_3L_2} \circ \gamma_{L_2L_1}  = \gamma_{L_3L_1} + \tau(L_1,L_2,L_3). 
\tag 4.1.3.2 
$$ 
Then one defines $\beta$ by the formula  $\beta(n) = n + \gamma_{L_1,L_2}$  for  
$n \in \text{Hom}_{{\Cal T}_V{'''}}(L_2,L_1) = {\Bbb Z}$ (recall that  
$\text{Hom}_{{\Cal T}''_V}(L_2,L_1)$ is a ${\Bbb Z}$-torsor by 1.1.2). 
 
To define  $\gamma_{L_1L_2}$  consider the subset  $U_{L_1L_2} \subset \Lambda$ 
that consists of $L$'s such that  $L_1 + L_2 \supset L \supset L_1 \cap L_2 = L \cap  
L_1 = L \cap L_2$.  A plane  $L \in U_{L_1,L_2}$  defines a quadratic form   
$\varphi_L$  on  $L_1/L_1 \cap L_2$  by the formula  $\varphi_L(a) = \langle b,a\rangle$ 
where  $b \in L_2$  is a vector such that  $b+a \in L$.  In this way one gets a  
1-1 correspondence between  $U_{L_1L_2}$  and the set of all non-degenerate forms 
on $L_1/L_1 \cap L_2$.  Let  $U^+_{L_1L_2} \subset U_{L_1L_2}$  be the subspace  
that corresponds to positive-definite forms, so  $U^+_{L_1L_2}$  is contractible. 
Now  $\gamma_{L_1,L_2}$  is the unique homotopy path from  $L_2$ to $L_1$  which,  
apart from its ends,  lies in  $U^+_{L_1L_2}$.  One verifies (4.1.3.2)  immediately. 
 
\medpagebreak 
 
\noindent 
4.1.4\ Below we will denote by ${\Cal T}_V$ either of the groupoids  
${\Cal T}_V', {\Cal T}_V'',{\Cal T}_V'''$  identified via (4.1.2.2), (4.1.3.1). 
In case $V = 0$, the groupoid ${\Cal T}_V$, by definition, has a single object $0$  
with End 0 = ${\Bbb Z}$.  For any $V$ and  $y \in {\Cal T}_V$ we will denote by  
$\gamma_0$ the generator  $1 \in {\Bbb Z} = \text{Aut}\ y$. 
 
\medpagebreak 
 
\noindent 
4.1.5\  The groupoid  ${\Cal T}_V$  has the following functorial properties.  Let 
$V$  be a symplectic space,  $N \subset V$   a vector subspace such that  
$\langle\quad \rangle\vert_N = 0$, and  let $N^\perp$  be the $\langle\quad \rangle$-orthogonal  
complement to $N$.  Then  $N^\perp/N$  has an obvious symplectic structure.   
Since the  
pre-image of a Lagrangian plane in  $N^\perp/N$  is a Lagrangian plane in $V$,  
we have an embedding $\Lambda_{N^\perp/N} \hookrightarrow \Lambda_V$, which defines a  
canonical equivalence of groupoids  ${\Cal T}''_{N^\perp/N}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
{\Cal T}_V$. 
 
\medpagebreak 
 
\noindent 
4.1.6\ Now let  $V_1,V_2$    be symplectic spaces.  One has an obvious map\  
$\Lambda_{V_1} \times \Lambda_{V_2} \longrightarrow \Lambda_{V_1 \oplus V_2}$, 
$(L_1,L_2) \longmapsto L_1 \oplus L_2$, and a similar map  $\widetilde{H}_{V_1}   
\times \widetilde{H}_{V_2}  \longrightarrow \widetilde{H}_{V_1 \oplus V_2}$, which  
comes from multiplication  $\det^{\otimes 2}L_1 \times \det^{\otimes 2}L_2  
\longrightarrow \det^{\otimes 2}L_1 \otimes \det^{\otimes 2}L_2 =  
\det^{\otimes 2}(L_1 \oplus L_2)$.  These define morphisms between corresponding  
fundamental groupoids, compatible with the canonical equivalences (4.1.2.2).  Hence 
we have a canonical morphism  ${\Cal T}_{V_1} \times {\Cal T}_{V_2}  \longrightarrow 
{\Cal T}_{V_1 \oplus V_2}$. 
 
\medpagebreak 
 
\subheading{4.2\ The Teichm\"uller groupoid}  This groupoid appears in two equivalent versions: a  
``combinatorial" or ``topological" version, and a ``holomorphic" version. 
 
\noindent 
4.2.1\ An object of the ``topological"  Teichm\"uller groupoid $Teich'$ is an oriented  
surface $S$ (possibly non-connected and with   boundary) together with a set of  
points  $P_S = \{ x_\alpha\}$ of the boundary  $\partial S$  such that each connected  
component of $\partial S$  contains exactly one  $x_\alpha$  (we will denote this 
component  $\partial S_{x_\alpha})$.  The morphisms are isotopy classes of  
diffeomorphisms.   
 
Let us define an ``enhanced" groupoid  $\widetilde{Teich}'$.  For a surface $S$  
denote by  $H(S)$  the image of the canonical map  $H^1_c(S,{\Bbb R}) \longrightarrow  
H^1(S,{\Bbb R})$  (which is the same as cohomology of a smooth compactification  
of $S$).  An orientation of $S$ defines a symplectic structure on  $H(S)$   
(intersection pairing).  Now an object of  $\widetilde{Teich}'$  is a triple  
$(S, P_S,y)$,  where  $(S, P_S) \in Teich'$  and  $y \in {\Cal T}_{H(S)}$.  A  
morphism  $(S, P_S,y) \longrightarrow (S',P_{S'},y')$  is a pair $(\varphi,\gamma)$, 
where  $\varphi: (S,P_S) \longrightarrow (S',P_{S'})$  is a morphism in  
$Teich'$, and $\gamma: \varphi_*(y) \longrightarrow y'$  is a morphism in  
${\Cal T}_{H(S')}$; the composition of morphisms is obvious.   
 
The projection  $\widetilde{Teich}' \to Teich'$, $(S, P_S, y) \longmapsto (S,P_S)$,  
is surjective.  For any  $(S, P_S,y) \in \widetilde{Teich}'$ the group  
Aut$_{\widetilde{Teich}'}(S, P_S,y)$  is a central extension of   
Aut$_{Teich'}(S,P_S)$  by  ${\Bbb Z}(= \text{Aut}_{{\Cal T}_{H(S)}}(y))$.  So we 
 may say that  $\widetilde{Teich}'$  is a central extension of $Teich'$  by  
${\Bbb Z}$.  We will denote the generator of this ${\Bbb Z}$  by $\gamma_0$. 
 
Consider the functor  $Teich' \longrightarrow Sets$, $(S, P_S) \longmapsto P_S =$ 
set of boundary components of $S$.  Clearly  $Teich'$  is a fibered category  
over the groupoid of finite sets.  For a finite set $A$ denote by $Teich'_A$  the  
fiber over $A$ (the objects of this groupoid are pairs  $((S,P_S), \nu)$, where  
$(S,P_S) \in Teich'$, and $\nu: P_S\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\  A$  is a  
bijection).  For a bijection $f: A\ {\lower 1ex\hbox{$\overrightarrow{\sim}\vrule  
height 1ex width 0pt$}}\  B$, $X \in Teich'_A$, $Y \in Teich'_B$  we will denote  
by  Hom$_f(X,Y)$  the set of $f$-morphisms (i.e., the ones that induce $f$ on the  
sets of boundary components).  We put  Aut$^0(S,P_S) = \text{Aut}_{id_{P_S}}(S,P_S)$. 
We will use the same notations for $\widetilde{Teich}'$. 
 
For  $(S,P_S) \in Teich'$  and  $x_\alpha \in P_S$  we denote by  
$d_{x_\alpha} \in \text{Aut}^0(S,P_S)$  the Dehn twist around $\partial S_{x_\alpha}$. 
Since  $d_{x_\alpha}$  acts as the identity on  $H(S)$  it lifts to the element 
$(d_{x_\alpha}, id_y) \in \text{Aut}^0_{\widetilde{Teich}'}(S,P_S,y)$, which we will also 
denote by  
$d_{x_\alpha}$.  These  $d_{x_\alpha}$  lie in the center.  In  
particular, we have a canonical morphism  ${\Bbb Z}^{P_S} \longrightarrow  
\text{Aut}^0(S,P_S)$, $(n_{x_\alpha}) \longmapsto \prod d_{x_\alpha}^{n_{x_\alpha}}$; 
${\Bbb Z} \times {\Bbb Z}^{P_S} \longrightarrow \text{Aut}^0(S,P_S,y),\ \   
(n_y, n_{x_\alpha}) \longmapsto \gamma_0^{n_y} \times \prod d_{x_\alpha}^{n_{x_\alpha}}$. 
 
 
\medpagebreak 
 
\noindent 
4.2.2\ Here is a ``holomorphic" definition of the Teichm\"uller groupoid.  An object 
of  $Teich''$  is a complex curve $C$ (smooth, projective, possibly reducible)  
together with a finite set of points  $P_C = \{ y_\alpha\} \subset C$  equipped 
with  non-zero co-tangent vectors  $\nu_\alpha \in \Omega^1_{C,y_\alpha}$.  The morphisms 
are 1-parameter  $C^\infty$-class families of such objects connecting two given  
ones, these families being considered up to homotopy.  In other words, $Teich''$   
is the Poincar\'e groupoid of the modular stack ${\Cal M}$  of the above  
structures.  In the same way, 
$\widetilde{Teich}''$  is the Poincar\'e groupoid of the modular stack  
$\widetilde{\Cal M}$  of the data $(C, y_\alpha, \nu_\alpha, y)$, where   
$(C, y_\alpha,\nu_\alpha) \in {\Cal M}$, and  $y \in  
\det^{\otimes 2}(H^0(C, \Omega^1_C))\setminus \{ 0\}$.  Clearly, the second modular stack  
is a ${\Bbb C}^*$-fibration over the first one, hence  $\widetilde{Teich}''$  is 
a ${\Bbb Z}(= \pi_1({\Bbb C}^*))$-extension of $Teich''$.   
 
\medpagebreak 
 
\noindent 
4.2.3\ The groupoids $Teich'$  and $Teich''$,  are canonically equivalent,  
as are $\widetilde{Teich}'$  and $\widetilde{Teich}''$.  To define this  
equivalence,  take  $(S,P_S) \in Teich'$.  Consider the data  
$(\mu; \{ r_{\alpha}\})$, where $\mu$ is a complex structure on $S$, and   
$r_\alpha: S^1 = \{ z \in {\Bbb C}:   
|z| = 1\}\ {\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
\partial S_{x_\alpha}$  is a parametrization such that  $r_\alpha(1) = x_\alpha$ 
and $r_\alpha$  extends $\mu$-holomorphically to the ring  $\{ z \in {\Bbb C}:  
1 \le |z| \le 1 + \epsilon\}$.  We may glue a collection of unit discs  
$D_\alpha = \{ z \in {\Bbb C}: |z| \le 1\}$  (with their standard complex  
structure) to $S$ using $r_\alpha$.  Denote the corresponding complex curve  
$C = C(S,P_S; (\mu, r_\alpha))$.  It is equipped with the set of points   
$y_\alpha = 0 \in D_\alpha$, and the cotangent vectors  $\nu_\alpha = dz_0 \in  
\Omega^1_{C,O}$.  Hence  $C(   S,P_S; (\mu, r_\alpha)) \in Teich''$.  It is easy  
to see that for given  $(S,P_S)$ the data  $(\mu; \{ r_\alpha\})$  form a  
 contractible space.  So $(S,P_S) \in Teich'$  defines a canonical ``homotopy 
point" in $Teich''$.  In this way we get a morphism of groupoids  
$Teich' \longrightarrow Teich''$  which is an equivalence of categories. 
 
To lift this equivalence to $\widetilde{Teich}' \longrightarrow \widetilde{Teich}''$, 
note that  $H(S) = H^1(C, {\Bbb R})$.  The complex structure on $C$ defines the  
Hodge subspace  $H^0(C, \Omega^1_C) \subset H(S)_{\Bbb C}$, which is a point  
$h_C$  on  the corresponding Siegel half plane (see 4.1.1).  Now let us interpret  
${\Cal T}_{H(S)}$  as a fundamental groupoid of the space denoted by  ${\Cal H}$ 
in (4.1.1.1).  For  $y \in {\Cal T}_{H(S)}$  put  $y_C := y(h_C) \in  
\det^{\otimes 2}(H^0(C, \Omega^1_C))\setminus \{ 0\}$.  Our equivalence   
$\widetilde{Teich}' \longrightarrow \widetilde{Teich}''$  is given by the formula  
$(S, P_S,y) \longmapsto  (C, y_\alpha, \nu_\alpha, y_C)$.   
 
\medpagebreak 
 
\noindent 
4.2.4\  The above equivalence transforms  $\gamma_y$  to the loop $\theta \longmapsto  
(C, y_\alpha, \nu_\alpha, e^{i\theta} y)$, and transforms the Dehn twist $d_{x_\beta}$ 
to the loop  $\theta \longmapsto (C, y_\alpha, e^{i\theta} \delta^\alpha_\beta \nu_\alpha, y)$.  
 
\bigpagebreak 
 
\subheading{4.3\ Operations in Teich}  We will need the following ones: 
 
\noindent 
(i)\ One has a functor ``{\it disjoint union}" $\coprod: Teich \times Teich \to  
Teich$.  According to 1.1.6 it lifts in a canonical way to a functor  
$\coprod: \widetilde{Teich} \times \widetilde{Teich} \to \widetilde{Teich}$.   
Clearly  $Teich, \widetilde{Teich}$  are strictly commutative monoidal categories, 
and the projection  $Teich \to Sets$, $(S,P_S) \longmapsto P_S$, commutes with  
$\coprod$.   
 
\noindent 
(ii) {\it Deleting of a point}.  For a finite set $A$ and $\alpha \in A$  one has a  
canonical functor  $del_\alpha: Teich_A \to Teich_{A\setminus \{ \alpha \}}, 
\widetilde{Teich}_A \longrightarrow \widetilde{Teich}_{A\setminus \{ \alpha\}}$. 
In ``holomorphic" language (4.2.2) this functor just deletes $y_\alpha, \nu_\alpha$. 
In ``topological" language (4.2.1) one should delete the component  
$\partial S_{\chi_\alpha}$  by glueing a ``cup" to $\partial S_{\chi_\alpha}$. 
 
\medpagebreak 
 
\noindent 
(iii) {\it Sewing.}  Let  $A$  be a finite set, and  $\alpha,\beta \in A$,  
$\alpha \ne \beta$, two elements.  One has a canonical Sewing Functor  
${\Cal S}_{\alpha,\beta}: Teich_A \to Teich_{A\setminus \{ \alpha,\beta\} }$, 
$\widetilde{Teich}_A \to \widetilde{Teich}_{A\setminus \{ \alpha,\beta\} }$. 
Let us define  ${\Cal S}_{\alpha,\beta}$  in combinatorial language first.  For a  
surface  $(S,A) \in Teich'$  choose a diffeomorphism $\varphi:  
\partial S_{x_\alpha}  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
\partial S_{x_\beta}$, $\varphi(x_\alpha) = x_\beta$, reversing orientations. 
Our ${\Cal S}_{\alpha,\beta}(S,A) \in Teich'_{A\setminus \{ \alpha,\beta\} }$  is 
$S$  with two boundary components identified by means of $\varphi$.  Since the 
$\varphi$'s form a contractible space, this surface does not depend on the choice  
of $\varphi$.  Note that either  $H(S) = H({\Cal S}_{\alpha,\beta}(S,A))$  (if  
$\alpha$  and $\beta$  lie in different connected components of $S$), or $H(S)$ 
coincides with a subquotient of $H({\Cal S}_{\alpha,\beta}(S,A))$  in a manner  
described in 4.1.5.  In any case one has a canonical equivalence 
${\Cal T}_{H(S)}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\    
{\Cal T}_{H({\Cal S}_{\alpha,\beta}(S,A))}$.  This defines ${\Cal S}_{\alpha,\beta}: 
\widetilde{Teich}'_A \to \widetilde{Teich}'_{A\setminus \{ \alpha,\beta\} }$. 
 
\medpagebreak 
 
\noindent 
4.3.1\ To define   ${\Cal S}_{\alpha,\beta}$  in holomorphic language, take  
$(C,y_\gamma, \nu_\gamma) \in Teich''_A$.  Consider a curve $C_{\alpha,\beta}$   
with a single quadratic singularity obtained from  $C$  by  ``clutching"  
$y_\alpha$ and $y_\beta$ together.  One knows that curves with a single quadratic  
singularity form a smooth part of the divisor of singular curves in the modular  
stack  $\overline{\Cal M}_{A\setminus \{ \alpha,\beta\}}$ of curves with at most 
quadratic singularities.  The fiber of the normal bundle $N$ to this divisor at 
$C_{\alpha,\beta}$  is canonically identified with  $T_{C,y_\alpha} \otimes  
T_{C,y_\beta}$.  Hence  $\nu_\alpha^{-1} \cdot \nu_\beta^{-1}$  is a non-zero  
vector of this normal bundle.  It defines a ``point at infinity" of the modular stack 
${\Cal M}_{A\setminus \{ \alpha,\beta\}}$  of smooth curves (for a detailed  
account of ``points at infinity" see [D]), which is a correctly defined (up  
to unique canonical isomorphism) object  ${\Cal S}_{\alpha,\beta}(C,y_{\gamma},  
\nu_\gamma) \in Teich''_{A\setminus \{ \alpha,\beta\}}$.  To lift   
${\Cal S}_{\alpha,\beta}$  to a functor between $\widetilde{Teich}''$'s, notice  
that the line bundle $\lambda$ over  ${\Cal M}$  with fibers  
$\lambda_C := \det H^0(C, \Omega^1_C)$  extends canonically to a line bundle  
$\lambda$ over  $\overline{\Cal M}$:  if $C'$  has quadratic singularities,  
one has  $\lambda_{C'} := \det H^0(C, \omega_{C'})$, where  $\omega_{C'}$  is the  
dualizing sheaf.  Define the  ${\Bbb C}^*$-bundle  
${\buildrel {\simeq}\over{\Cal M}} \to \overline{\Cal M}$  to be  
$\lambda^{\otimes 2}\setminus \{ \text{zero\ section}\}$.  Recall that for any  
$C' \in \overline{\Cal M}$  one has a canonical isomorphism  
$\lambda^{\otimes 2}_{C'} = \lambda^{\otimes 2}_{\widetilde{C}'}$, where  
$\widetilde{C}'$  is the normalization of $C'$  (recall that  
$\omega_{C'}/\omega_{\widetilde{C}'}$  is a skyscraper sheaf, supported at  
singular points, trivialized canonically up to sign using residues).  Hence the  
fibers of  $\widetilde{\Cal M}$  over  $(C, y_\gamma, \nu_\gamma)$  and  
${\Cal S}_{\alpha,\beta}(C,y_\gamma,\nu_\alpha)$  are nearby fibers of the  
 same       ${\Bbb C}^*$-fibration, and therefore one has a canonical identification  
of their fundamental groupoids.  This defines the desired lifting\  
${\Cal S}_{\alpha,\beta}: 
\widetilde{Teich}''_A \to \widetilde{Teich}''_{A\setminus \{ \alpha,\beta\} }$.   
It is easy to verify that the equivalence 4.2.3 identifies the above  
``topological" and ``holomorphic" constructions of  ${\Cal S}_{\alpha,\beta}$. 


 
\medpagebreak 
 
\noindent 
4.3.2\ It is convenient to consider both sewing and deleting of points  
simultaneously.  To do this, consider a category, $Sets^\#$, whose objects  
are finite sets, and whose morphisms  $f: A \to B$  are pairs  $(i_f, \phi_f)$, where 
$i_f: B \hookrightarrow A$  is an embedding, and  $\phi_f = \{ \phi_{f\delta}\}$ 
is a collection of two-element mutually non-intersecting subsets  
$\phi_{f\delta}$ of  $A\setminus i_f(B)$.  The composition is obvious:  if 
$g: B \to C$  is another morphism, then  $g \circ f = (i_f \circ i_g,  
\phi_f \cup \phi_g)$.  For $f$ as above we put  $A^1_f := {\dsize\coprod_\delta}  
\phi_{f\delta},\ A^0_f = A\setminus (i_f(B) \cup A^1_f)$,\ so $A = i_f(B)  
\coprod A^0_f \coprod A^1_f$.   
 
Now for any morphism  $f: A \to B$  we have a canonical functor  $f_*: Teich_A  
\to Teich_B$, $\widetilde{Teich}_A \to \widetilde{Teich}_B$  that deletes points  
in  $A^0_f$  and  sews pairwise points in all $\phi_{f\delta}$'s.  One has  
$(g \circ f)_* = g_* \circ f_*$, and each  $f_*$  is a composition of elementary  
deletings of a single point, and glueing of a single pair.  Clearly these   
$f_*$'s define a cofibered categories $Teich^\#, \widetilde{Teich}^\#$ over  
$Sets^\#$  with old fibers  $Teich_A, \widetilde{Teich}_A$, respectively. 
 
Note that all these categories are strictly commutative monoidal categories with  
respect to ``disjoint union" operation $\coprod$; all the functors commute with  
$\coprod$.   
 
\medpagebreak 
 
\subheading{4.4\ Representations of Teich; central charge}  Let  $A$  be a finite 
set.  Denote by  ${\Cal R}_A$	the category of finite dimensional  
${\Bbb C}$-representations of  $Teich_A$  (i.e., the objects of  ${\Cal R}_A$  
are functors  $L: Teich_A \to Vect$), and by  $\widetilde{{\Cal R}}_A$  the same  
for $\widetilde{Teich}_A$.  More generally, if  $Q$  is a component (i.e., a strictly full 
subcategory) of $Teich_A$, we denote by  ${\Cal R}_{A,Q}$  the category of  
representations of $Q$, identified with the full subcategory of  ${\Cal R}_A$  that 
consists of representations supported on $Q$ (i.e., vanish off $Q$).  For a representation   
$V \in \widetilde{\Cal R}_A$  and  $X \in \widetilde{Teich}_A$  we denote by  
$V_X$  the value of $V$ at $X$.   
 
\medpagebreak 
 
\subheading{4.4.1\ Definition}  A representation  $V \in \widetilde{\Cal R}_A$  
has multiplicative central charge  $a \in {\Bbb C}^*$  if for any   
$X \in \widetilde{Teich}$  the canonical element  $\gamma_0 \in Aut X$  acts on  
$V_X$  as multiplication by  $a$. \hfill $\square$ 
 
For any  $a \in {\Bbb C}^*$   denote by  ${\Cal R}_{aA} \subset \widetilde{\Cal R}_A$ 
the full subcategory of representations of central charge $a$.  In particular,  
${\Cal R}_{1A} = {\Cal R}_A$.   
 
For any morphism  $f: A \to B$  in $Sets^\#$ the functor   
$f_*: \widetilde{Teich}_A \to \widetilde{Teich}_B$  defines the corresponding  
functor  $f^*: \widetilde{\Cal R}_B \to \widetilde{\Cal R}_A$; one has   
$f^*({\Cal R}_{aB}) \subset {\Cal R}_{aA}$.  The functors  $f^*$  define a  
category  $\widetilde{\Cal R}^*$  fibered over $Sets^\#$  with fibers  
$\widetilde{\Cal R}_A$, together with fibered subcategories ${\Cal R}_a^\#  
\subset \widetilde{\Cal R}^\#$  with fibers ${\Cal R}_{aA}$.   
 
\medpagebreak 
 
\noindent 
4.4.2\ Here is an explicit description of representations.  From a combinatorial  
point of view a representations  $V \in \widetilde{\Cal R}_A$  assigns to each  
surface  $(S,A) \in Teich_A$  a local system $V_S$  on the Lagrangian Grassmannian  
$\Lambda_{H(S)}$    (see 4.1.2), and to each  $\varphi \in \text{Hom}((S,A),(S',A))$  a  
lifting of the corresponding diffeomorphism  $\Lambda_{H(S)}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
\Lambda_{H(S')}$  to  $V_S\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\  V_{S'}$. 
This  $V$  lies in  ${\Cal R}_{aA}$  if the monodromy matrix of the loop  
$\gamma_0 = 1 \in {\Bbb Z} = \pi_1(\Lambda_{H(S)})$  coincides with  
multiplication by $a$.   
 
\medpagebreak 
 
\noindent 
4.4.3\ From a holomorphic point of view our $V$ is a local system on the modular 
stack  $\widetilde{\Cal M}_A$;   $V$  lies in ${\Cal R}_{aA}$  if the monodromy 
around the fiber of the projection  $\pi: \widetilde{\Cal M}_A \to {\Cal M}_A$   
equals multiplication by $a$. 
 
Recall that  ${\Bbb C}$-local systems on smooth algebraic manifolds can be  
identified with algebraic vector bundles with integrable connections (= lisse  
$D$-modules) having regular singularities at infinity (see [D], [Bo]).  So our 
$V$  is a lisse $D$- module on  $\widetilde{\Cal M}_A$  with regular singularities 
at $\infty$.  Assume that  $V \in {\Cal R}_{aA}$.  Choose $c \in {\Bbb Z}$   
(``additive central charge") such that  $exp(2\pi i c) = a$.  Let  $D_{\lambda^c} =  
{\Cal D}_{c{\Cal A}(\lambda)}$  be the ring of differential operators on the ``line bundle" 
$\lambda^{\otimes c}$.  This is a twisted differential operator ring on  
${\Cal M}_A$ (see 3.2.6-3.2.8).  Recall that  $D_{\lambda^c}$-modules can be  
identified canonically with $D$-modules on  $\widetilde{\Cal M}_A$, monodromic 
along the fibers of $\pi$  with monodromy $a$ (see, e.g., [V]).  In particular, 
$V$ is a lisse $D_{\lambda^c}$-module on  ${\Cal M}_A$  having regular  
singularities at $\infty$. 
 
\medpagebreak 
 
\subheading{4.5\ Axioms of a fusion category}  We will start with  
preliminary data.   
 
\noindent 
4.5.1\ Let  ${\Cal A}$  be an abelian ${\Bbb C}$-category (``category of  
modules").  We assume that ${\Cal A}$ is semisimple, for $X \in {\Cal A}$ the  
${\Bbb C}$-vector space  $End X$  is finite dimensional, and there are finitely  
many isomorphism classes of irreducibles.  Denote by  $Irr A$  the set of  
isomorphism classes of irreducible objects in ${\Cal A}$.   
 
We should also have the following data: 
 
-- a contravariant functor (``duality") $*: {\Cal A}^{\circ} \to {\Cal A}$ together with 
a natural isomorphism  $**\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
id_{\Cal A}$   
 
-- a distinguished irreducible object (``vacuum module") $\Bbb 1$  together  
with an isomorphism $\nu: \Bbb 1 \  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\ *\Bbb 1$ 
such that  $*(\nu) \circ \nu = id_{\Bbb 1}$.   
 
-- an automorphism $d$ of the identity functor  $id_{\Cal A}$, called the Dehn  
automorphism, such that $d* = *d$ and $d_{\Bbb 1} = 1$. Clearly to give $d$  is  
the same as giving a collection of numbers  $d_j = d_{I_j} \in {\Bbb C}^*$  for   
$j \in Irr {\Cal A}$  
(here  $I_j$  is an irreducible object of class $j$; recall that   
$Aut I_j = {\Bbb C}^*$). 
 
\medpagebreak 
 
\noindent 
4.5.2\ For any finite set $B$  we have a category  ${\Cal A}^{\otimes B}$:  this 
is an abelian ${\Bbb C}$-category equipped with a polylinear functor  
$\otimes:  {\Cal A}^B = {\dsize\prod_{b\in B}} A_b \longrightarrow\  
{\Cal A}^{\otimes B}$, $(X_b)_{b\in B} \longrightarrow\ {\dsize\bigotimes_{b\in B}} 
X_b$, which is universal in an obvious sense (see [D] \S\quad    for an extensive 
discussion in a less trivial  situation).  The category  ${\Cal A}^{\otimes B}$  
is semisimple.  Its irreducible objects are tensor products of irreducibles in  
${\Cal A}$, so $Irr {\Cal A}^{\otimes B} = (Irr {\Cal A})^B$.  Any isomorphism 
$\varphi: B \to B'$  induces a canonical equivalence  ${\Cal A}^{\otimes B} \to  
{\Cal A}^{\otimes B'}$, $\otimes X_b \longmapsto \otimes X_{\varphi^{-1}(b')}$. 
 
\noindent 
4.5.3\   We put  
$A^{\otimes \emptyset} = Vect$.  One may identify  ${\Cal A}^{\otimes\{ 1,2\}} =  
{\Cal A}^{\otimes 2}$  with the category of ${\Bbb C}$-linear functors  
$F = {\Cal A}^0 \to {\Cal A}$.  Namely, to an object  $X \otimes Y \in  
{\Cal A}^{\otimes 2}$  there corresponds the functor  $F_{X\otimes Y}$ defined 
by formula  $F_{X \otimes Y}(Z) = Hom(Z,X) \otimes Y$.  We define a canonical 
object (``regular representation")  $R \in {\Cal A}^{\otimes 2}$  as an object 
that corresponds to the functor  $*: {\Cal A}^0 \to {\Cal A}$.  Here is an  
explicit construction of $R$.  For each  $j \in Irr {\Cal A}$  pick an  
irreducible object  $I_j$  of class $j$.  Then one has a canonical isomorphism  
$R = {\bigoplus_{j \in Irr {\Cal A}}} I_j \otimes * I_j$.  Note that $R$ is  
symmetric:  for the transposition  $\sigma = \{ 1,2\}$ acting on  
${\Cal A}^{\otimes 2}$    one has a  
canonical isomorphism  $\sigma(R) = R$.  So for any two element set $B$ we have  
a canonical object  $R_B \in {\Cal A}^{\otimes B}$. 
 
\medpagebreak 
 
\noindent 
4.5.4\ For finite sets  $A,B$  and a morphism  $f: A \to B$  in $Sets^\#$   
(see 4.3.2) we define a ${\Bbb C}$-linear functor  $f^*: {\Cal A}^{\otimes B}  
\to {\Cal A}^{\otimes A}$  by the formula   
$$ 
f^*(\bigotimes_{b\in B} X_b) = \left[ \bigotimes_{a\in i_f(B)}  
X_{i^{-1}_f(a)}\right] \otimes \left[ \bigotimes_{a\in A^0_f} {ident}_a\right]  
\otimes \left[ \bigotimes_{\phi_{f\delta} \in \phi_f} R_{\phi_{f\delta}}\right]. 
$$ 
Clearly  $(g \circ f)^* = f^* \circ g^*$, so the $f^*$'s define a fibered category  
${\Cal A}^\#$  over $Sets^\#$  with fibers ${\Cal A}^\#_A = {\Cal A}^{\otimes A}$. 
The tensor product functor  $\otimes:  {\Cal A}^{\otimes B_1} \times  
{\Cal A}^{\otimes B_2} \longrightarrow {\Cal A}^{\otimes (B_1 \coprod B_2)}$  
defines on  ${\Cal A}^\#$  the structure of commutative monoidal category such  
that the projection  ${\Cal A}^\# \to Sets^\#$ is a monoidal functor. 
 
\medpagebreak 
 
\proclaim{4.5.4\ Definition}\ A\ {\bf fusion\ structure}\ on ${\Cal A}$  is a collection  
of functors  $\langle\quad \rangle: {\Cal A}^{\otimes A} \times \widetilde{Teich}_A  
\longrightarrow Vect$,\quad $(X,S) \longmapsto \langle X\rangle_S$  (here $A$ is any  
finite set), together with natural isomomorphisms (i), (ii): 
 
\item{(i)}\ $\langle X \otimes Y\rangle_{S\sqcup T} = \langle X\rangle_S  
\otimes \langle Y\rangle_T$  for  $X \in {\Cal A}^{\otimes A}, Y \in  
{\Cal A}^{\otimes B}, S \in \widetilde{Teich}_A, T \in \widetilde{Teich}_B$. 
 
\item{(ii)}\ $\langle f^*X\rangle_T = \langle X\rangle_{f_*T}$  for any morphism 
$f: A \to B$  in $Sets^\#$, $X \in {\Cal A}^{\otimes B}, T \in \widetilde{Teich}_A$. 
 
 These isomorphisms should be compatible in an obvious sense.  We also demand  
that: 
 
\item{a.}\ For fixed $S \in \widetilde{Teich}_A$  the functor  
$\langle\quad \rangle_S:  {\Cal A}^{\otimes A} \longrightarrow Vect$ is additive. 
 
\item{b.}\ $\langle\quad \rangle$  transforms Dehn automorphism to Dehn twist,  
i.e., for a finite set $A$, an element  $\alpha \in A$  and a collection of  
objects  $X_\gamma \in {\Cal A},\ \gamma \in A$, the automorphisms of\  
$\langle \otimes X_\gamma\rangle_S$  induced by  ${\dsize\bigotimes_{\gamma\ne\alpha}}  
id_{X_\gamma} \otimes d_{X_\gamma} \in Aut \otimes X_\gamma$  and by   
$d_\alpha \in Aut S$  coincide.   
 
\item{c.}\ $\langle \quad\rangle$  is non degenerate in the sense that for any  
non-zero  $X \in {\Cal A}$  there exists  $Y \in {\Cal A}$  such that  
$\langle X \otimes Y\rangle_{S_0} \ne 0$  where  $S_0$  is a 2-sphere with two  
punctures. 
 
We will say that  $({\Cal A},\langle\quad \rangle)$  is a fusion category of  
multiplicative central charge $a \in {\Bbb C}^*$  if for any $X \in  
{\Cal A}^{\otimes A}$  the representation  $\langle X\rangle$  of  
$\widetilde{Teich}$ lies in ${\Cal R}_{aA}$. \hfill $\square$ 
\endproclaim 
 
\medpagebreak 
 
\noindent 
4.5.5\ Clearly (ii) just means that  $X \longmapsto \langle X\rangle$ is a  
cartesian functor  ${\Cal A}^\# \to \widetilde{\Cal R}^\#$  between categories  
fibered over  $Sets^\#$.  Since any morphism in $Sets^\#$  is a successive  
deleting of points and sewing of couples of points, we may rewrite (ii) as two  
compatibilities.  Namely 
 
\item{(ii)$'$}\ $\langle X\rangle_{del_\alpha S} = \langle X \otimes  
{ident}_\alpha\rangle_S$  for any finite set $A$, $\alpha \in A,  
X \in {\Cal A}^{\otimes A\setminus\{ \alpha\}}$, $S \in \widetilde{Teich}_A$. 
 
\item{(ii)$''$}\ $\langle X\rangle_{{\Cal S}_{\alpha,\beta}S} =  
\langle X \otimes R_{\alpha\beta}\rangle_S$  for any finite set $A$,  
a pair of elements $\alpha,\beta \in A, \alpha \ne \beta,  
X \in {\Cal A}^{\otimes A\setminus\{ \alpha,\beta\}}$,  
$S \in \widetilde{Teich}_A$. 
 
\medpagebreak 
 
\noindent 
4.5.6\ Here is a reformulation of 4.5.5(ii)$''$ in ``holomorphic" language  
4.4.3.  For  $X \in {\Cal A}^{\otimes A\setminus\{ \alpha,\beta\}}$  our 
$\langle X\rangle$  is a lisse $D_{\lambda^c}$-module with regular singularities 
at infinity.  As was explained in 4.3.1 we have a canonical surjective smooth  
map  $\pi:  {\Cal M}_A \to N\setminus \{ \text{zero\ section}\}$, where $N$ is the   
normal bundle to the (smooth part of) the divisor at infinity of  
${\Cal M}_{A\setminus \{\alpha,\beta\}}$.  We have the canonical specialization  
function  $Sp$  that assigns to a lisse $D_{\lambda^c}$-module with regular  
singularities at infinity on  ${\Cal M}_{A\setminus \{ \alpha,\beta\}}$,  
the one on  
$N\setminus\{ \text{zero\ section}\}$.  Hence we have the   
$D_{\lambda^c}$-module $\pi^*Sp\langle X\rangle$ on ${\Cal M}_A$, and  
4.5.5 (ii)$'$  is an isomorphism  $\pi^*Sp\langle X\rangle =  
\langle X \otimes R_{\alpha\beta}\rangle$. 
 
\bigpagebreak 
 
\subheading{4.6\ Fusion functors}  Let  $({\Cal A}, \langle\quad \rangle)$  be a 
fusion category.  Let  $A,B$  be finite sets.  Any object  $S \in  
\widetilde{Teich}_{A\sqcup B}$  defines a functor  ${\Cal F}_S = {\Cal F}_S^{A,B}: 
{\Cal A}^{\otimes A} \to {\Cal A}^{\otimes B}$  by the formula  $Hom({\Cal F}_S(X),Y)  
= \langle X \otimes *Y\rangle^*$, $X \in {\Cal A}^{\otimes A}, Y \in {\Cal A}^{\otimes B}$. 
We will call  ${\Cal F}_S$  the fusion functor along $S$.  The automorphisms of  
$S$ act as automorphisms of ${\Cal F}_S$.  Note that if $B = \emptyset$ then  
$A^{\otimes B} = Vect$ and ${\Cal F}_S = \langle\quad \rangle_S$.  If  $A =  
\emptyset$, then ${\Cal F}$ is a functor  $\widetilde{Teich}_B \to  
{\Cal A}^{\otimes B}$, i.e., an  ${\Cal A}^{\otimes B}$-valued representation of  
$\widetilde{Teich}_B$.   
 
Let  $C$  be a third finite set, $T \in \widetilde{Teich}_{B\sqcup C}$.  We  
define $T \circ S \in \widetilde{Teich}_{A \sqcup C}$  as the surface obtained from 
$T \sqcup S$  by sewing the $B$-boundary components. 
 
\medpagebreak 
 
\proclaim{4.6.1\ Lemma}\ There is a canonical isomorphism of functors  
${\Cal F}_{T\circ S} = {\Cal F}_S \circ {\Cal F}_T:  {\Cal A}^{\otimes A} \to  
{\Cal A}^{\otimes C}$. 
\endproclaim 
 
\demo{Proof}  For  $X \in {\Cal A}^{\otimes A}, Z \in {\Cal A}^{\otimes C}$  one 
has   
$$ 
\aligned 
Hom({\Cal F}_{T\circ S}(X), Z) &= \langle X \otimes *Z\rangle^*_{T\circ S}\  
{\lower 1.5ex\hbox{$\buildrel =\over {\scriptstyle 4.5.4(ii)}$}}\  
\langle X \otimes R^{\otimes B} \otimes *Z\rangle^*_{T\sqcup S}\\  
& {\lower 1.5ex\hbox{$\buildrel =\over {\scriptstyle 4.5.4(i)}$}}\ 
{\bigoplus_{I_{\vec{j}} \in Irr {\Cal A}^{\otimes B}}} \langle X \otimes  
*I_{\vec{j}}\rangle^*_S \otimes \langle I_{\vec{j}} \otimes *Z\rangle^*_T \\ 
&=  \bigoplus Hom({\Cal F}_S(X), I_{\vec{j}}) \otimes Hom({\Cal F}_T(I_{\vec{j}}),Z)  
= Hom({\Cal F}_T \circ {\Cal F}_S(X), Z). 
\endaligned 
$$ 
The last equality comes since  
$$ 
{\Cal F}_S(X) = \oplus Hom({\Cal F}_S(X), I_{\vec{j}})^* \otimes I_{\vec{j}}. 
$$ 
\hfill $\square$ 
\enddemo 
 
\medpagebreak 
 
Now assume that $A = \{ 0\}$, $B = \{ \infty\}$  are one point sets.  Let  
$Teich^{'0}_{\{ 0,\infty\}} \subset Teich'_{\{ 0,\infty\}}$  be the full  
subcategory of ``cylinders".  So $Teich^{'0}_{\{ 0,\infty\}}$   
is a connected groupoid; for $(S,0,\infty) \in  
Teich^0_{\{ 0,\infty\}}$  the group (of its automorphisms) is a free abelian  
group with generator  $d_0 = d_\infty^{-1}$.  Denote by  $S_0 = (S_0,0,\infty)$  
the object of $Teich^0_{\{ 0,\infty\}}$  such that for any $(S,0,\infty) \in  
Teich^0_{\{ 0,\infty\}}$  one has $Hom(S_0,S) = \{$ set of homotopy classes of  
paths in $S$ connecting $0$ and $\infty\}$.  This is a canonical object of  
$Teich^{'0}_{\{ 0,\infty\}}$.  Its ``holomorphic" counterpart is  
$({\Bbb P}^1,0,\infty,dt(0),dt^{-1}(\infty)) \in Teich^{''0}_{\{ 0,\infty\}}$,  
where  $t$  is a standard parameter on  ${\Bbb P}^1$.  
One identifies  this point of $Teich''$ with  $S_0$  canonically  
by drawing the path  ${\Bbb R}_{\ge 0}$  from $0$ to $\infty$.   
Note that since  $H(S) = 0$  for $S \in Teich^{'0}_{\{ 0,\infty\}}$  we have  
an obvious embedding $Teich^{'0}_{\{ 0,\infty\}} \hookrightarrow  
\widetilde{Teich}'_{\{ 0,\infty\}}$; the ``holomorphic" counterpart of this  
section comes since the line bundle $\lambda$ is canonically trivialized over  
the ``moduli space" of genus zero curves.  So we will consider  $S_0$  as a canonical object  
of  $\widetilde{Teich}_{\{ 0,\infty\}}$.  Note that if $A$ is any finite set  
and  $T \in \widetilde{Teich}_{A \sqcup \{ 0\}}$, then one has an obvious  
canonical isomorphism  $S_0 \circ T = T$.  According to 4.6.1 this gives a  
canonical isomorphism of functors  ${\Cal F}_{S_0} \circ {\Cal F}_T = {\Cal F}_T$.  In  
fact, one has  
 
\proclaim{4.6.2\ Lemma}  There is a canonical identification of the functor  
${\Cal F}_{S_0} : {\Cal A} \to {\Cal A}$ with the identity functor $id_{\Cal A}$ that generates the  
above isomorphisms\ ${\Cal F}_{S_0} \circ {\Cal F}_T = {\Cal F}_T$ for all  
$T \in \widetilde{Teich}_{A\sqcup \{0\}}$. 
\endproclaim 
 
\demo{Proof} Assume that we know that ${\Cal F}_{S_0}$  is an equivalence of  
categories.  Then the desired isomorphism ${\Cal F}_{S_0} = id_{\Cal A}$  would 
be ${\Cal F}_{S_0}^{-1}({\Cal F}_{S_0} \circ {\Cal F}_{S_0} = {\Cal F}_{S_0})$.   
Since  ${\Cal A}$  is semi-simple, to see that ${\Cal F}_{S_0}$ is an equivalence  
it suffices to prove that ${\Cal F}_{S_0}$ induces the identity map of the 
Grothendieck  group $K({\Cal A})$.  The irreducible $I_i$ form a basis in  
$K({\Cal A})$.   
Put ${\Cal F}_{S_0}(I_i) = f^j_i I_j$; we have to show that  $f^j_i = \delta^j_i$.  
We know that  $f^j_i \in {\Bbb Z}_{\ge 0}$.  Since  $f^j_i = \langle I_j \otimes 
* I_i\rangle^*_{S_0}$  we see, by 4.5.4c, that any row or column of the matrix  
$f^j_i$  is non-zero.  Since ${\Cal F}_{S_0}^2 = {\Cal F}_{S_0}$, these  
properties imply that ${\Cal F}_{S_0} = id_{K({\Cal A})}$ (just note that  
${\Cal F}_{S_0}^2(I_i) = {\Cal F}_{S_0}(I_i)$  implies ${\Cal F}_{S_0}$  induces 
a transposition of the set of those  $I_j$'s that  $f^j_i \ne 0$; hence   
${\Cal F}_{S_0}$  is a surjective endomorphism of  $K({\Cal A})$, and hence it  
is the identity).\hfill $\square$ 
\enddemo 
 
\medpagebreak 
 
\noindent 
4.6.3\ Assume now that  $S$  is a connected surface of genus $0$ and $B$ is a  
one point set.  Then the corresponding functors ${\Cal F}_S: {\Cal A}^{\otimes A}  
\longrightarrow {\Cal A}$, together with $*$ and $d$ from 4.5.1, define on  
${\Cal A}$  the structure of a balanced rigid tensor category (see, e.g. [K]). 
Here are some details.  Denote by  $S_n$  the surface obtained from a unit disc 
by cutting out $n$ holes with centers on the real line; the marked points lie on 
the real line to the right: 
\vskip .2in 
$$ 
S_3:\qquad\qquad  {O}^{x_1}\qquad {O}^{x_2}\qquad {O}^{x_3}\  
\qquad\quad  x_\infty 
$$ 
\vskip .3in 
 
Put ${\Cal F}_{S_n}(X_1 \otimes \cdots \otimes X_n) = X_1 \widehat{\otimes} \cdots 
\widehat{\otimes} X_n$.  The axiom 1.5.4 (ii)$''$ implies immediately that the  
operation  $\widehat{\otimes}:  {\Cal A}^n \to {\Cal A}$  is strictly associative: 
one has  $X_1 \widehat{\otimes} X_2 \widehat{\otimes} X_3 = (X_1 \widehat{\otimes}  
X_2) \widehat{\otimes} X_3 = X_1 \widehat{\otimes} (X_2 \widehat{\otimes} X_3)$.   
Consider the following diffeomorphism  $\sigma$ of $S_2$  that fixes   
$\partial S_{2x_\infty}$  and interchanges $\partial S_{2x_1}$  and   
$\partial S_{2x_2}$  (we move the holes in a way that the marked point remain  
on the very right of the hole): 
\vskip .2in 
$$ 
\phantom{S} 
$$ 
\vskip .3in 
 
This diffeomorphism induces a natural isomorphism $\sigma_{X_1X_2}:  
X_1 \widehat{\otimes} X_2\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
X_2 \widehat{\otimes} X_1$.  It is easy to see that  $\sigma$  satisfies the braid 
relations, and also one has a relation  $\sigma^2 = d_{x_\infty} d^{-1}_{x_1}  
d^{-1}_{x_2}$ in $Aut S_2$.  These imply the hexagon axiom for  $\widehat{\otimes}$, 
and the axiom  $\sigma^2_{X_1,X_2} = d_{X_1 \widehat{\otimes} X_2} \circ  
(d_{X_1} \widehat{\otimes} d_{X_2})^{-1}$  of balanced tensor categories.   
 
\medpagebreak 
 
\subheading{4.7\ The fusion algebra}  The above tensor structure on ${\Cal A}$  defines 
a commutative ring structure on the Grothendieck group  $K({\Cal A})$.   
One calls  
$K({\Cal A})$  the fusion algebra of  ${\Cal A}$.  Note that $K({\Cal A})$  has a  
distinguished basis $\{ I_j\}$  of irreducibles.  By 4.5.5 (ii)$'$ the base  
element 1 that corresponds to vacuum module is the unit in $K({\Cal A})$. 
 
Now 4.6.2 implies that $(K({\Cal A}), \{ I_j\})$  is a {\it based ring} in the sense 
of [L] 1.1.  According to [L] 1.2, $K({\Cal A}) \otimes {\Bbb Q}$  is a semisimple 
algebra.  Hence  $K({\Cal A}) \otimes {\Bbb C}$  has another canonical basis --  
the one that consists of mutually orthogonal idempotents. 
 
Let $T$ be a torus (= oriented genus one surface).  Choose a basis $\gamma_1,\gamma_2$ 
in $H_1(T,{\Bbb Z})$ compatible with the orientation, so that $\gamma_1,\gamma_2$  are 
cycles on $T$ that intersect at one point $a$.  Consider the vector space  
$\langle {\Bbb 1}\rangle_T$.  Note that if we cut $T$ along $\gamma_1$, then  
$\gamma_2$  will become a path that connects two copies of $a$ on the components 
of the boundary, hence it identifies this surface with the surface $S_0$  of 4.6.2.   
According to 4.5.5 (ii)$''$, 4.6.2, the corresponding decomposition 4.5.5(ii)$''$ 
gives the basis in  $\langle {\Bbb 1}\rangle_T$ numbered by irreducibles in  
${\Cal A}$, i.e., we have the isomorphism  $i_{\gamma_1,\gamma_2}: K({\Cal A})  
\otimes {\Bbb C} \to \langle \Bbb 1\rangle_T$   that transforms  $I_j$'s to this 
basis.  Interchanging $\gamma_1$ and $\gamma_2$  we get the isomorphism  
$i_{\gamma_2,-\gamma_1}: K({\Cal A}) \otimes {\Bbb C}\  
{\lower 1ex\hbox{$\overrightarrow{\sim}\vrule height 1ex width 0pt$}}\   
\langle {\Bbb 1}\rangle_T$.  The composition  $i^{-1}_{\gamma_2,-\gamma_1} \circ 
i_{\gamma_1,\gamma_2} \in Aut K({\Cal A}) \otimes {\Bbb C}$  is called the  
{\it Fourier transform}.  According to the Verlinde conjecture, proved by  
Moore-Zeiberg, the Fourier transform maps a canonical basis $\{ I_j\}$  of  
irreducibles to a basis proportional to the one given by the idempotents. 

 
\newpage 
 
\pageno=66 
 
\heading 
\S6.\ Algebraic field theories 
\endheading 
 
\subheading{6.1\ Axioms}  Let $c \in {\Bbb C}$  be any complex number.  An  
{\bf algebraic rational field theory} (in dimension 1) of central charge $c$ consists 
of data (i) - (iv) subject to axioms a-g below:   
 
\noindent 
6.1.1  
\item{(i)} A fusion category ${\Cal A}$  of multiplicative central charge  
exp$(2\pi i c)$  (see 4.5.4) 
 
\item{(ii)}  An additive ``realization" functor $r: {\Cal A} \to  
(\widetilde{\Cal T}, {\Cal V}_1)_c$-mod (see 3.4.7). 
 
We assume that for any  $X \in {\Cal A}$ 
 
 \itemitem{a.}  {\it $r(X)$  is a higher weight module}, i.e., the ``coordinate  
module"  $r(X)_{{\Bbb C}((t)),dt(o)}$  is a (direct) sum of generalized  
eigenspaces $r(X)_{{\Bbb C}((t)),\lambda} = \{ m \in r(X)_{{\Bbb C}((t))}:$ 
\newline 
$(L_0 - \lambda)^Nm = 0$  for $N \gg 0\}$  for the operator $L_0$  (see 3.4.7,  
7.3.1).  {\it Each $r(X)_{{\Bbb C}((t))\lambda},\ \lambda \in {\Bbb C}$, is a  
finite dimensional vector space}. 
 
\itemitem{b.} {\it $r(d_X) = T_{r(X)}$, where  $d_X$  is the Dehn automorphism}   
(see 4.5.1) {\it and $T$ is the monodromy automorphism} (see 7.3.2). 
 
Note that these axioms imply that $r({\Bbb 1})$\  is actually a  
$(\widetilde{\Cal T}, {\Cal V})_c$-module (since $T_{r(\Bbb 1)} =  
id_{r(\Bbb 1)}$. 
 
\item{(iii)} A fixed ``vacuum" vector $1 \in \text{Hom}_{\Cal V}({\Bbb C},  
r({\Bbb 1}))$. 
     
We assume that  
 
\itemitem{c.}  {\it 1 is a non-zero vector invariant with respect to the action 
of $s_{{\Cal O}_F}({\Cal T}_{-1F}) \subset \widetilde{\Cal T}_F$} (see 3.4.1). 
 
\medpagebreak 
 
\noindent 
6.1.2.\ Now let  $S$  be a smooth scheme, $\pi: C_S \to S$  a family of smooth 
projective curves,  $A \subset C_S(S)$  a finite disjoint set of sections, and 
$\{ \nu_a\}_{a\in A}$  1-jets of parameters at points in $A$.  This collection  
defines $S$-localization data  $\psi_c$  for $(\widetilde{\Cal T}^A_c,  
{\Cal V}^A_1)$ (see 3.4.7, 3.4.5).  The corresponding algebra of twisted differential  
operators  $D_{\psi_c}$  coincides with  $D_{\lambda^c}$  (see  
3.5.6).  Hence, by 3.3.5, we have the $S$-localization functor  
$\Delta_{\psi_c} \circ r^{\otimes A}: {\Cal A}^{\otimes A} \longrightarrow  
D_{\lambda^c}$-mod.  On the other hand, by 4.5.4, 4.4.3, the fusion structure  
on ${\Cal A}$  defines the functor  $\langle\quad \rangle_{C_S}:  
{\Cal A}^{\otimes A} \longrightarrow D_{\lambda^c}$-mod such that for any  
$\otimes X_a \in {\Cal A}^{\otimes A}$  the corresponding  $D_{\lambda^c}$-module  
$\langle \otimes X_a\rangle_{C_S}$  is lisse with regular singularities at  
infinity.  Our next piece of data is  
 
\item{(iv)} A morphism of functors  $\gamma: \Delta_{\psi_c} \circ r^{\otimes A}  
\longrightarrow \langle\quad \rangle_{C_S}$. 
 
For $X \in {\Cal A}^{\otimes A}$  denote by  $r(X)_{A,C_S} = r(X)_{A,\nu_A,C_S}$  
the ${\Cal O}_S$-module that corresponds to the $S$-object ``formal completion of 
$C_S$  at $A$ with 1-jet of parameters $\nu_A$" of ${\Cal V}^A_1$  (see 3.4.3,  
3.4.6, 3.4.7).  If  $X = \otimes X_a$, then $r(X)_{A,C_S} = \otimes_{{\Cal O}_S} 
r(X_a)_{a,C_S}$.  Recall that $\Delta_{\psi_c} \circ r^{\otimes A}(X)$, considered 
as an ${\Cal O}_S$-module, is a quotient of  $r(X)_{A,C_S}$.  For any section  
$\varphi$ of $r(X)_{A,C_S}$  put $\langle \varphi\rangle_{C_S} =  
\gamma(\varphi) \in \langle X\rangle_{C_S}$.  This is the ``correlator of the field 
$\varphi$  along $C_S$".   
 
The following axioms should hold: 
 
\itemitem{d.}  {\it $\gamma$ commutes with base change}, i.e., $\gamma$  is a  
morphism of  $D_{\lambda^c}$-modules on the modular stack  ${\Cal M}_A$. 
 
\itemitem{e.} {\it For $a \in A$, objects  $X \in {\Cal A}^{\otimes A\setminus\{ a\}}$ 
and a section  $\varphi \in r(S, r(X)_{A,C_S})$  one has  
$\langle \varphi\rangle_{C_S} = \langle \varphi \otimes 1_a\rangle_{C_S}$.}  Here 
$\langle \varphi\rangle_{C_S}$  is a section of  $\langle X\rangle_{C_S}$  (we  
forget about the point $a$), and  $\langle \varphi \otimes 1_a\rangle_{C_S}$  is a  
section of  $\langle X \otimes\ {\Bbb 1}_a\rangle_{C_S}$; the two   
$D_{\lambda_c}$-modules are identified via 4.5.5 (ii)$'$. 
 
\medpagebreak 
 
\noindent 
6.1.3  Now consider the two pointed curve  $C_0 = ({\Bbb P}^1,0,\infty, dt(0),  
dt^{-1}(\infty))$.  We have coordinates $t$ at $0$ and  $t^{-1}$  at $\infty$. 
For any object  $X \in {\Cal A}$  consider the pairing 
$$ 
\langle~~~\rangle_{C_0}: r(*X)_{{\Bbb C}((t))} \otimes r(X)_{{\Bbb C}((t^{-1}))}  
= r(*X)_{C_{0{\hat{0}}}} \otimes r(X)_{C_0{\hat{\infty}}} \longrightarrow  
\langle *X \otimes X\rangle_{C_0}\ {\lower 1.5ex\hbox{$\buildrel =\over  
{\scriptstyle{4.6.2}}$}}\ \text{End}\  X 
$$ 
Here we write simply  ${\Bbb C}((t))$  for $({\Bbb C}((t)), dt(0)) \in {\Cal V}_1$. 
This pairing is a morphism of  
 
\noindent 
End $X$-bimodules, hence it defines a linear map 
$$ 
i: r(*X)_{{\Bbb C}((t))}\ \longrightarrow\ \text{Hom}_{\text{End}\ X} 
(r(X)_{{\Bbb C}((t^{-1}))}, \text{End}\ X) =: r(X)^*_{{\Bbb C}((t^{-1}))}. 
$$ 
 
Note that $r(X)^*_{{\Bbb C}((t^{-1}))}$  is a $\widetilde{\Cal T}_{{\Bbb C}((t^{-1}))}$-
module in an obvious manner.  Denote by  
 
\noindent 
$*r(X)_{{\Bbb C}((t^{-1}))} \subset r(X)^*_{{\Bbb C}((t^{-1}))}$  the sum of  
generalized eigenspaces of the operator  $L_0~\in~\widetilde{\Cal T}_{{\Bbb C}((t))}$. 
The pairing  $\langle~~~\rangle_{C_0}$  is   
${\Cal T}({\Bbb P}^1\setminus \{0,\infty\})$-invariant (by definition of  
$\Delta_\psi$, see 3.4.4), hence  $i$  commutes with the the $L_0$-action.  By axiom  
$a$  above we see that  $i(r(*X)_{{\Bbb C}((t))} \subset *r(X)_{{\Bbb C}((t^{-1}))}$. 
Our next axiom is 
 
\itemitem{f.}  {\it The map}  $i: r(*X)_{{\Bbb C}((t))} \longrightarrow  
*r(X)_{{\Bbb C}((t^{-1}))}$  {\it is an isomorphism of vector spaces}. 
 
It suffices  to verify  $f$  for irreducible  $X$'s only. 
 
\medpagebreak 
 
\noindent 
6.1.4\ Our final axiom $g$  (``factorization at infinity") describes the  
asymptotic expansion of correlators near the boundary of the moduli space.  So  
consider the following situation. 
 
Let  $\pi: C_S \to S = \text{Spec}\ {\Bbb C}[[q]]$  be a proper flat family of  
curves such that the generic fiber  $C_\eta$  is smooth and the special fiber  
$C_0$  has exactly one singular point which is quadratic.  Let  $B = \{ b_i\}$   
be a finite non-empty set of sections of $\pi$  such that the points  
$b_i(0) \in C_0$  are pairwise different, and let $\nu_i \in b_i^*\omega_{C_S/S}$   
be a 1-jet of coordinates at the  $b_i$'s.  Then  ${\Cal C} = (C_\eta, b_i,\nu_i)$  
is a ${\Bbb C}((q))$-point of ${\Cal M}_B$.   
 
Let  $t_1,t_2$  be formal coordinates at $a$  such that  $t_1t_2 = q$.   
According to 3.6.1 we get a smooth  $S$-curve  $C^\vee_S$  with points   
$a_1,a_2 \in C_S^\vee(S)$  and formal coordinates  $t_i$  at $a_i$.  Put  
$A = B \bigsqcup \{ a_1,a_2\}$.  Then  ${\Cal C}^\vee = (C^\vee_\eta, b_i,  
a_1,a_2; \nu_i; q^{-1}dt_1(a_1), dt_2(a_2))$  is a ${\Bbb C}((q))$-point of  
${\Cal M}_A$.   
 
The $S$-curves $C_S$  and  $C^\vee_S$  define the corresponding determinant  
line bundles on $S$.  According to 3.6.3 their ratio is canonically stratified,  
hence the corresponding rings of differential operators are canonically identified; 
we denote this algebra  $D_{\lambda^c}$.   
 
For any object  $X \in {\Cal A}^{\otimes B}$  we get the lisse $D_{\lambda^c}$-modules 
$\langle X\rangle_{\Cal C}$  and $\langle X \otimes R\rangle_{{\Cal C}^\vee}$ on 
$\eta$  with regular singularities at  $q = 0$.  According to 4.5.6 we have a  
canonical isomorphism between their specializations to $q = 0$  (these are  
$D$-modules on the punctured tangent line at  $q = 0$).  Since  Sp$_0$  is an  
equivalence of categories, we have a canonical isomorphism of  $D_{\lambda^c}$-modules 
$\langle X\rangle_{\Cal C} = \langle X \otimes R\rangle_{{\Cal C}^\vee}$. 
 
To formulate axiom $g$  we need to consider a special vector in  $r(R)$.   
Recall that  $R = {\bigoplus_{I_j\in {\text{Irr}}\ A}} I_j \otimes * I_j$.   
Choose a basis  $\{ e^K_j\}$  in each  $r(I_j)_{{\Bbb C}((t))}$  compatible  
with grading by generalized eigenspaces of  $L_0$.  Here, as above, we write  
simply  ${\Bbb C}((t))$	 for  $({\Bbb C}((t)),dt(0)) \in {\Cal V}_1$. 
 
Below we will use the following notation:  if  $F \in {\Cal V}$  is any local  
field,  $t_F$  a parameter in  $F$,  $X \in {\Cal A}$  and   
$e \in r(X)_{{\Bbb C}((t))}$, then  $e_{(F,t_F)} \in r(X)_{F,dt_F(0)}$  is a  
vector that corresponds to  $e$  via the isomorphism  
$({\Bbb C}((t)), dt(0))\  
{\lower .5ex\hbox{$\vrule height 1ex width 0pt\overrightarrow{\sim}$}}\  
(F, dt_F(0)),\ t \longmapsto t_F$. 
 
According to axiom f. above, we get the dual basis  $\{ *e^K_j\}$  of   
$r(*I_j)_{{\Bbb C}((t))}$, namely $*e^K_j = i^{-1}e_j^{K*}$, where  
$e_j^{K*} \in *r(I_j)_{({\Bbb C}((t^{-1})),t^{-1})}$  is the dual basis to   
$e^K_{j({\Bbb C}((t^{-1})), t^{-1})}$.   
 
Now let  $\varphi = \varphi(q)$  be any section of  $r(X)_{B,\nu_B,C} =  
r(X)_{B,\nu_B,C^\vee}$  over $S$.  Consider the correlator  
$a^K_j = \langle \varphi \otimes e^K_{j({\Bbb C}((t_1)), q^{-1}t_1)} \otimes  
*e^K_{j({\Bbb C}((t_2)),t_2)}\rangle_{{\Cal C}^\vee}$:  this is a section of  
$\langle X \otimes I_j \otimes *I_j\rangle_{{\Cal C}^\vee}$.  Note that 
$\langle X \otimes I_j \otimes * I_j\rangle_{{\Cal C}^\vee}$  is a  
finite dimensional  ${\Bbb C}((q))$-vector space.  One has  
 
\proclaim{6.1.5\ Lemma}  The series  ${\dsize\sum_K} a^K_j$  converges; its  
limit  $\langle \varphi \otimes c_j\rangle_{{\Cal C}^\vee} \in  
\langle X \otimes I_j \otimes *I_j\rangle_{{\Cal C}^\vee}$  does not depend  
on a particular choice of basis  $\{ e^K_j\}$.  \hfill $\square$ 
\endproclaim 
 
Assuming the lemma, our final axiom is 
 
\itemitem{g.}  {\it One has}\ $\langle\varphi\rangle_{\Cal C} =   
\langle \varphi \otimes {\dsize\sum_j} C_j\rangle_{{\Cal C}^\vee} =  
{\dsize\sum_j} \langle \varphi \otimes C_j\rangle_{{\Cal C}^\vee}$   
{\it via the above canonical isomorphism}\  
$$ 
\langle x\rangle_{\Cal C} = \langle X \otimes R\rangle_{{\Cal C}^\vee} =  
\oplus  \langle X \otimes I_j \otimes * I_j\rangle_{{\Cal C}^\vee}. 
$$ 
 
\demo{Proof of 6.1.5}  The independence of a choice of basis is straightforward. 
To prove that our series converges it is convenient to add a parameter $u$, and  
consider a base scheme  $\widetilde{S} = \text{Spec}({\Bbb C}[u,u^{-1}]) \times  
S$  together with an $\widetilde{S}$-point of ${\Cal M}_A$  defined by the family  
${\Cal C}^\vee_u = (C^\vee_{\widetilde{S}}, b_i, a_1,a_2; \nu_i, u dt_1,dt_2)$.   
We get the lisse $D_{{\lambda^c}}$-module  
$\langle X \otimes I_j \otimes * I_j\rangle_{{\Cal C}^\vee_u}$ on $\widetilde{S}$,  
and a collection of sections  $a^K_j(u,q) = \langle \varphi(q) \otimes  
e^K_{j({\Bbb C}((t_1)), ut_1)} \otimes  
* e^K_{j({\Bbb C}((t_2)),t_2)}\rangle_{C^\vee_u} \in  
\Gamma(\widetilde{S}, \langle X \otimes I_j \otimes * I_j\rangle_{{\Cal C}^\vee})$.   
The old picture is just the restriction of this one to the diagonal  $u = q^{-1}$.   
Our $D$-module has regular singularities along the divisor  $u = \infty$,  
so we may extend it to a vector bundle  $V$  to  
$\widetilde{S}^- = \text{Spec}({\Bbb C}[u^{-1}]) \times S$  invariant with  
respect to operator  $u\partial_u$.  Our lemma would follow if we show that  
for any  $N \gg 0$  one has  $a^K_j(u,q) \in u^{-N}V$  for all but finitely  
many $K$'s.  The action of the operator $u\partial_u$ on  $a^K_j(u,q)$  was  
computed in 3.4.7.1.  Namely, we have $u\partial_u(a^K_j(u,q)) =  
\langle \varphi(q) \otimes L_0(e^K_j)_{({\Bbb C}((t_1)), ut_1)} \otimes  
* e^K_j\rangle_{{\Cal C}^\vee_u}$, hence  $a^K_j(u,q)$  is a generalized  
eigenvector of  $u\partial_u$  with eigenvalue equal to an eigenvalue of  
$L_0$  at $e^K_j$.  Axiom a. above implies that for any   
$\overline{\mu} \in {\Bbb C}/{\Bbb Z}$  and 
$c \in {\Bbb R}$  the space ${\dsize\bigoplus \Sb\\ \mu = \overline{\mu}\  
\text{mod}\ {\Bbb Z} \\ \text{Re}\ \mu > c\endSb} r(I_j)_{{\Bbb C}(t)\mu}  
\subset r(I_j)_{{\Bbb C}((t))}$  is finite dimensional.  On the other hand,  
since $\langle X \otimes I_j \otimes * I_j\rangle_{{\Cal C}^\vee_u}$  is a  
lisse module, there are only finitely many $\overline{u} \in {\Bbb C}/{\Bbb Z}$ 
such that one has a section  which is a generalized eigenvector of $u\partial_u$  
with eigenvalue  mod ${\Bbb Z}$  equal to  $\overline{u}$.  This implies that 
for any  $c \in {\Bbb R}$  all but finitely many  $a^K_j$'s  are generalized  
eigenvectors of  $u\partial_u$  with Re (eigenvalue) $< c$.  This implies that 
all but finitely many of them lie in  $u^{-N}V$.  \hfill $\square$ 
\enddemo 
 
\demo{6.1.6\ Remark}  We may consider the situation when a smooth curve degenerates 
to a curve with several quadratic singular points. One trivially reformulates  
axiom $g$ for this situation; it is easy to see that this generalized version  
follows from axiom g. above (the case of one singular point).  
\enddemo 
 
\noindent 
6.1.7\ Here is an example of how axiom $g$ works.  Let  $C$  be a fixed  
curve, $A \subset C$ a finite set, $\{ \nu_a\},\ a \in A$,  1-jets of  
coordinates at $a$'s, $X \in {\Cal A}^{\otimes A}$, and $\varphi \in  
r(X)_{a,C}$.  Let  $x \in C\setminus A$  be a point, $t_x$  a  
parameter at $x$ and $\lambda_1,\ldots, \lambda_n \in {\Bbb C}$   
distinct complex numbers.  Let  $x_i(q)$  be ${\Bbb C}[[q]]$  points of  $C$   
defined by the formula  $x_i(o) = x, t_x(x_i(q)) = \lambda_iq$.  Put  
$t_i = t_{x/q} - \lambda_i$:  these are parameters at  $x_i$'s for $q \ne 0$.  
Let  $Y_1,\ldots, Y_n$  be objects in ${\Cal A}$, $\psi_i \in  
r (Y_i)_{{\Bbb C}((t))}$.  We would like to compute   
$\langle \varphi \in {\psi_1}_{({\Bbb C}((t_1)),t_1)} \otimes \cdots \otimes 
{\psi_n}_{({\Bbb C}((t_n)),t_n)}\rangle_C \in \langle X \otimes Y_1 \otimes  
\cdots \otimes Y_n\rangle_{(C,A,\{ x_i\}, \nu_A, dt_i(x_i)}$.  To do it one  
should blow up the point  $(x,0) \in C_S = C \times S$; denote this curve  
$C'_S$.  Clearly  $A, \{ x_i\}$  are  $S$-points of  $C'_S$, and we have  
parameters  $t_x, q/t_x$  at the (only) singular point of  $C'_0$.  The  
corresponding $S$-curve  $C_S^{'\vee}$ is constant: one has  
$C_S^{'\vee} = C_S \coprod {\Bbb P}^1_S$; the formal 
parameters at  $a_1 = x \in C_S,\ a_2 = \infty \in {\Bbb P}^1_S$  are   
$t_x$, $t^{-1}$, respectively.  We see that  $C'_S$  comes from   
$(C \coprod {\Bbb P}^1; x,\infty; t_s, t^{-1})$  via the construction 3.6.4. 
The points  $A, \{ x_i\}$  on  $C_S^{'\vee}$  are also constant, as well as  
coordinates  $t_i$:  one has  $x_i = \lambda_i \in {\Bbb P}^1, t_i =  
t-\lambda_i$.  Hence   
$$ 
\aligned 
&\langle X \otimes Y_1 \otimes \cdots \otimes Y_n\rangle_{(C;A,\{ x_i\};  
\nu_A, dt_i(x_i))}  
= {\dsize\bigoplus_j} \langle Y_1 \otimes \cdots \otimes  
Y_n \otimes I_j\rangle_{({\Bbb P}^1;\lambda_i,\infty;  
dt(\lambda_i), q^{-1}dt^{-1}(\infty))}\\  
&\qquad \otimes \langle *I_j \otimes  
X\rangle_{(C;x,A; dt_x(x), \nu_A)} 
\endaligned 
$$   
 
\noindent 
and  
$$ 
\aligned 
&\langle \varphi \otimes {\psi_1}_{({\Bbb C}((t_1)),t_1)} \otimes \cdots \otimes 
{\psi_n}_{({\Bbb C}((t_n)),t_n)}\rangle_C =  
\langle {\psi_1}_{({\Bbb C}((t-\lambda_1)),t-\lambda_1)} \otimes \cdots \otimes 
{\psi_n}_{({\Bbb C}((t-\lambda_n)),t-\lambda_n)} \\ 
&\quad \otimes  
e^K_j({\Bbb C}((t^{-1})), q^{-1}t^{-1})\rangle_{{\Bbb P}^1} \otimes   
\langle * e^K_{j\ ({\Bbb C}((tx)),tx)} \otimes \varphi \rangle_C. 
\endaligned 
$$ 

 
\subheading{6.2\ Global vertex operators}\ Assume we have an algebraic field  
theory as in 6.1.  Let  $C$ be a smooth compact curve, $A \subset C$  a  
finite set of points and  $\nu_a$, $a \in A$,  a 1-jet of parameters at  
$a$'s.   
 
\noindent 
6.2.1\ For an object  $X \in {\Cal A}^{\otimes A}$  we have a finite dimensional  
vector space  $\langle X\rangle_C$  and a linear map   
$\langle {\phantom X}\rangle_C: r(X)_{A_C} \longrightarrow \langle X\rangle_C$.   
Also for any $n$-tuple of points  $x_1,\cdots, x_n \in C\setminus A$,  
$x_i \ne x_j$  for  $i \ne j$, we have a linear map   
$\langle {\phantom X}\rangle_C: r(X)_{A,C} \otimes r({\Bbb 1})_{x_1,C}  
\otimes \ldots \otimes r({\Bbb 1})_{x_n,C} = $ 
 
\noindent 
$r(X\otimes {\Bbb 1} \otimes \cdots \otimes {\Bbb 1} )_{A \cup  
\{ x_1,\cdots, x_n\},C} \longrightarrow  
\langle X\otimes {\Bbb 1} \otimes \cdots \otimes {\Bbb 1} \rangle_C =  
\langle X\rangle_C$, where the last equality is 4.5.5 (ii)$'$.  Note that we  
need not  fix here 1-jets of parameters at  $x_i$'s since  $r({\Bbb 1})$  
is a $(\widetilde{\Cal T}, {\Cal V})_c$-module (see axiom b).  We may rewrite  
this as a linear map  
$$ 
V^A_{x_1,\cdots,x_n}: \otimes r(\Bbb 1)_{x_i,C} \longrightarrow  
r(X)^*_{A,C} \otimes \langle X\rangle_C. 
$$ 
 
\medpagebreak 
 
This construction may be rearranged in several ways: 
 
\noindent 
6.2.2\ Let the points  $x_1,\cdots,x_n$  vary. On $C^n$  we have a locally free 
${\Cal O}_{C^n}$-module  $r({\Bbb 1})^{\otimes n}_{C^n}$  with fibers  
$r({\Bbb 1})^{\otimes n}_{C^n(x_1,\cdots,x_n)} = \otimes  
r({\Bbb 1})_{x_i,C}$.  On  $U = (C\setminus A)^n\setminus\{\text{diagonals}\}$  
we have a morphism  $V^A: r{\Bbb 1})^{\otimes n}_U \longrightarrow  
\text{Hom}_{\Bbb C}(r(X)_{A,C}, \langle X\rangle_C \otimes {\Cal O}_U)$   
of  ${\Cal O}_U$-modules such that the value of  $V^A$  at $(x_1,\cdots,x_n)$ 
coincides with  $V^A_{x_1,\cdots,x_n}$.  For any open set  $W \subset U$  we  
get a map  
$$ 
V^A_H:  \Gamma(W, r({\Bbb 1})_W^{\otimes n} \otimes \Omega^n_W)  
\longrightarrow r(X)^*_{A,C} \otimes \langle X\rangle_C \otimes H^n_{DR}(W) 
$$ 
which is a composition of  $V \otimes id_{\Omega^n_W}$  and the canonical  
projection  $\Gamma(W, \Omega^n_W) \to H^n_{DR}(W)$.   
 
\medpagebreak 
 
\noindent 
6.2.3\ Assume that  $A = A_1 \sqcup A_2$  and $X = X_1 \otimes  X_2,\  
X_i \in {\Cal A}^{\otimes A_i}$.  Then  $r(X)_{A,C} =  
r(X_1)_{A_1,C} \otimes r(X_2)_{A_2,C}),\quad  r(X)^*_{A,C} =  
\text{Hom}(r(X_1)_{A_1,C}, r(X_2)^*_{A_2,C})$. Let us fix a formal  
parameter  $t_a$  at $\alpha$ such that  $dt_a(a) = \nu_a$.  These identify  
$r(X_i)_{A_i,C}$  with ``coordinate modules"  
$r(X_i)_{{\Bbb C}((t_{A_i}))}$  and  $r(X_2)^*_{A_i,C}$  with a  
completion  $r(*X_2)^{{}^\wedge}_{{\Bbb C}((t_{A_2}))}$  of\quad    
$r(*X_2)_{{\Bbb C}((t_{A_2}))}$.  So we may rewrite the above  
$V_{x_1,\cdots,x_n}$  as  
$$ 
V^{A_1,A_2}_{x_1,\cdots,x_n}:\ \otimes r(\Bbb 1)_{x_i,C} \otimes 
\langle X_1 \otimes X_2\rangle^*_C \longrightarrow  
\text{Hom}(r(X_1)_{{\Bbb C}((t_{A_1}))},  
r(*X_2)^{{}^\wedge}_{{\Bbb C}((t_{A_2}))})   . 
$$ 
  
 
The linear operators in the image of this map are called vertex operators. 
 
\noindent 
6.2.4\ Now assume that  $X_1 = Y,\  X_2 = * {\Cal F}_C^{A_1,A_2}(Y)$, where  
${\Cal F}_C^{A_1,A_2}:  {\Cal A}^{\otimes A_1} \to {\Cal A}^{\otimes A_2}$  is 
the fusion functor from 4.6.  Then  $\langle X_1 \otimes X_2\rangle^*_C =  
\text{Hom}({\Cal F}_C^{A_1,A_2}(X_1), *X_2)$  has a canonical element  
id$_{*X_2}$; hence we get  
$$ 
V^{A_1,A_2}_{x_1,\cdots,x_n} :  \otimes r(\Bbb 1)_{x_i,C}\  
\longrightarrow\ \text{Hom}(r(Y)_{{\Bbb C}((t_{A_1}))},  
r({\Cal F}_C^{A_1,A_2}(Y))^{{}^\wedge}_{{\Bbb C}((t_{A_2}))}). 
$$ 
 
Here are the first properties of vertex operators in this setting, that follow 
directly from the axioms.   
 
\noindent 
6.2.5\ For  $j \in \{ 1,\ldots,n\}$  and  $\varphi \in {\bigotimes_{i\ne j}}  
r(\Bbb 1 )_{x_i,C}$\ one has   
$V^{A_1,A_2}_{x_1,\ldots,\widehat{x}_j,\ldots,x_n}(\varphi) =  
V^{A_1,A_2}_{x_1,\ldots,x_n} (\varphi \otimes 1_{x_j})$. 
 
\noindent 
6.2.6\ Put  ${\Cal T}(C\setminus A, x_1,\ldots,x_n) = \{ \tau \in  
{\Cal T}(C\setminus A):  \tau(x_i) = 0\} \subset {\Cal T}(C\setminus A)$.  Then 
the linear map  $V^{A_1,A_2}_{x_1,\ldots,x_n}$  commutes with the 
${\Cal T}(C\setminus A,x_1,\ldots,x_n)$-action.  Here   
${\Cal T}(C\setminus A, x_1,\ldots,x_n)$ acts on the left hand side via  
${\Cal T}(C\setminus A, x_1,\ldots,x_n) \to {\Cal T}_{(x_i)o} \subset 
\widetilde{\Cal T}_{(x_i)}$ ($=$ Virasoro algebra at $x_i$) and on the right  
hand side via the map  ${\Cal T}(C\setminus A) \to \widetilde{\Cal T}_{(A)}$  
from 2.3.4.  In particular, any vertex operator $F$ transforms via a  
finite dimensional representation of  ${\Cal T}(C\setminus A, x_1,\ldots,x_n)$  
and  $F$ is fixed by a Lie subalgebra of  ${\Cal T}(C\setminus A)$  that  
consists of fields vanishing to sufficiently high order at the $x_i$'s.   
 
\noindent 
6.2.7\ Let  $C'$  be another curve,  $A' = A_2 \sqcup A_3 \subset C'$  a  
finite set of points, $t_{a'}$  formal parameters at  $a' \in A'$, and  
$\{ x'_1, \cdots, x'_m\} \subset C'\setminus A'$.  Let  $(C \circ C')_q$  be  
the   ${\Bbb C}[[q]]$-curve with zero fiber obtained from  $C \sqcup C'$  by 
clutching together the points of $A_2$  in  $C,C'$, and where the $q$-deformation  
comes from using parameters  $t_{a_2},t_{a'_2}$  according to 3.6.4.  Then  
$A_1 \sqcup A_3 \sqcup \{ x_1,\ldots,x_n\} \sqcup \{ x'_1,\ldots, x'_m\}$  is a 
finite set of  ${\Bbb C}[[q]]$-points of  $(C \circ C')_q$,  and hence we have our 
vertex operators map  $V^{A_1,A_3}_{x_1,\ldots,x_n,x'_1,\ldots,x'_m}:   
\otimes r(\Bbb 1 )_{x_i,C} \otimes r(\Bbb 1)_{x'_j,C'}\  
\longrightarrow\ \text{Hom}(r(Y)_{{\Bbb C}((t_{A_1}))}$, 
 
\noindent  
$r({\Cal F}^{A_1,A_3}_{(C\circ C')_q}(Y)^{{}^\wedge}_{{\Bbb C}((t_{A_3}))})$.   
On the other hand, it is easy to see that ``topologically" $(C \circ C')_q$  
coincides with ``topological" composition $C_q \circ C'$  from 4.6.1, where  
$$ 
C_q = (C, dt_{a_1}(a_1), q^{-1}dt_{a_2}(a_2)) \in {\Cal M}_A,\  
a_1 \in A_1, a_2 \in A_2. 
$$ 
Hence, by 4.6.1, one has   
${\Cal F}^{A_1,A_3}_{(C\circ C')_q} = {\Cal F}^{A_2,A_3}_{C'}   
\circ {\Cal F}^{A_1,A_2}_{C_q}$.   
 
Our next property, that follows directly from axiom g, is:   
 
for any $\varphi \in \otimes r(\Bbb 1)_{x_i,C},\ \varphi' \in  
\otimes r(\Bbb 1 )_{x'_j,C'}$  one has   
$$ 
V^{A_1,A_3}_{x_1,\cdots,x_n,x'_1,\cdots,x'_m} (\varphi \otimes \varphi') =  
V^{A_2,A_3}_{x'_1,\cdots,x'_m}(\varphi') \circ V^{A_1,A_2}_{x_1,\cdots,x_n}, 
$$ 
where composition of ``infinite matrixes" is understood in a way similar to  
6.1.5.   
\vskip .2in 
 
 
\subheading{6.3\ Local vertex operators}\ Assume we have a field theory as in 
6.1.   
 
\noindent 
6.3.1\ Let  $C$  be a smooth curve.  Denote by  $\widetilde{C}$  the cotangent 
bundle of $C$  with zero section removed; so a point of  $\widetilde{C}$  is a  
pair $(x, \nu_x)$,  $x \in C$, $\nu_x$  is a 1-jet of coordinates at $x$.  Any  
object  $X \in {\Cal A}$  defines a locally free ${\Cal O}_{\widetilde{C}}$-module 
$r(X)_{\widetilde{C}}$  with fibers  $r(X)_{(x,\nu_x)} =  
r(X)_{x,\nu_x,C}$.  A choice of a family of local parameters defines a  
trivialization of  $r(X)_{\widetilde{C}}$.  More precisely, let $t$  be a  
function on a formal neighbourhood of the diagonal  $\Delta: \widetilde{C}\  
\hookrightarrow\ \widetilde{C} \times C$, $\Delta(x,\nu_x) = (x, \nu_x,x)$,  
such that  $t|_\Delta = 0$, $d_{x_2} t(x,\nu_x,x) = \nu_x$  (so   
$t_{(x,\nu_x)} = t(x,\nu_x,\cdot)$  is a formal parameter at $x$);  such a $t$ 
defines a trivialization  $s^t: r(X)_{\widetilde{C}})\   
{\lower .5ex\hbox{$\vrule height 1ex width 0pt\overrightarrow{\sim}$}}\  
r(X)_{{\Bbb C}((t))} \otimes {\Cal O}_{\widetilde{C}}$.   
 
This $r(X)_{\widetilde{C}}$ is a $D_{\widetilde{C}}$-module in a canonical 
way; the $D$-module structure comes from the ${\Cal T}_{{\Bbb C}((t))^{-1}}$-action  
on  $r(X)_{{\Bbb C}((t))}$.  Explicitly, a vector field  
$\tau \in {\Cal T}_{\widetilde{C}} \subset D_{\widetilde{C}}$  acts on  
$r(X)_{\widetilde{C}}$  as follows.  Choose (locally) a family $t$ of local  
parameters as above.  Let  $\nabla_0$  be the flat connection that corresponds  
to the trivialization  $S^t$.  Let  $\widetilde{\tau}^t \in  
\widetilde{\Cal T}_{{\Bbb C}((t))} \otimes {\Cal O}_{\widetilde{C}}$  be the  
section defined by formula  $\widetilde{\tau}^t =  
{\Cal S}_{{\Bbb C}[[t]]}({\Cal T}_{(x_1,\nu_{x_1})} (t)\partial_t)$:  here  
${\Cal T}_{(x_1,\nu_{x_1})}$  is a vector field on $\widetilde{C} \times C$   
equal to $\tau$ in the $\widetilde{C}$-directions and to $0$ in the $C$ directions  
(hence  ${\Cal T}_{(x_1,\nu_{x_1})}(t)$  is a function on the formal neighbourhood  
of $\Delta$), and  ${\Cal S}_{{\Bbb C}[[t]]}:  {\Cal T}_{{\Bbb C}[[t]]}	\to 
\widetilde{\Cal T}_{{\Bbb C}((t))}$  was defined in 3.4.1.  Now for a section  
$\varphi$  of  $r(X)_{\widetilde{C}}$  one has $\tau(\varphi) =  
\nabla_0(\tau)(\varphi) - \widetilde{\tau}^t(\varphi)$, where  
$\widetilde{\tau}^t(\varphi)$  is the $\widetilde{\Cal T}_{{\Bbb C}((t))}$-action  
on  $r(X)_{{\Bbb C}((t))}$. 
 
\medpagebreak 
 
\demo{6.3.2\ Remarks}\ (i)  One may explain the  $D_{\widetilde{C}}$-module  
structure on  $r(X)_{\widetilde{C}}$ as follows.  We have two natural  
actions of the Lie algebra  ${\Cal T}_C$  on  $r(X)_{\widetilde{C}}$.  The  
first one -- ``Lie derivative" -- comes since  $r(X)_{\widetilde{C}}$  
is a natural sheaf, hence symmetries of $C$  (and infinitesimal ones also)     
act on it.  The second is an ${\Cal O}$-linear action that comes because the  
fibers of  $r(X)_{\widetilde C}$  
are Virasoro modules (using the splitting  ${\Cal S}_{{\Cal O}^\wedge_x}$).  Now 
the $D$-module action of vector fields is the difference of these two actions. 
 
\noindent 
(ii)\ For any \'etale map  $f: C' \to C$  one has a canonical isomorphism  
$f^*_r(X)_{\widetilde{C}} = r(X)_{\widetilde{C}'}$  of   
$D_{\widetilde{C}'}$-modules. 
 
\noindent 
(iii)\ If  $d_X = id_X$  (see 4.5), e.g., if $X =\Bbb 1$, then $r(X)$  is 
actually a  $(\widetilde{\Cal T}, {\Cal V})$-module, hence  $r(X)_{\widetilde{C}}$ 
comes from a canonical $D$-module  $r(X)_C$  on $C$.   
 
\medpagebreak 
 
\noindent 
6.3.3\ For  $X_1,\cdots,X_n \in {\Cal A}$  consider the $D$-module  
$\boxtimes_i r(X_i)_{\widetilde{C}} = r(X_1)_{\widetilde{C}} \boxtimes  
\ldots \boxtimes r(X_n)_{\widetilde{C}}$  on  $\widetilde{C}^n$.  If  $C$ 
is compact, we also have a lisse $D$-module  $\langle X_1 \otimes \cdots \otimes 
X_n\rangle_{\widetilde{C}}$ on $\widetilde{C}\setminus \{ \text{diagonals}\}$  
with regular singularities along the diagonals; the fiber of  
$\langle X_1 \otimes \cdots \otimes X_n\rangle_{\widetilde C}$ over  
$(x_1,\nu_1,\cdots,x_n,\nu_n)$  is  
$\langle X_1 \otimes \cdots \otimes X_n\rangle_{(C,\{ x_i\},\{\nu_i\})}$.  By 
6.1.2 we have a canonical morphism of  $D_{\widetilde{C}^n}$-modules  
$\langle\quad \rangle_{\widetilde{C}}:  \boxtimes r(X_i)_{\widetilde{C}}\  
\to\ j_*\langle \otimes X_i\rangle_{\widetilde{C}}$, where   
$j:  \widetilde{C}^n\setminus \{ \text{diagonals}\} \hookrightarrow  
\widetilde{C}$. 
 
\noindent 
6.3.4\ For a moment let us drop the compactness assumption on $C$; we will work  
locally.  For  $X \in {\Cal A}$  let  $r(X){{}^\wedge}_{{\Cal C},C^n}$  be the  
completion of  $r(X)_{\widetilde{C}} \boxtimes {\Cal O}_{C^n}$  around the  
diagonal  $\Delta: \widetilde{C} \to \widetilde{C} \times C^n$,\  
$\Delta(x,\nu_x) = (x,\nu_x; x,\cdots,x)$.  A choice of a family of local  
parameters $t = (t_{x,\nu_x})$  identifies sections of  
$r(X)^{\wedge}_{\widetilde{C}, C^n}$  with formal power series  
$\Sigma m_{i_1,\cdots,i_n} t^{i_1}_1 \cdots t^{i_n}_n$, where  
$m_{i_1,\cdots,i_n}$  are sections of  $r(X)_{\widetilde{C}}$  and  
$t_i(x_0,\nu_{x_0},x_1,\cdots,x_n) = t_{(x_0,\nu_{x_0})}(x_i)$.  Then  
$r(X)^\wedge_{\widetilde{C},C^n}$  is a (non quasicoherent)   
$D_{\widetilde{C} \times C^n}$-module in an obvious manner. Let  
${\Cal O}^\#_{\widetilde{C} \times C^n} \supset  
{\Cal O}_{\widetilde{C} \times C^n}$  denote the sheaf of functions having  
(meromorphic) singularities at diagonals  $x_i = x_j,$\ $i,j \ge 0$.  Put 
$r(X)^\#_{\widetilde{C},C^n} := {\Cal O}^\#_{\widetilde{C} \times C^n}  
\otimes_{{\Cal O}_{\widetilde{C}\times C^n}} r(X)^\wedge_{\widetilde{C},C^n}$:   
this is also a  $D_{\widetilde{C} \times C^n}$-module.  A section of  
$r(X)^\#_{\widetilde{C} \times C^n}$  is a formal series  
$$ 
\prod (t_i -t_j)^{-a_{ij}}(\Sigma m_{i_1\cdots i_n} t_1^{i_1} \cdots t_n^{i_n}),\  
a_{ij} \ge 0. 
$$ 


 
Now let us define the ``local" vertex operators: 
 
\proclaim{6.3.5\ Lemma}  There is a canonical morphism of   
$D_{\widetilde{C} \times C^n}$-modules 
$$ 
\mu:  r(\Bbb 1 )_C \boxtimes \cdots \boxtimes r(\Bbb 1 )_C  
\boxtimes r(X)_{\widetilde{C}}\ \longrightarrow\  
r(X)^\#_{\widetilde{C},C^n}  
$$ 
such that (assuming $C$ is compact) for any   
$(x,\nu_x; y_1,\nu_{y_1};\cdots ; y_m,\nu_{y_m}) \in \widetilde{C} \times  
\widetilde{C}^m$,\ $x \ne y_i$,  $y_i \ne y_j$  for  $i \ne j$,  objects  
$Y_i \in {\Cal A}$, an element  $\psi_x \in r(X)_{x,\nu_x},  
\psi_{y_i} \in r(Y_i)_{y_i,\nu_{y_i}}$  and a section   
$\varphi_1,\cdots ,\varphi_n$  of $r(\Bbb 1)_C$  in a  
neighbourhood of $x$ one has  
$$ 
\langle \varphi_1 \otimes \cdots \otimes \varphi_n \otimes \psi_x \otimes  
\cdots \otimes \psi_{y_m}\rangle_{\widetilde{C}} =  
\langle \mu (\varphi_1 \otimes \cdots \otimes \varphi_n \otimes \psi_x) \otimes  
\psi_{y_1} \otimes \cdots \otimes \psi_{y_n}\rangle_{\widetilde{C}} 
$$ 
(as meromorphic functions on a formal neighbourhood of $(x,\ldots,x) \in C^n$  
with values in $\langle X \otimes Y_1 \otimes \cdots \otimes  
Y_m\rangle_{(C,\{ x,y_i\},\{\nu_x,\nu_{y_i}\})}$  identified with  
$\langle {\Bbb 1} \otimes \cdots \otimes \Bbb 1 \otimes X \otimes  
Y_1 \otimes \cdots \otimes Y_m\rangle$  via 4.5.5 (ii)$'$). 
\endproclaim 
 
\demo{Proof - construction}  We will write an explicit formula for $\mu$.  To do  
this consider first  ${\Bbb P}^1$  with the standard parameter $t$.  So $t$  defines  
a family of local parameters  $t_x = t-x$  on ${\Bbb P}^1\setminus \{\infty\}$,  
and hence we have a trivialization   
$s^t: r(\Bbb 1_{{\Bbb P}^1\setminus \{\infty\}} =  
r(\Bbb 1)_{{\Bbb C}((t))} \otimes  
{\Cal O}_{{\Bbb P}^1\setminus \{\infty\} }$.  For  $\varphi \in   
r(\Bbb 1)_{{\Bbb C}((t))}$  we denote by  $\varphi^t$  the  
corresponding ``constant" section of   
$r(\Bbb 1)_{{\Bbb P}^1\setminus \{\infty\}}$. 
\enddemo 
 
Now for  $\varphi_1,\cdots,\varphi_n \in r(\Bbb 1)_{{\Bbb C}((t))}$  
and  $x_1,\ldots,x_n \in {\Bbb P}^1\setminus \{\infty\},\ x_i \ne x_j$ for  
$i \ne j$, consider the vertex operator  
$V^{0,\infty}_{x_1,\cdots,x_n}(\varphi^t_1 \otimes \cdots \otimes \varphi^t_n): 
r(X)_{{\Bbb C}((t))} \longrightarrow r(X)^{{}^\wedge}_{{\Bbb C}((t))}k$  
from 6.2.4 (here we identified the module  $r(X)_{{\Bbb C}((t^{-1}))}$   
at $\infty$  with  $r(X)_{{\Bbb C}((t))}$  via  $t^{-1} \longmapsto t)$. 
In fact, this operator lies in  End $r(X)$. 
 
\medpagebreak 
 
\demo{[Proof}  For any  $a \in {\Bbb C}^*$  one has  $t_{ax} = a(t-x)$; hence the 
automorphism  $x \longmapsto ax$  of  ${\Bbb P}^1$  acts on  
$r({\Bbb 1})_{{\Bbb P}^1}$ (according to 6.3.2) by the formula   
$\varphi^t \longmapsto (a^{L_0}\varphi)^t$.   This implies immediately that  
if  $L_0\varphi_i = n_i\varphi_i$, then  
$V^{0,\infty}_{x_1,\cdots, x_n}(\otimes \varphi^t_i)(L_0e) =  
(L_0 + n_1 + \cdots + n_n) V^{0,\infty}_{x_1,\cdots, x_n}(e)$.  Hence  
$V^{0,\infty}_{x_1,\cdots, x_n}(\otimes \varphi^t_i)$  maps $L_0$-generalized  
eigenspaces in  $r(X)_{{\Bbb C}((t))}$  to ones in  
$r(X)^\wedge_{{\Bbb C}((t))}$; since the sum of these equals  
$r(X)_{{\Bbb C}((t))}$, we see that   
$V^{0,\infty}_{x_1,\cdots, x_n}(\otimes \varphi^t_i)$ maps  
$r(X)_{{\Bbb C}((t))}$ to $r(X)_{{\Bbb C}((t))}$.] 
 \enddemo
 
\medpagebreak 
 
Clearly, $V^{0,\infty}_{x_1,\cdots, x_n}(\varphi^t_1 \otimes \cdots \otimes  
\varphi^t_n)$  is a meromorphic function on  
$({\Bbb P}^1\setminus \{ 0,\infty\})^n\setminus \{ \text{diagonals}\}$  with  
values in End $r(X)_{{\Bbb C}((t))}$.  Put  
$\mu(\varphi^t_1 \otimes \cdots \otimes \varphi^t_n \otimes \psi_0) =  
V^{0,\infty}_{x_1,\cdots, x_n}(\varphi^t_1 \otimes \cdots \otimes  
\varphi^t_n)(\psi_0)$  for  $\psi_0 \in r(X)_{{\Bbb C}((t))}$:  we will 
consider $\mu(\quad )$ as a formal power series in variables  
$t_1,\cdots, t_n, t_i = t(x_i)$, with poles along diagonals  $t_i = t_j$,  
with values in $r(X)_{{\Bbb C}((t))}$.   
 
Now consider our curve  $\widetilde{C}$.  Choose a family of parameters  $t$.   
It defines a trivialization  $r({\Bbb 1})_C \boxtimes \cdots \boxtimes  
r({\Bbb 1})_C \boxtimes r(X)_{\widetilde{C}}\   
{\buildrel \sim\over\longrightarrow}\ r({\Bbb 1})_{{\Bbb C}((t))}^{\otimes n}  
\otimes r(X)_{{\Bbb C}((t))} \otimes {\Cal O}_{\widetilde{C} \times C^n}$ 
in a formal neighbourhood of the diagonal.  We put   
$\mu(\varphi^t_1 \otimes \cdots \otimes \varphi^t_n \otimes \psi_{x,t}) =  
\mu(\varphi^t_1 \otimes \cdots \otimes \varphi^t_n \otimes \psi_{{\Bbb C}((t))t})_{x,t}$ 
in obvious notations (so we write down the above $\mu$ on our curve in the 
coordinates  $t_x$  for each  $x \in C$).  It is easy to see that $\mu$,  
so defined, is independent of choice of the family of parameters and is a  
morphism of $D$-modules.   
 
To prove the correlators formula in 6.3.5 one proceeds as in 6.1.7:  we should  
consider the curve  $C'_c$ as in 6.1.7 over  ${\Bbb C}[[q]]$  and apply  
axiom g.  \hfill $\square$ 
 \enddemo
 
\medpagebreak 
 
We will often write $\mu(\varphi_1 \otimes \cdots \otimes \varphi_n \otimes  
\psi) = \varphi_1(x_1) \cdots \varphi_n (x_n)\psi(x) \in  
{\dsize\prod_{i,j}} (x_i-x_j)^{-N} {\Bbb C}[[x_1-x,\cdots, x_n-x]] \otimes  
r(X)_x$.  The composition property 6.2.7 for global vertex operators  
implies this associativity property of $\mu$: 
 
\noindent 
6.3.6\ One has  
 
$\varphi_1(x_1) \cdots \varphi_n(x_n)\psi(x) = $ 
  
$\varphi_1(x_1)(\varphi_2(x_2)(\cdots (\varphi_n(x_n)\psi(x)) \cdots) \in  
{\Bbb C}((x_1-x((\cdots ((x_n-x))\ldots ))) \otimes r(X)_x. $ 
 
Also if one of the $\varphi_i$'s is equal to 1, we may delete it.  
 
\bigpagebreak 
 
\subheading{6.4\ Chiral algebra}  Consider the three step complex   
${\Cal L}_{C^\bullet} = ({\Cal L}_2 \to {\Cal L}_1 \to {\Cal L}_0)$  of  
sheaves for the  Zariski or \'etale topology of $C$.  Here ${\Cal L}_2 =  
r({\Bbb 1})_C$, ${\Cal L}_1 = \omega \otimes_{{\Cal O}_C}  
r({\Bbb 1})_C$, the differential  $d: {\Cal L}_2 \to {\Cal L}_1$  is the  
de Rham differential, and  ${\Cal L}_0 = {\Cal L}_1/d{\Cal L}_2 =  
{\Cal H}^1_{DR}(r({\Bbb 1})_C)$  is the sheaf of de Rham cohomology 
with coefficients in the   $D_C$-module  $r({\Bbb 1})_C$, and  
$d: {\Cal L}_1 \to {\Cal L}_0$  is the projection.  
 
\medpagebreak 
 
\noindent 
6.4.1\ For sections  $\gamma_1, \gamma_2$  of  ${\Cal L}_1$  we define a  
section  $\gamma_1 * \gamma_2$ of ${\Cal L}_1$  by the formula $\gamma_1 * \gamma_2  
= Res_1 \mu(\gamma_1 \otimes \gamma_2)$, and a section  $\{ \gamma_1,\gamma_2\}  
\in {\Cal L}_2$  by the formula $\{ \gamma_1,\gamma_2\} =  
\widetilde{Res} \mu(\gamma_1 \otimes \gamma_2)$.  Here $\gamma_1 \otimes \gamma_2$ 
is a section of  ${\Cal L}_1 \boxtimes {\Cal L}_1 = \Omega^2_{C \times C}  
\otimes_{{\Cal O}_{C\times C}} (r({\Bbb 1})_C \boxtimes r({\Bbb 1})_C)$, 
$\mu(\gamma_1 \otimes \gamma_2)$  is a section of  $\omega_C \boxtimes {\Cal L}_1  
= \Omega^2_{C\times C} \otimes p^*_2r({\Bbb 1})_C$  with poles along the  
diagonal, Res$_1$  is residue around the diagonal along the first variable, and  
$\widetilde{Res}$  was defined in 2.2.4.  Now the lemma 6.3.5 implies immediately 
that  $d(\{ \gamma_1,\gamma_2\}) = \gamma_1 * \gamma_2 + \gamma_2 * \gamma_1$   
and for  $\varphi \in {\Cal L}_2$  one has $(d\varphi) * \gamma = 0$.  Define the  
bracket $[\ ,\ ]: {\Cal L}_\bullet \otimes {\Cal L}_\bullet \to {\Cal L}_\bullet$  
by the formula $[d\gamma_1,d\gamma_2]_{0,0} = d(\gamma_1 *\gamma_2)$,\  
$[d\gamma_1,\gamma_2]_{0,1} = - [\gamma_2,d\gamma_1]_{1,0} = \gamma_1 * \gamma_2$, 
$[\gamma_1,\gamma_2]_{1,1} = \{ \gamma_1,\gamma_2\}$ for $\gamma_i \in {\Cal L}_1$. 
The associativity property 6.3.6 implies 
 
\proclaim{6.4.2\ Lemma}  This bracket provides ${\Cal L}$ with the structure of  
DG Lie algebra.\hfill $\square$ 
\endproclaim 
 
This DG Lie algebra (or rather its zero component ${\Cal L}_0$) is called the  
chiral Lie algebra of our field theory. 
 
\medpagebreak 
 
\noindent 
6.4.3\ Consider a canonical embedding  $i: {\Cal O}_C \to r({\Bbb 1})_C$  
of  $D_C$-modules, $i(f) = f \cdot 1$.   
 
Denote by $C_\bullet$  the three step complex  $C_2 = {\Cal O}_C\  
{\buildrel d\over\longrightarrow}\ C_1 = \omega_C \to C_0 = {\Cal H}$; here  
${\Cal H} = {\Cal H}^1_{DR}$ and the differential  $C_1 \to C_0$  is the canonical  
projection.  We get a canonical morphism  $i: C_\bullet \to {\Cal L}_\bullet$  
of complexes, $i(f) = f \cdot 1$.  One may see that $i$  is actually an  
embedding (for  $i_0$  this will follow from 6.4.6), and obviously $i(C_\bullet)$ 
lies in the center of the chiral algebra. 
 
\medpagebreak 
 
\noindent 
6.4.4\ For any  $x \in {\Cal A}$  consider the  $D_{\widetilde{C}}$-module  
$r(X)_{\widetilde{C}}$.  The formula  $\gamma(m) = Res_1 \mu(\gamma  
\otimes m)$  for  $\gamma \in {\Cal L}_0$, $m \in r(X)_{\widetilde{C}}$  
defines a canonical action of ${\Cal L}_0$  on $r(X)_{\widetilde{C}}$   
that commutes with the $D_{\widetilde{C}}$-action. 
 
\medpagebreak 
 
\noindent 
6.4.5\ For any local field $F$ we may consider the ``local" version  
${\Cal L}_{F^{\bullet}}$ of the above  ${\Cal L}_{C^{\bullet}}$.  This is  
a differential graded Lie 
algebra constructed in a way similar to 6.4.1.  If  $F = F_x$  is a  
local field at a point  $x \in C$, then ${\Cal L}_{F_x^2} = F_x  
\otimes_{{\Cal O}_C} {\Cal L}_{C^2}$,  ${\Cal L}_{F_x^1} = F_x  
\otimes_{{\Cal O}_C} {\Cal L}_{C^1}$, ${\Cal L}_{F_x^0} = H^1_{DR}(F_x,  
r({\Bbb 1})_C) = {\Cal L}_{F_x^1}/d{\Cal L}_{F_x^2}$.  For any $X \in  
{\Cal A}$  we have a canonical map  ${\Cal L}_{F^0} \otimes r(X)_F \to  
r(X)_F$, $\gamma \otimes m \longmapsto \gamma(m) = Res_0 \mu(\gamma \otimes  
m)$.  Here $\mu(\gamma \otimes m)  \in H^1_{DR}(F) \otimes r(X)_F$  and one  
has (cf. 6.4.4): 
 
\medpagebreak 
 
\proclaim{6.4.6\ Lemma}  This map defines a representation of the Lie algebra  
${\Cal L}_{F^0}$ on  $r(X)_F$.  The central subalgebra   
${\Bbb C} {\buildrel i\over\hookrightarrow}\ {\Cal L}_{F^0}$, $i(a) =  
a {\frac{dt}{t}}$, (see 6.4.3) acts on  $r(X)_F$  by the formula  
$i(a)(m)~=~am$.\hfill $\square$				 
\endproclaim 
 
\medpagebreak 
 
In particular, $i({\Bbb C}) \ne 0$; this implies, by degeneration arguments,  
that  $i: C_0 \to {\Cal L}_0$  is an embedding in the ``global" situation. 
 
Now assume that $C$ is compact, $x_1,\cdots,x_n \in C$, $x_i \ne x_j, \nu_i$  
are 1-jets of parameters at  $x_i$'s, and $X_1,\cdots, X_n \in {\Cal A}$.   
Put $U = C\setminus \{ x_1,\cdots, x_n\}$.  Consider the pairing\  
$\langle\quad \rangle_C: r(X_1)_{x,\nu_1,C} \otimes \cdots \otimes  
r(X_n)_{x_n,\nu_n,C} \longrightarrow \langle X_1 \otimes \cdots \otimes  
X_n\rangle_{C,x_i,\nu_i}$.  We have an obvious ``localization" morphism 
${\Cal L}_0(U) \to {\Cal L}_0(F_{x_i})$, hence a natural action of  
${\Cal L}_0(U)$  on $\otimes r(X_i)_{x_i,\nu_i,C}$.   
 
\medpagebreak 
 
\proclaim{6.4.7\ Lemma}  The morphism $\langle\quad \rangle_C$  is  
${\Cal L}_0(U)$-invariant. 
\endproclaim 
 
\demo{Proof}  Stokes formula:  we rewrite for $\ell \in {\Cal L}_0(U) =  
\Omega^1 \otimes r({\Bbb 1})_U$  the sum \newline $\Sigma\langle \varphi_1 \cdots 
\ell (\varphi_i) \cdots \varphi_n\rangle$  as $\Sigma\ \text{Res}_{x=x_i}  
\langle \ell (x) \varphi(x_1) \cdots \varphi(x_n)\rangle$. \hfill $\square$ 
\enddemo 
 
\bigpagebreak 
 
\subheading{6.5\ Stress-energy tensor} TO BE REWRITTEN! POSSIBLE MISTAKES!

 For any local field $F$ consider the  
linear map  ${\Cal T}_{F-2}/{\Cal T}_{F-1} \to r({\Bbb 1})_F/{\Bbb C} \cdot 1$, 
$\tau \longmapsto \tau(1)$  (see 3.4.1; recall that $1$ is fixed by  
${\Cal T}_{F-1}$ by axiom c).  The one-dimensional space   
${\Cal T}_{F-2}/{\Cal T}_{F-1}$ canonically coincides with the fiber at $0$ of 
${\Cal T}^{\otimes 2}$.  Tensoring this map with the dual line, we get for any  
curve $C$ a canonical section $T$ of  $\omega^{\otimes 2}_{C} \otimes {{\Cal O}_C}  
(r(\Bbb 1)_C/{\Cal O}_C)$.  This section is called the stress-energy tensor. 
Multiplication by $T$ defines a canonical map  ${\Cal T}_C \to  
\omega_{C} \otimes {{\Cal O}_C} (r(\Bbb 1)_C/{\Cal O}_C) =  
{\Cal L}_1/C_1\ {\buildrel d\over\longrightarrow} {\Cal L}_0/C_0$  (see 6.4.3). 
 
\medpagebreak 
 
\proclaim{6.5.1\ Lemma}  (i)  The composition  ${\Cal T} \to {\Cal L}_0/C_0$   
is a morphism of Lie algebras. 
 
\noindent 
(ii) The corresponding ``local" projective action  (see 6.4.5, 6.4.6) of  
${\Cal T}_F \subset  {\Cal L}_{0F}/{\Bbb C}$  on $r(X)_F$  coincides  
with the canonical Virasoro action. 
\endproclaim 
 
\demo{Remark}  One should have a canonical isomorphism between the induced  
extension of ${\Cal T}$  by  $C_0 = {\Cal H}$ and the Virasoro extension from  
\S 2, but we do not know how to establish it at a moment. 
\enddemo 
 
\demo{Proof}  Let us sketch a proof of (ii); one proves (i) in a similar way.   
We may assume that  $F = {\Bbb C}((t))$.  Let us compute the action of the  
operator  $L_K := t^{K+1} \partial_t \cdot T \subset {\Cal L}_{{\Bbb C}((t))^o}/{\Bbb C}$ 
on  $r(X)_{{\Bbb C}((t))}$.  Take  $e \in r(X)_{{\Bbb C}((t))},  
e^* \in r(*X)_{{\Bbb C}((t^{-1}))}$.  Consider the function   
$\nu(z) = \langle {\frac{1}{t-z}} \partial_{t-z} (1_z) \cdot e \cdot  
e^*\rangle_{{\Bbb P}^1}$;   here $z \in {\Bbb P}^1\setminus \{ 0,\infty\}$,  
$\langle\quad \rangle_{{\Bbb P}^1}$  is the correlator for fields  
${\frac{1}{t-z}} \partial_{t-z} (1_z) \in r(\Bbb 1)_{{\Bbb C}((t-z)),t-z}, 
e, e^*$  at points  $z,0,\infty$.  By definition, the matrix coefficient  
$\langle L_K(e),e^*\rangle $  is equal to  Res$_{z=0} z^{K+1} \nu(z)dz$.  We  
have the invariance property  $\langle {\frac{1}{t-z}} \partial_{t-z} (1_z)  
\cdot e \cdot e^*\rangle + \langle (1_z) \cdot {\frac{1}{t-z}} \partial_{t}e  
\cdot e^*\rangle + \langle (1_z) \cdot e \cdot {\frac{1}{t-z}} \partial_{t}  
e^*\rangle = 0$.  Deleting $1_z$  by $ax \cdot e$, we get   
$\langle L_K(e), e^*\rangle = - Res_{z=0}(\langle {\frac{1}{t-z}} \partial_te  
\cdot e^*\rangle + \langle e \cdot {\frac{1}{t-z}} \partial_t e^*\rangle)  
\cdot Z^{K+1} dz$.  To compute  ${\frac{1}{t-z}} \partial_te$  one should  
expand  ${\frac{1}{t-z}}$ around $t = 0$, and to compute  
${\frac{1}{t-z}} \partial_t e^*$  one should expand  ${\frac{1}{t-z}}$  at  
$t = \infty$.   
 
\medpagebreak 
 
Hence   
$$ 
\langle L_K e,e^*\rangle = - Res_{z=0} z^{K+1}(- \langle \sum_{n\ge 0}  
z^{-n-1} t^n \partial_te, e^*\rangle + \langle e,  
\sum_{n\ge 0} z^nt^{-n-1}\partial_t e^*\rangle)dz = 
\langle t^{K+1} \partial_te, e^*\rangle, 
$$ 
since  $\langle t^a\partial_t e,e^*\rangle + \langle e,t^a \partial_t e^*\rangle  
= 0$.  We see that  $L_K = t^{K+1} \partial_t$, q.e.d. \hfill $\square$ 
 \enddemo
 
\bigpagebreak 
 
\subheading{6.6\ Theta functions}  Consider the vector spaces  $\langle \Bbb 1\rangle_C$, 
where  $C$  is a smooth connected compact curve (with empty set of distinguished  
points).  They are fibers of a lisse $\lambda^c$-twisted $D$-module  
$\langle\Bbb 1 \rangle$  on the moduli space of smooth curves.  For a point   
$x \in C$  we have   
$\langle \Bbb 1\rangle_C = \langle \Bbb 1_x\rangle_{C,x}$, hence one has a  
canonical map  $\gamma_x: r(\Bbb 1 )_{x,C} \to \langle \Bbb 1\rangle_C$.  The  
image $\gamma_{{}_C} = \gamma_x({\frac{1}{x}})$  is independent of the choice of  
$x$ (since $\partial_x (\gamma_x (1_x)) = 0$).  As $C$  varies, the $\gamma_C$  
form a holomorphic section of  $\langle \Bbb 1 \rangle$.   
 
Here is an explicit formula for $\gamma$  on the moduli space of elliptic curves.  
Consider the usual uniformization of the moduli space by the upper half plane  
$H$ with  
parameter $z$; then  $q = exp(2 \pi iz)$  is the standard parameter at infinity.  
The family of elliptic curves degenerates when $q \to 0$  in the standard way  
described in 3.6.6.  Hence on $H$  we get a canonical trivialization  
$\langle \Bbb 1\rangle_H = \oplus {\Bbb C}_{I_j}$, horizontal with respect to the  
trivialization of  $\lambda^c$  described in 3.6.6.  In this trivialization  
we have  $\gamma(q) = \sum \gamma^\vee_{I_j}(q)$, where $\gamma^\vee_{I_j}(q)  
= tr_{I_{j{\Bbb C}((t))}} q^{-L_0}$  by axiom g.  The ``global"  
trivialization of $\lambda^c$  given by  $\eta(q)^c$  differs from the above  
trivialization by  $q^{c/24}$  (see 3.6.6).  In this global $\eta$-trivialization  
the components of $\gamma$ are  $\gamma_{I_j}(q) = q^{c/24}  
tr_{I_{j{\Bbb C}((t))}} q^{-L_0}$.  We see that these are holomorphic  
functions on $H$ and for any  ${\pmatrix a & b\\ c & d\endpmatrix} \in  
SL_2({\Bbb Z})$  the function  $\gamma_{{}_{I_j}} ({\frac{az+b}{cz+d}})$  is a linear 
combination with constant coefficients of other  $\gamma_{{}_{I_i}}$'s.   

 
\newpage 
 
\pageno=79 
 
\heading \S7.\ Lisse representations \endheading 
\vskip .2in 
 
\subheading{7.1\ Singular support, lisse modules}  Let  ${\frak g}$  be a  
Lie algebra, and $U = U({\frak g})$  its universal enveloping algebra.  Then  
$U$  is a filtered algebra  $(U_0 = {\Bbb C}, U_1 = {\Bbb C} + {\frak g},  
U_i = U^i_1$ for  $i > 0$), $gr U = \oplus_i U_i/U_{i-1} = S^\bullet({\frak g})$. 
For  $\varphi \in U_i$  its symbol  $\sigma_i(\varphi)$  is $\varphi\  
\text{mod}\ U_{i-1} \in S^i{\frak g}$; if $\varphi \in U_i\setminus U_{i-1}$  we  
will write  $\sigma(\varphi) = \sigma_i(\varphi)$.  
 
\medpagebreak  
 
\noindent 
7.1.1  Let  $M$  be a finitely generated  ${\frak g}$-module.  Recall that a  
good filtration   $M_\bullet$  on $M$  is a  $U_\bullet$-filtration such that  
$M = \cup M_i$, $\cap M_i = 0$  and  $gr M_\bullet$  is a finitely generated  
$S^\bullet({\frak g})$-module.  For example, if  $M_0 \subset M$  is a  
finite dimensional vector subspace that generates $M$, then $M_i = U_iM_0$  is  
a good filtration.   Any two good filtrations  $M_\bullet, M'_\bullet$  on $M$  
are comparable, i.e., for some $a$  one has $M_{\bullet -a} \subset M'_\bullet  
\subset M_{\bullet +a}$.   
 
Define the singular support $SSM$ of  $M$  to be the support of the   
$S^\bullet({\frak g})$-module $gr M_\bullet$, where  $M_\bullet$  is a good 
filtration on $M$.  This is a Zariski closed canonical subset of   
$Spec S^\bullet({\frak g}) = {\frak g}^*$;  it does not depend on the choice of  
a good filtration $M_\bullet$.  If  $\eta$  is a generic point of  $SSM$, then the 
length of the $S^\bullet({\frak g})$-module  $(gr M_\bullet)_\eta$  only depends  
on $M$; denote it  $\ell_\eta(M)$.  We will say that $M$ is finite at  
$\eta$  if $\ell_\eta(M) < \infty$:  this means that  $(gr M_\bullet)_\eta$  is  
killed by an ideal of finite codimension in  $S^\bullet({\frak g})_\eta$. 
 
\demo{7.1.2\ Remarks} (i)  If $M$ is generated by a single vector,  
$M \simeq U/I$, then  $SS(M)$  is the zero set of symbols of elements of $I$. 
 
\noindent 
(ii)  A more precise way to speak about this subject needs the microlocalization  
language, see e.g. [La], Appendix. 
\enddemo 
 
The algebra $gr U = S^\bullet({\frak g})$  carries a Poisson bracket defined by the  
formula  $\{ f_i, g_j\} = \widetilde{f}_i \widetilde{g}_j - \widetilde{g}_j  
\widetilde{f}_i$  mod $U_{i+j-2}$; here $f_i \in S^i({\frak g}),  
\widetilde{f}_i \in U_i, f_i = \widetilde{f}_i$ mod $U_{i-1}$, and the same for  
$g_j$,\ $\{ f_i,g_j\} \in S^{i+j-1}({\frak g})$.  One has the following  
integrability theorem, due to O. Gabber [Ga]: 
 
\proclaim{7.1.3\ Theorem}  Let  $M$  be a finitely generated $U$-module finite  
at any generic point of  $SSM$.  Then  $SSM$  is involutive, i.e., if   
$f,g \in S^\bullet({\frak g})$ vanish on $SSM$, then so does $\{ f,g\}$.  
\hfill $\square$ 
\endproclaim 
 
\medpagebreak 
 
\proclaim{7.1.4\ Definition}  A finitely generated module $M$ is {\it lisse} if  
$SSM = \{ 0\}$. More generally, we will say that $M$ is lisse along a vector  
subspace  $\ell \subset {\frak g}$  if  $SSM \cap \ell^\perp = \{ 0\}$. 
\hfill $\square$ 
\endproclaim 
 
\medpagebreak 
 
Note that any quotient of a lisse module is lisse.  Any extension of a lisse  
module by a lisse module is lisse.  Any finite dimensional $M$ is lisse; the  
converse is true if  dim ${\frak g} < \infty$.   
 
Explicitly, a module $M$ is lisse if amd only if  for a finite dimensional subspace  
$V \subset M$  that generates $M$ and any  $g \in {\frak g}$  there exists  
$N \gg 0$  such that  $g^NV \subset U_{N-1}V$.   
 
\bigpagebreak 
 
\subheading{7.2\ Finiteness property}  Let  $k \subset {\frak g}$  be a Lie 
subalgebra.  We will say that a ${\frak g}$-module $M$ is a $({\frak g},k)$-module  
if  $k$  acts on $M$ in a locally finite way (i.e., for any  $x \in M$  one has 
$dim U(k)x < \infty$).  If such an $M$ is finitely generated, then it carries a  
good $k$-invariant filtration (e.g., take a finite dimensional $k$-invariant  
subspace  $M_0 \subset M$  that generates $M$ and put $M_i = U_i M_0$).  Hence  
$SSM \subset k^\perp = ({\frak g}/k)^* \subset {\frak g}^*$. 
 
\medpagebreak  
 
\proclaim{7.2.1\ Lemma}  Let  $M$  be a finitely generated $({\frak g},k)$-module  
and  $n \subset {\frak g}$  be a vector subspace such that  
$dim {\frak g}/n+k < \infty$  and  $M$  is lisse along $n$.  Then   
$dim M/nM < \infty$. 
\endproclaim 
 
\demo{Proof}  Let  $M_\bullet$  be a $K$-invariant good filtration on $M$, so  
$gr M_\bullet$  is a finitely generated  $S^\bullet({\frak g}/k)$-module.  Consider 
the induced filtration on  $M/nM$.  It suffices to see that  
$dim gr(M/nM) < \infty$.  But $gr(M/nM)$  is a quotient of $gr M/n gr M$  (since 
$gr_i M/nM = M_i/M_{i-1} + (M_i \cap NM)$, $(gr M/n gr M)_i = M_i/M_{i-1} + nM_{i-1})$. 
The latter is a finitely generated module with zero support over the finitely generated  
algebra $S^\bullet({\frak g}/k + n)$, hence it is finitely generated.  
\hfill $\square$ 
\enddemo 
 
We will use 7.3.1 as follows.  Assume we are in a situation 3.3, so we have  
a Harish-Chandra pair  $({\frak g}, K)$, an $S$-localization data  
$\psi = (S^\#,N,\varphi,\varphi_0)$  for   $({\frak g}, K)$  and the  
corresponding $S$-localization functor   
$\Delta_\psi: (\widetilde{\frak g},K)$-mod  
$\to {\Cal D}_\psi$-mod.  Certainly, any  $(\widetilde{\frak g},K)$-module $M$ is a  
$(\widetilde{\frak g},k)$-module and $SSM$  is an Ad $K$-invariant closed  
subset of $k^\perp$.  Now 7.2.1 (together with 3.3.4) implies: 
 
\proclaim{7.2.2\ Corollary}\ Assume that the following finiteness condition holds: 
 
\noindent 
(*)  The sheaf  ${\frak g}^\#_S/k^\#_S + \varphi(N_{(0)})$ is ${\Cal O}_S$-coherent. 
 
Then for a lisse  $(\widetilde{\frak g},K)$-module $M$ the ${\Cal D}_\psi$-module 
$\Delta_\psi(M)$  is lisse (see 3.2.7).  More generally, if a  
$(\widetilde{\frak g},K)$-module $M$ is lisse along any subspace   
$\varphi_0(N_{(0)s}) \subset \widetilde{\frak g}$,\ $s \in S^\#$, then   
$\Delta_\psi(M)$  is a lisse ${\Cal D}_\psi$-module.  \hfill $\square$ 
\endproclaim 
 
The following corollaries of 7.1.3 will be useful. 
 
\proclaim{7.2.3\ Lemma}  Let $M$  be a $({\frak g},k)$-module such that $SSM$  
has finite codimension in  $k^\perp$.  Then  $SSM$ is involutive.  
\hfill $\square$ 
\endproclaim 
 
\proclaim{7.2.4\ Corollary}  Assume that a Harish-Chandra pair $({\frak g},K)$  
has the property that any Zariski closed Ad $K$-invariant subset of  $k^\perp$   
is either $\{ 0\}$  or has finite codimension.  Then for any $({\frak g},K)$-module  
$M$ the $SS(M)$  is involutive.  \hfill $\square$ 
\endproclaim 
 
\bigpagebreak 
 
\subheading{7.3\ Lisse modules over Virasoro algebra}  Consider the Virasoro  
algebra  $\widetilde{\Cal T}_c$:  this is the central ${\Bbb C}$-extension of  
Lie algebra  ${\Cal T} = {\Bbb C}((t))$  that corresponds to the 2 cocycle  
$\langle f\partial_t, g\partial_t\rangle_c = c Res(f'''g {\frac{dt}{t}})$. 
It carries the filtration  $\widetilde{\Cal T}_{cn}:$  for $n \ge 1$,  
$\widetilde{\Cal T}_{cn} = t^{n+1} {\Bbb C}[[t]]\partial_t$, for  $n \le 0$,  
$\widetilde{\Cal T}_{cn} = {\Bbb C} + t^{n+1} {\Bbb C}[[t]]\partial_t$.  Put 
$L_i := t^{i+1} \partial_t \in {\widetilde{\Cal T}}_c$.  One also has the  
following Lie subalgebras of  $\widetilde{\Cal T}_c$: 
$$ 
n_+ = \widetilde{\Cal T}_{c1} \subset b_+ = {\Bbb C}[[t]]t\partial_t \subset P_+  
= {\Bbb C}[[t]]\partial_t,\quad n_- = {\Bbb C}[t^{-1}]\partial_t \subset  
b_- = {\Bbb C}[t^{-1}]t\partial_t, 
$$ 
so  $b_+ = Lie K$, $n_+ =$ Lie $K_1$ (see 3.4.1).  One has $b_+ \oplus n_-  
\oplus {\Bbb C} = \widetilde{\Cal T}_c$, $b_+ \cap b_- - f = {\Bbb C} L_0$. 
 
\noindent 
7.3.1  A higher weight ${\Cal T}$-module of central charge $c$ is a  
$(\widetilde{\Cal T}_c, b_+)$-module $M$ such that  $1 \in {\Bbb C} \subset  
\widetilde{\Cal T}_c$  acts as  $id_M$  and any $m \in M$  is killed by some  
$\widetilde{\Cal T}_{cn}$  for  $n \gg 0$.  Denote by  ${\Cal T}_{c+}$-mod  
the category of such modules.  Note that any  $M \in {\Cal T}_{c+}$-mod is a 
$(\widetilde{\Cal T}_c, K_1)$-module.  We will say that $M$ is $L_0$-diagonalizable 
if $M$ coincides with the direct sum of  $L_0$-eigenspaces. 
 
Let  $M$  be a higher weight module.  Denote by $*M$  the space of those linear  
functionals  $\varphi$  on $M$ that are finite with respect to the action of  
${}^tL_0$.  The operators  $L_i := {}^tL_{-i}$  define the $\widetilde{\Cal T}_c$-action  
on $*M$.  Clearly  $*M$  is a higher weight module called the (contravariant) dual  
to $M$.  One has an obvious morphism $M \to **M$  which is an isomorphism if amd  
only if  the  
generalized eigenspaces of $L_0$  on $M$ are finite dimensional.  In particular 
this holds when $M$ is a finitely generated module. 
 
\medpagebreak 
 
\demo{7.3.2\ Remark}  For  $M \in {\Cal T}_{c+}$-mod consider the monodromy   
operator  $T = exp(2\pi i L_0)$.  Clearly $T$ commutes with the Virasoro action,  
i.e., $T \in Aut M$.  Hence one has a canonical direct sum decomposition   
$M = {\bigoplus_{\overline{a}\in {\Bbb C}/{\Bbb Z}}} M_{\overline{a}}$, where  
$M_{\overline{a}}$  is the generalized  $exp(2\pi i a)$-eigenspace of $M$.  Denote 
by  ${\Cal T}_{c+\overline{a}}$-mod the subcategory of those $M$'s that   
$M = M_{\overline{a}}$.  Clearly  ${\Cal T}_{c+}$-mod  
$= {\prod_{a\in {\Bbb C}/{\Bbb Z}}} {\Cal T}_{c+\overline{a}}$-mod. 
\enddemo 
 
\medpagebreak 
 
\proclaim{7.3.3\ Lemma}  For any finitely generated $M \in {\Cal T}_{c+}$-mod  
there are exactly three possibilities for $SSM$:  it  is either equal to  
$\{ 0\}$, or to  $\widetilde{\Cal T}_{c0}^\perp = ({\Bbb C} + b_+)^\perp$, or  
to  $\widetilde{\Cal T}^\perp_{c-1} = ({\Bbb C} + P_+)^\perp$. 
\endproclaim 
 
\demo{Proof}  Clearly  $SSM \subset \widetilde{\Cal T}_{c0}^\perp$.  It is  
Ad $K$-invariant (the Ad $K_1$-invariance is obvious; for any $t \in {\Bbb C}$ 
the operator $exp(t L_0)$  acts on $M$, hence $SSM$  is also Ad $exp(t L_0)$-invariant). 
It is easy to see that any Ad $K$-invariant Zariski closed subset of  
$\widetilde{\Cal T}^\perp_{c0}$  is either  $\{ 0\}$  or coincides with  
one of the vector spaces  $\widetilde{\Cal T}^\perp_{c-n}$,\ $n \ge 0$.   
According to 7.2.4 this $\widetilde{\Cal T}_{c-n}$  is the Lie subalgebra of  
$\widetilde{\Cal T}_c$; this implies 7.3.3.  \hfill $\square$ 
\enddemo 
 
\medpagebreak 
 
For a higher weight module $M$  consider the subspace  $M^{{\frak n}_+}$  of singular  
vectors.  Clearly  $M^{{\frak n}_+} \ne 0$ and it is $L_0$-invariant, so we have  a decomposition  $M^{{\frak n}_+} = {\dsize\bigoplus_{h\in {\Bbb C}}} M^{{\frak n}_+}_{(h)}$  
by  
generalized eigenspaces of  $L_0$.  We will say that a singular vector   
$v$  has generalized weight $h$ if  $v \in M^{{\frak n}_+}_{(h)}$ (i.e., if   
$(L_0 - h)^nv = 0$  for $n \gg 0$), and that  $v$  has weight $h$ if  
$L_0v = hv$.  As usual, the Verma module $M_{ch} = M_h \in \widetilde{\Cal T}_{c+}$-mod 
is a module generated by a single ``vacuum" singular vector  $v_h$  of weight  
$h$ with no other relations.  This  $M_h$  is the free $U({\frak n}_-)$-module generated  
by  $v_h$, hence any submodule of $M_h$  generated by a singular vector is a Verma 
module.  Denote by  $L_{ch} = L_h$  the  (only) irreducible quotient of  $M_h$.   
Any irreducible higher weight module is isomorphic to some $L_h$, and the  
$L_h$'s with different $h$'s non-isomorphic.  One has  $*L_h = L_h$.   
 
The following basic facts are due to Feigin-Fuchs   [FF]. 
 
\medpagebreak 
 
\proclaim{7.3.4\ Proposition}  Let  $M = M_h$  be a Verma module, $N \subset M$  
is a non-zero submodule.  Then  
 
\item{(i)}  $N$ is generated by $\le 2$  singular vectors, i.e., $N$ is either a  
Verma submodule or a sum of two Verma submodules. 
 
\item{(ii)}  $N$ is an intersection of $\le 2$ Verma submodules. 
 
\item{(iii)} $M/N$  has finite length. 
 
\item{(iv)} The spaces  $M^{{\frak n}_+}_{(h')}$  have dimension  $\le 1$,  
therefore, by (i),  the irreducible constituents of $M$ have multiplicity 1. 
\hfill $\square$ 
\endproclaim
  
\bigpagebreak 

 
\proclaim{7.3.5\ Lemma}  Let  $P \in \widetilde{\Cal T}_{c+}$-mod be a finitely  
generated module.  Then  
 
\item{(i)} $P$ admits a filtration of finite length $\ell$ with successive  
quotients isomorphic to a quotient of a Verma module. 
 
\item{(ii)} The maximal semisimple quotient of $P$ has length $\le \ell$. 
 
\item{(iii)} Any submodule of $P$ is finitely generated. 
\endproclaim					        
 
\demo{Proof} Note that $P$ is a quotient of some module $Q$ induced from a  
finite dimensional $b_+$-module.  Such $Q$ has a filtration with successive  
quotients isomorphic to Verma modules.  This implies (i) and reduces  
(ii), (iii) to the case of Verma module which follows from 7.3.4 (i). 
\hfill$\square$ 
\enddemo 
 
\bigpagebreak 
 
\proclaim{7.3.6\ Lemma}  Let  $M = M_h$  be a Verma module,  $N \subset M$  be  
a non-zero submodule, $L = M/N$.  One has 
 
\item{(i)} $SSM = \widetilde{\Cal T}^\perp_{c0} = {\frak n}^*_-$ 
 
\item{(ii)} $SSL$ is either $\{ 0\}$  or equals to $\widetilde{\Cal T}^\perp_{c-1}$ 
 
\item{(iii)} If  $SSL = 0$, then $L$ is irreducible and $N$ is generated by  
two singular vectors. 
 
\item{(iv)} If $N$ is a proper Verma submodule, then the coinvariants  
$L_{[{\frak n}_-,{\frak n}_-]}$  are infinite dimensional.   
\endproclaim 
 
\demo{Proof}  (i) is obvious.  To prove (ii) take a non-zero  
$\varphi \in U({\frak n}_-)$  such that $\varphi v_h \in N$.  
The symbol  $\sigma(\varphi)$  vanishes on $SSL$, hence  $SSL \ne {\frak n}^*_-$,  
and we are done by 7.3.3.   
 
(iii) By 7.3.4 (iii) any reducible $L$ has a quotient such that the corresponding  
$N$ is a Verma submodule.  Since a quotient of a lisse module is lisse, (iii) is  
reduced to a statement that for any proper Verma submodule  $N = M_{h'} \subset  
M_h$  one has  $SS M_h/M_{h'} \ne 0$.  By 7.2.1 this follows from (iv). 
 
(iv) The commutant $[{\frak n}_-,{\frak n}_-]$  is Lie subalgebra of ${\frak n}_-$  
with basis $L_{-3}, L_{-4}, L_{-5},\ldots$.  The quotient ${\frak n}_-/[{\frak n}_-, 
{\frak n}_-]$  is abelian Lie algebra with basis  $L_{-1},L_{-2}$.  To prove  
(iv) note that  $M_{h[{\frak n}_-,{\frak n}_-]}$  is a free module over  
$U({\frak n}_-/[{\frak n}_-,{\frak n}_-]) ={\Bbb C}[L_{-1},L_{-2}]$ with  
generator  $\overline{v}_h$, and  $(M_h/M_{h'})_{[{\frak n}_-,{\frak n}_-]}$  is  
a quotient of  $M_{h[{\frak n}_-,{\frak n}_-]}$ modulo the  
${\Bbb C}[L_{-1},L_{-2}]$ submodule generated by the image $\overline{v}_{h'}$ of 
$v_{h'}$  (since $M_{h'} = U({\frak n}_-)v_{h'}$).  Since   
$\overline{v}_{h'} = P \overline{v}_h$, where $P$ is a polynomial of weight  
$h' - h \ne 0$, we see that our coinvariants $(M_h/M_{h'})_{[{\frak n}_-,{\frak n}_-]}  
= {\Bbb C}[L_{-1},L_{-2}]/P{\Bbb C}[L_{-1},L_{-2}]$  are infinite dimensional. 
\hfill$\square$ 
\enddemo 
 
\bigpagebreak 
 
\noindent 
7.3.7\ We will say that an irreducible module $L_h \in \widetilde{\Cal T}_{c+}$-mod 
is minimal, or a Belavin-Polyakov-Zamolodchikov module, if the conditions (i), (ii)  
below hold: 
 
\item{(i)} For some integers $p,q$  such that $1 < p < q$, $(p,q) = 1$, one has 
$$ 
c = c_{p,q} = 1 - 6(p-q)^2/pq 
$$ 
(clearly $p,q$ are uniquely defined by $c$) 
 
\item{(ii)} For some integers $n,m$, $0 < n < p, 0 < m < q$ one has 
$$ 
h = h_{n,m} = {\frac{1}{4pq}} [(nq-mp)^2 - (p-q)^2]. 
$$ 
 
Clearly $h_{n,m} = h_{p-n,q-m}$.  For given $c = c_{p,q}$  there is exactly  
${\frac{1}{2}} (p-1)(q-1)$  different minimal irreducible modules.  Note that  
$L_{c_{p,q},0}$  is always minimal (since $0 = h_{1,1}$).   
 
\proclaim{7.3.8 Proposition} ([FF]\quad)  An irreducible module $L_h$  is minimal  
iff both the following conditions hold: 
 
\item{(i)}  $L_h$  is dominant which means that  $L_h$  is not isomorphic to a  
subquotient of any  $M_{h'}, h' \ne h$. 
 
\item{(ii)} The kernel $N_h$ of the projection  $M_h \to L_h$  is generated by  
exactly 2 singular vectors (see 7.3.4 (i)).  \hfill$\square$ 
\endproclaim 
 
\demo{7.3.9\ Remarks} (i) For $h = h_{nm}, c = c_{pq}$ the singular vectors from 
7.3.8 (ii) have weights  $h - nm, h - (p-n)(q-m)$.  They are different by  
7.3.4 (iv) (or by a direct calculation). 
 
(ii) It is easy to see, using contravariant duality, that $L_h$  is dominant iff  
$M_h$  is a projective object in the category of $L_0$-diagonalizable higher weight  
modules.  Equivalently, this means that  $M_h^\wedge =  
{\dsize\lim_\leftarrow} M_h^{(n)}$   
is a projective covering of $L_h$  in the category  $\widetilde{\Cal T}_{c+}$-mod.   
Here $M_h^{(n)}$  is the higher weight module generated by the singular vector $v$ 
that satisfies the only relation $(L_0 - h)^nv = 0$. 
\enddemo 
 
\proclaim{7.3.10 Proposition}  For an irreducible module $L = L_h = M_h/N_h$   
the following conditions are equivalent: 
 
\item{(i)} $L$ is lisse 
 
\item{(ii)} $L$ is minimal 
 
\item{(iii)} The coinvariants  $L_{[{\frak n}_-,{\frak n}_-]}$  are  
finite-dimensional 
 
\item{(iv)} The invariants  $L^{[{\frak n}_-,{\frak n}_-]}$ are finite dimensional 
 
\item{(v)} For some non-zero $\varphi \in U([{\frak n}_-,{\frak n}_-])$  one has  
$\varphi v_h \in N_h$ 
\endproclaim 
 
\demo{Proof}  One has (i) $\Longrightarrow$ (iii) by 7.2.1, (iii) $\Longleftrightarrow$  
(iv) by contravariant duality, (ii) $\Longleftrightarrow$ (iii) by [FF],  
(v) $\Longrightarrow$ (i) by 7.3.5 (ii) (since  $\sigma(\varphi)$  vanishes on  
 $SSL$, one has   $SSL \ne \widetilde{\Cal T}^\perp_{c-1}$.  It remains to show  
that  (ii) $\Longrightarrow$ (v).  So let  $L_h$  be minimal.  Put  
$T = U({\frak n}_-,{\frak n}_-])v_h \subset M_h$.  We wish to see that the  
projection  $T \to    L_h$  is not injective.  This follows since the asymptotic  
dimension of $T$ is larger than the one of $L_h$.  Precisely, according to the  
character formula for $L$ (see [K] prop. 4) the function $\log tr_L(exp(2\pi t L_0))$  
is asymptotically equivalent as  $t \to 0$ to  $\pi\alpha/12t$  for some  
constant  $\alpha < 1$.  On the other hand, one has  $\log tr_T (exp(-2\pi t L_0))  
= \log tr_{M_h}(exp(2\pi t L_0)) + \log(1 - exp(-2 \pi t)) + \log(1 - exp(-4\pi t))$ 
(since as $L_0$-module  $M_h$  is isomorphic to  $v_h \otimes S(L_{-1},L_{-2},\cdots)$, 
where the generators  $L_{-i}$  of the symmetric algebra have weights $i$, and  $T$  
is isomorphic to  $v_h \otimes S(L_{-3},L_{-n},\cdots)$).  This function is  
asymptotically equivalent to  $\pi/12t$.  Since the spectrum of  $L_0$  is real,  
this implies that  $T \to L_h$  is not injective. \hfill $\square$ 
\enddemo 
 
\demo{7.3.11 Remark}  For  $c = c_{p,q}, h = h_{11} = 0$  one may prove that  
(ii) $\Longrightarrow$ (i) in a very elementary way.  Namely, by 7.3.8 (ii) one  
knows that  $L_0$  is minimal iff $N_0$  does not coincide with the submodule  
$N'$ of $M_0$  generated by  $L_{-1} v_0$.  Choose minimal $i$ such that for  
certain  $\varphi \in U({\frak n}_-)_i$  one has  $\varphi v_0 \in N_0\setminus N'$.   
Then the symbol of $\varphi$ is prime to $L_{-1}$, hence, by 7.3.5 (ii), $L_0$ is  
lisse.  This remark, due essentially to Drinfeld, was a starting point for the  
results of this paragraph.   \hfill $\square$ 
\enddemo 
 
\proclaim{7.3.12 Proposition} The following conditions on a higher weight module  
$M$ are equivalent 
 
\item{(i)} $M$ is a finitely generated lisse module 
 
\item{(ii)} $M$ is isomorphic to a finite direct sum of minimal irreducible  
modules. 
 
\item{(iii)} One has  dim $M^{[{\frak n}_-,{\frak n}_-]} < \infty$ 
\endproclaim 
 
\demo{Proof}  By 7.3.10 we know that  (i) $\Longleftarrow$ (ii) $\Longrightarrow$  
(iii).  We will use the following facts: 
 
\item{(*)}  Let $L_h$  be a minimal irreducible module.  Then any quotient of  
length 2 of $M^{(n)}_h$  (see 7.3.9 (ii)) is actually a quotient of   
$M_h = M_h^{(1)}$  (i.e., is $L_0$-diagonalizable). 
 
\item{(**)} If  $L_{h_1},L_{h_2}$  are minimal and  $h_1 \ne h_2$, then  
$M_{h_1}$  and $M_{h_2}$  have no common irreducible component. 
 
Here (*) follows from the fact that  $N_h \subset M_h$  coincides with  
the 1st term of Jantzen filtration, see [FF]; for (**) see [FF].  Note that  
(*) implies, by 7.3.8, 7.3.9 (ii), that 
 
\item{(***)} $Ext^1(L_{h_1},L_{h_2}) = 0$  for any minimal $L_{h_1},L_{h_2}$. 
 
Now we may prove that (i) $\Longrightarrow$ (ii).  By 7.3.10 it suffices to show  
that a lisse module $M$ is semisimple.  Consider the maximal semisimple quotient  
$P = M/N$ (see 7.3.5 (ii)).  We have to show that $N = 0$.  By 7.3.5 (iii)  
there is an irreducible quotient  $Q = N/T$ of $N$, so we have a non-trivial  
extension      $0 \to Q \to M/T \to P \to 0$ with lisse  $M/T$.  According to  
7.3.9     (ii) and (**) we see that there exists at most one minimal $L_h$  such  
that  $Ext^1(L_h, Q) \ne 0$.  By (*) and 7.3.9 (i) for such $L_h$  one has  
dim $Ext^1(L_h,Q) = 1$.  This implies that  $M/T$  is isomorphic to a direct sum  
of minimal irreducible modules and a length 2 module which is a non-trivial  
extension of a minimal module  $L_h$  by $Q$.  By 7.3.9 (ii) and (*) this  
extension is a quotient of a Verma module.  By 7.3.5 (ii) it is non-lisse, hence  
$M/T$  is non-lisse.  Contradiction. 
 
 
 
\bigpagebreak 
 
Let us prove that (iii) $\Longrightarrow$ (ii).  Let $M$ be a module such that  
dim $M^{[{\frak n}_+,{\frak n}_+]} = r < \infty$.  Let  $M' \subset M$  be a  
maximal semisimple submodule of $M$.  By 7.3.10 $M'$  is a direct sum of minimal  
irreducible modules.  Clearly the length of $M'$ is $\le r$, so it suffices to  
show that$M' = M$.  Note that any non-zero submodule  $N \subset M$ intersects  
$M'$ non-trivially (if $N \cap M' = 0$  then, shrinking $N$ if necessary, we may  
assume that $N$ is a quotient of a Verma module.  If $N$ has finite length, then  
it contains an irreducbile submodule, which lies in $M'$.  If $N$ has infinite  
length, then, by 7.3.4, dim $N^{{\frak n}_+} = \infty$; since $N^{{\frak n}_+}  
\subset M^{[{\frak n}_+,{\frak n}_+]}$ this is not true).  Assume that  $M/M' \ne 0$.   
Replacing  $M$ by an appropriate submodule that contains $M$ we may assume that  $M/M'$   
is a quotient of a Verma module, in particular  $M/M'$  is $L_0$-diagonalizable.  
Consider the dual extension  $0 \to *(M/M') \to *M \to *M' \to 0$.  One has  
$*M' = \oplus L_{h_i}$, hence, by 7.3.8, 7.3.9 (ii) the projection   
$\oplus M_{h_i} \to \oplus L_{h_i} = *M'$  lifts to the map   
$\oplus M^{(2)}_{h_i} \to *M$.  This map is surjective (otherwise the dual to  
its cokernel would intersect $M'$ trivially), hence  $*M$  has finite length.   
Replacing  $*(M/M')$  by its irreducible quotient we may assume that  $M/M'$  is  
irreducible.   
 
As above (see the proof (i) $\Longrightarrow$ (ii))$*M$ is a direct sum of  
irreducible minimal modules plus a length two non-trivial extension of a minimal  
module  $L_h$.  By 7.39 (ii), 7.3.4 (ii) and (*) above this length two extension  
is a quotient of  $M_h$  by a Verma submodule.  By 7.3.6 (iv) the coinvariant  
$(*M)_{[{\frak n}_-,{\frak n}_-]}$  are of infinite dimension.  Since  
$(*M)_{[{\frak n}_-,{\frak n}_-]} = (M^{[{\frak n}_-,{\frak n}_-]})^*$, we are  
done. \hfill $\square$ 
\enddemo 
 
\noindent 
7.3.13  Now for  $n \ge 1$ consider the product of Virasoro algebras   
$\widetilde{\Cal T}^n_c$:  this is a central ${\Bbb C}$-extension of ${\Cal T}^n$ 
with cocycle  $\langle (f_i\partial_t), (g_i\partial_t)\rangle_c =  
{\dsize\sum_i}\langle f_i\partial_t, g_i\partial_t\rangle_c$  (see 3.4.1).  The 
above theory extends to  $\widetilde{\Cal T}^n_c$  in an easy manner.  Namely,  
we have a standard subalgebra  ${\frak n}_+ = \prod {\frak n}_{+i} \subset  
{\frak b}_+  = \prod {\frak b}_{+i} \subset {\frak p}_+ = \prod {\frak p}_{+i},\  
{\frak n}_{-i} \subset 	 {\frak b}_- = \prod {\frak b}_{-i}, {\frak f} =  
{\frak b}_+ \cap {\frak b}_- = {\Bbb C}^n$ etc. of $\widetilde{\Cal T}^n_c$. 
One defines the corresponding category  ${\Cal T}^n_{c+}$-mod of higher weight  
modules in an obvious manner.  We have an obvious functor  $\otimes:  
\prod {\Cal T}_{c+}$-mod $\to {\Cal T}^n_{c+}$-mod, $(M_1,\ldots,M_n)  
\longmapsto M_1 \otimes \ldots \otimes M_n$.  Clearly  $SSM_1 \otimes \ldots  
\otimes M_n = SSM_1 \times SSM_2 \times \cdots \times SSM_n$.   
 
For $\hbar = (h_i) \in {\Bbb C}^n$  we have the corresponding Verma module  
$M_{\hbar} = \otimes M_{h_i}$  and its unique irreducible quotient   
$L_{\hbar} = \otimes L_{h_i}$; any irreducible higher weight module is  
isomorphic to a unique  $L_{\hbar}$.  It follows from 7.3.4 (iv) that any  
submodule $N \subset M_{\hbar}$  is tensor product $\otimes N_i$  of submodules  
$N_i \subset M_{h_i}$, so the structure of $N$ is clear from 7.3.4.  The  
lemma 7.3.5 (with its proof) remains valid for  ${\Cal T}^n_{c+}$-mod.  The  
version of 7.3.6 for  $\widetilde{\Cal T}^n_c$  case (with obvious modifications)  
follows immediately from the case  $n = 1$.  A module  $L_{\hbar} = \otimes L_{h_i}$   
is called minimal if all  $L_{h_i}$  are minimal (see 7.3.7).  The analog of  
7.3.8 (with ``2 singular vectors" replaced by ``$2n$ singular vectors") remains  
obviously valid, as well as 7.3.9.  The proposition 7.3.10 remains valid and  
follows directly from the case $n = 1$.  The proposition 7.3.12 remains valid  
together with its proof. 

 
\newpage 
 
\pageno=88 
 
\heading \S8.\  Minimal models \endheading 
\vskip .2in 
 
These were defined by Belavin, Polyakov and Zamolodchikov [BPZ].  Let us start  
with a general representation-theoretic construction.   
 
\subheading{8.1 Fusion functors for Virasoro algebra}  Let  $C$  be a compact  
smooth curve, $A,B~\subset~C$  be two finite sets of points such that   
$A \cap B = \emptyset, A \ne \emptyset$.  For a central charge $c \in {\Bbb C}$   
we have Virasoro algebra  
$\widetilde{\Cal T}_c^A$  which is central ${\Bbb C}$-extension of   
${\Cal T}^A =  
{\dsize\prod_{a\in A}} {\Cal T}_a$ (where  ${\Cal T}_a =$ vector fields on  
punctured formal disc at a) and similar algebras $\widetilde{\Cal T}_c^B$,  
$\widetilde{\Cal T}_c^{A\sqcup B}$.  One has a canonical surjective map  
$\widetilde{\Cal T}_c^A \times \widetilde{\Cal T}_c^B \to \widetilde{\Cal T}_c^{A\sqcup B}$  
(which is factorization by $\{ (a,-a)\} \subset {\Bbb C} \times {\Bbb C}$);  
the morphisms  
$\widetilde{\Cal T}_c^A \to \widetilde{\Cal T}_c^{A\sqcup B} \longleftarrow  
\widetilde{\Cal T}_c^B$  are injective.  One also has the canonical embedding  
$i_{A\sqcup B}: {\Cal T}(U) \longrightarrow \widetilde{\Cal T}_c^{A\sqcup B}$,  
where $U = C\setminus (A\sqcup B)$, and the ones  $i_A: {\Cal T}(C\setminus A)  
\to \widetilde{\Cal T}_c^A, i_B: {\Cal T}(C\setminus B) \to \widetilde{\Cal T}_c^B$.   
There is also a canonical morphism  $j_B: {\Cal T}(C\setminus A) \to \widetilde{\Cal T}_c^B$  
which is composition of the obvious embedding  ${\Cal T}(C\setminus A)  
\to {\Cal T}^B_{-1}$  and the section   
$s_{{\Cal O}_B}: {\Cal T}^B_{-1} \to \widetilde{\Cal T}^B_c$.  The  
restriction  $i_{A\sqcup B}\big\vert_{{\Cal T}(C\setminus A) }:   
{\Cal T}(C\setminus A) \longrightarrow \widetilde{\Cal T}_c^{A\sqcup B}$  coincides  
with  $i_A + j_B$.   
 
\medpagebreak 
 
\noindent 
8.1.1  Assume we have a positive divisor  $d = \sum n_bb \ge 0$  supported on $B$. 
Let  ${\Cal T}(C\setminus A,d) \subset {\Cal T}(C\setminus A,d)$  be the Lie subalgebra  
of vector fields vanishing of order  $\ge n_b + 1$  at any  $b \in B$.  Clearly  
one has  ${\Cal T}(C\setminus A,d_1) \subset {\Cal T}(C\setminus A,d_2)$  for  
$d_1 \ge d_2$, and  ${\Cal T}(C\setminus A,0)/{\Cal T}(C\setminus A,d) =  
{\Cal T}^B_{0}/{\Cal T}^B_{d}$, where  ${\Cal T}^B_{d} = \prod {\Cal T}_{n_b,b}$.  
Let  $\epsilon_d:  \widetilde{\Cal T}_c^B \longrightarrow  
\widetilde{\Cal T}_c^A/i_A(\widetilde{\Cal T}(C\setminus A,d))$  be the  
composition 
$$ 
\widetilde{\Cal T}_c^B \longrightarrow  
\widetilde{\Cal T}_c^B/s_{{\Cal O}_B}({\Cal T}_{B,d}) \longrightarrow  
\widetilde{\Cal T}_c^{A\sqcup B}\big/i_{A\sqcup B}({\Cal T}(U)) +  
s_{{\Cal O}_B}({\Cal T}_{B,d})  
{\lower .5ex\hbox{${\buildrel \leftarrow\over\sim}$}}  
\widetilde{\Cal T}_c^A/i_A({\Cal T}(C\setminus A, d)). 
$$ 
The maps  $t_d$  are compatible, so we have   
$\epsilon = {\dsize\lim\Sb\\ \leftarrow\\ d\endSb} \epsilon_d:  
\widetilde{\Cal T}_c^B \longrightarrow {\dsize\lim\Sb\\\leftarrow\\d\endSb}  
{\Cal T}^A/i_A({\Cal T}(C\setminus A, d))$. 
 
\bigpagebreak 
 
 
 
\noindent 
8.1.2  Now we are able to define the (contravariant) fusion functor  
${\Cal F}_C: \widetilde{\Cal T}^A_c - \text{mod} \to  
\widetilde{\Cal T}^B_c-\text{mod}$. 
 
Let   $M$  be any  $\widetilde{\Cal T}_c^A$-module (so     $1 \in {\Bbb C}  
\subset \widetilde{\Cal T}_c^A$ acts as  $id_M$).  Put  ${\Cal F}_C(M) :=$ 
 
\noindent  
${\dsize\bigcup_d}\ M^* i_A({\Cal T}(C\setminus    A,d)) \subset M^*$;  
therefore an element of  ${\Cal F}_C(M)$  is a linear functional on $M$ invariant  
with respect to some $i_A({\Cal T}(C\setminus A, d))$.  For  
$\tau \in \widetilde{\Cal T}_c^B$, $\ell \in {\Cal F}_C(M)$ put  
${\tau}(\ell) = {}^t {\epsilon}(\tau)(\ell)$.  It is easy to see that this formula  
is correct, $\tau(\ell)$  lies in ${\Cal F}_C(M) \subset M^*$ and  
$(\tau,\ell) \longmapsto \tau(\ell)$ is $\widetilde{\Cal T}_c^B$-action on  
${\Cal F}_C(M)$.  This way  ${\Cal F}_C(M)$  becomes $\widetilde{\Cal T}_c^B$-module. 
One has an easy  
 
\medpagebreak 
 
\proclaim{8.1.3 Lemma}  
 
(i)  One has ${\Cal F}_C(M) = {\dsize\bigcup_\alpha} {\Cal F}_C(M)^{{\Cal T}_{B,d}}$,  
and  ${\Cal F}_C(M)^{{\Cal T}_{B,d}} = (M_{{\Cal T}(C\setminus A,d)})^*$. 
 
(ii) Let  $N$  be any  $\widetilde{\Cal T}_c^B$-module s.t. $N = {\dsize\bigcup_\alpha}  
N^{{\Cal T}_{B,d}}$.  Then Hom$(N, {\Cal F}_CM) =$  
 
\noindent 
$[(M \otimes N)_{{\Cal T}(U)}]^*$  
(here we consider  $M \otimes N$  as $\widetilde{\Cal T}_c^{A\sqcup B}$-module 
via the surjection  
 
\noindent 
$\widetilde{\Cal T}_c^A \times \widetilde{\Cal T}_c^B  
\longrightarrow \widetilde{\Cal T}_c^{A\sqcup B}$).  \hfill $\square$   
\endproclaim 
 
>From now on let us fix a central charge $c = c_{p,q}$  from the list 7.3.7(i).  We  
will assume that our virasoro modules have central charge $c$.  Let $M$ be a  
finitely generated higher weight $\widetilde{\Cal T}_c^A$-module. 
 
\medpagebreak 
 
\proclaim{8.1.4 Corollary} (i) ${\Cal F}_C(M)$  is finitely generated lisse  
higher weight $\widetilde{\Cal T}_c^B$-module. 
 
(ii) For any finitely generated higher weight $\widetilde{\Cal T}_c^B$-module $N$  
one has  $(M \otimes N)_{{\Cal T}(U)} = (M \otimes \overline{N})_{{\Cal T}(U)}$, 
where  $\overline{N}$  is the maximal lisse quotient of $N$. 
\endproclaim 
 
\demo{Proof}  (i)  Use 8.1.3 (i), 7.2.1, 7.3.12 (inversion 7.3.13). 
 
\item{(ii)} First note that the maximal lisse quotient $\overline{N}$  exists and  
has finite length by 7.3.5, 7.3.8, 7.3.12.  By 8.1.3 (ii), 8.1.4 (i) one has  
$(M \otimes N)^*_{{\Cal T}(U)} = \text{Hom}(N, {\Cal F}_C(M)) =  
\text{Hom}(\overline{N}, {\Cal F}_C(M)) = (M \otimes \overline{N})^*_{{\Cal T}(U)}$, 
q.e.d.  \hfill $\square$ 
\enddemo							    
 
For $h = (h_b) \in {\Bbb C}^B$ let  $L^B_h = {\dsize \bigotimes_{b\in B}}  
L_{c,h_b}$  be the irreducible $\widetilde{\Cal T}_c^B$-module of higher weight  
$h$.  
 
\medpagebreak 
 
\proclaim{8.1.5 Corollary} One has a canonical isomorphism   
$M_{{\Cal T}(C\setminus A)} = (M \otimes L^B_{0})_{{\Cal T}(U)}$. 
\endproclaim 
 
\demo{Proof}  Clearly  $M_{{\Cal T}(C\setminus A)} =  
(Ind^{{\Cal T}(U)}_{{\Cal T}(C\setminus A)} M)_{{\Cal T}(U)}$.  But  
$Ind^{{\Cal T}(U)}_{{\Cal T}(C\setminus A)} M$  coincides, as ${\Cal T}(U)$-module,  
with $\widetilde{\Cal T}_c^{A\sqcup B}$-module  $M \otimes P^B_{o}$, where  
$P_{oB} = {\dsize\bigotimes_{b\in B}} P_{c,o,b}, P_{c,o}$  is a quotient of Verma 
module  $M_{c,0}$ modulo relation  $L_{-1}v_0 = 0$.  Clearly  $L^B_o$  is maximal  
lisse quotient of  $P^B_{o}$ (see 7.3.8), and 8.1.5 follows from 8.1.4 (ii).  
\hfill $\square$ 
\enddemo 
 
\medpagebreak 
 
\proclaim{8.1.6 Corollary}  Let  $d_1$  be the divisor  ${\dsize\sum_{b\in B}}b$.  
Consider the action of Lie algebra  
 
\noindent 
${\Cal T}(C\setminus A,0)/{\Cal T}(C\setminus A,d_1)  
= {\Cal T}^B_{0}/{\Cal T}^B_{d_1} = {\Bbb C}^B$  on coinvariants   
$M_{{\Cal T}(U,d_1)}$.  This action is semisimple.  For  
$h = (h_b) \in {\Bbb C}^{B}$   
the  $(h_b)$-component  $M^{(h_b)}$  is equal to the coinvariants   
$(M \otimes L^B_h)_{{\Cal T}(U)}$.  This space vanishes unless all  $h_b$   
lie in the list 7.3.7  (ii). 
\endproclaim 
 
\demo{Proof}Similar to 8.1.5; the semi-simplicity of ${\Bbb C}^B$-action follows  
from 7.3.12 (ii).  \hfill $\square$ 
\enddemo 
 
\medpagebreak 
 
\proclaim{8.1.7 Corollary}  Assume that $B$ consists of two points  $b_1,b_2$.   
Let  ${\Cal T}(C\setminus A, B)'$  
 
\noindent 
$\subset {\Cal T}(C\setminus A,0)$  be the Lie 
 subalgebra of vector fields that project to $\{ (a,-a)\} \subset {\Bbb C}^2$   
via the projection to  ${\Cal T}(C\setminus A,0)/{\Cal T}(C\setminus A,d_1) =  
{\Bbb C}^2$.  Then  $M_{{\Cal T}(C\setminus A,B)'} = \oplus (M \otimes  
L_{c,h b_1} \otimes L_{c,h b_2})_{{\Cal T}(U)}$, where  $L_{ch}$  runs the  
list 7.3.7 (ii) of irreducible lisse modules.   
\endproclaim 
 
\demo{Proof}  Similar to 8.1.6.  \hfill $\square$ 
\enddemo    
 
\bigpagebreak 
 
\subheading{8.2 Localization of lisse modules}  Let  $\pi: C_S \to S$  be a family  
of smooth projective curves, $A \subset C_S(S)$  be a finite non-empty disjoint 
set of sections, $\nu_a$  are 1-jets of parameters at $a \in A$.  By  
3.4.3-3.4.7 these define the $S$-localization data for  $(\widetilde{\Cal T}_c^A,  
v_1)$.  Consider the corresponding $S$-localization functor  $\Delta_{\psi_c}:  
(\widetilde{\Cal T}_c^A,v_1)_c$-mod $\to D_{\lambda^c}$-modules on $S$.  Assume as  
above that $M$ is a lisse $(\widetilde{\Cal T}_c^A,v_1)_c$-module. 
 
\proclaim{8.2.1 Lemma}  The $D_{\lambda^c}$-module $\Delta_{\psi_0}(M)$  is lisse  
with regular singularities at infinity.   
\endproclaim 
 
\demo{Proof}  Lissing follows from 7.2.2; the statement on regular singularities  
follows from 8.2.5 below.  \hfill $\square$ 
\enddemo 
 
\bigpagebreak 
 
\noindent 
8.2.2 Assume now that  $S = Spec {\Bbb C}[[q]], \pi: C_S \to S$  be a projective  
family of curves such that the generic fiber  $C_\eta$ is smooth and the closed  
fiber  $C_0$  has the only singular point $b$ which is quadratic, $A \subset C_S(S)$  
be a finite non-empty disjoint set of sections, and  $\{ \nu_a\}$  be a 1-jet of  
coordinates at $a \in A$.   
 
This collection defines an $S$-localization data ``with logarithmic singularities  
at $q = 0$"  for $(\widetilde{\Cal T}_c^A,v_1)$.  (The definition of ``$S$-loc. data  
$\psi$ with log. sing. at $q = 0$" coincides with 3.3.3 but we replace the  
condition that $N$ is a transitive Lie algebroid by the one that a canonical map  
$\sigma: N \to {\Cal T}_S$  has image equal to  ${\Cal T}^0_S = q {\Cal T}_S =  
{\Bbb C}[[q]]q \partial_q$.  As in 3.3 such data defines an ${\Cal O}_S$-extension  
${\Cal A}^0_{\psi_c}$  of  ${\Cal T}^0_S$  and the corresponding associative  
algebra  $D^0_{\psi_c}$  which is isomorphic to the subalgebra of $D_{{\Bbb C}[[q]]}$   
generated by ${\Bbb C}[[q]] = {\Cal O}_S$  and  $q\partial_q$.  We have the  
corresponding $S$-localization functor  $\Delta_{C_S}:  
(\widetilde{\Cal T}_c^A, v_1)$-mod  
$\to D^0_{\psi_c}$-mod.  The definition of this $\psi$ repeats word-for-word  
3.4.3-3.4.7:  we get the loc. data with logarithmic singularities just because  
${\Cal T}^0_S$  consists precisely of those vector fields that could be lifted to  
$C_S\setminus A(S)$.  Note that the ``vertical" part  $N_{(0)} = \ker \sigma \subset  
N$  is a free ${\Cal O}_S$-module and  $N_{(0)}/qN_{(0)}$  coincides with the Lie  
algebra  ${\Cal T}(C^\vee_0\setminus A,B)'$, where  $C^\vee_0$  is the normalization  
of $C_0$  and  $B = \{ b_1,b_2\}$ is the preimage of $b$ (see 8.1.7).  According to  
3.5 the algebra  $D^0_{\psi_C}$  coincides with the algebra $D^0_{\lambda^c_{C_S}}$  
of differential operators on the determinant bundle  $\lambda^c_{C_S}$  
generated by ``$q \partial_q$" and ${\Cal O}_S$.   
 
\bigpagebreak 
 
Now let  $t_1,t_2$  be formal coordinates at $b$ such that $q = t_1t_2$.  Let  
$C^\vee_S$  be the corresponding smooth $S$-curve (our $b$'s are the $a$'s in 3.6.1).  We have canonical points  
$b_1,b_2 \in C^\vee_S(S)$  with parameters $t_1,t_2$.  Take 1-jets of parameters  
$q^{-1}dt_1,dt_2$ (see 6.1.4) at $b$'s.  These, together with $A,\nu_A$, define  
${\Bbb C}((q))$-localization data for $(\widetilde{\Cal T}_c^{A\sqcup B}, v_1)$.   
The corresponding algebra coincides with  $D_{\lambda^c_{C_\eta^\vee}}$, so we have  
the localization functor $\Delta_{C_\eta^\vee}:   
(\widetilde{\Cal T}_c^{A\sqcup B}, v_1)$-mod  
$\to D_{\lambda^c_{C_\eta^\vee}}$-mod. 
 
\bigpagebreak 
 
\noindent 
8.2.3  Let  ${\Cal H}$  be a lisse $D_{\lambda^c_{C_\eta^\vee}}$-module, i.e. a  
finite dimensional ${\Bbb C}((t))$-vector space with  $D$-action.  An $h$-lattice  
${\Cal H}_h \subset {\Cal H}$, where  $h \in {\Bbb C}$, is a ${\Bbb C}[[t]]$-lattice  
in ${\Cal H}$ invariant with respect to the action of $D^0_{\lambda^c_{C_S^\vee}}$  
and such that the operator  $q \partial_q \in D^0_{\lambda^c_{C_S^\vee}}/q$   acts  
on  ${\Cal H}_h/q{\Cal H}_h$  as multiplication by $h$.  Certainly, such  
${\Cal H}_h$  exists iff  ${\Cal H}$  has regular singularities at $0$ with  
monodromy equal to $h$ mod ${\Bbb Z}$; if  ${\Cal H}_h$  exists, it is unique,  
so we'll call it ``the" $h$-lattice. 
 
From now on let $M$ be a lisse $\widetilde{\Cal T}_c^A$-module. 
 
\proclaim{8.2.4 Lemma}  For any  $h \in {\Bbb C}$,  
$\Delta_{C_\eta^\vee}(M \otimes L_{hb_1} \otimes L_{hb_2})$  
is a lisse module that admits the $h$-lattice  $\Delta_{C_\eta^\vee}(M \otimes  
L_h \otimes L_h)_h$. 
\endproclaim 
 
\demo{Proof}  ``lisse" follows from 8.1.4 (ii), 7.2.1.  The existence of  
$h$-lattice follows easily from 3.4.7.1. \hfill $\square$ 
\enddemo 
 
\bigpagebreak 
 
According to 3.6.3 we have a canonical isomorphism  $D_{\lambda^c_{C_S}} =  
D_{\lambda^c_{C_S^\vee}}$.  Denote this algebra  $D_{\lambda^c}$.  So, by  
8.2.4, we have for any  $h \in {\Bbb C}$  a $D^0_{\lambda^c}$-module   
$D_{\lambda^c_{C_\eta^\vee}} (M \otimes L_n \otimes L_h)_h$, which is zero if  
$L_h$  is not lisse (i.e. if  $h \ne h_{nm}$ from 7.3.7 (ii)) by 8.1.4 (ii). 
 
On the other hand, we have the $D^0_{\lambda^c}$-module  $\Delta_{C_S}(M)$. 
 
\proclaim{8.2.5\ Proposition} There is a canonical isomorphism of  $D^0_{\lambda^c}$-modules  
$$ 
\Delta_{C_S}(M) = \bigoplus_h  \Delta_{C_\eta^\vee} (M \otimes L_h \otimes L_h)_h. 
$$ 
\endproclaim 
 
\demo{Proof}  First, note that  $\Delta_{C_S}(M)$  is a coherent ${\Cal O}_S$-module  
by a version of 7.2.2 ``with logarithmic singularities".  Namely,   
$\Delta_{C_S}(M)$ is a coherent  $D^0_{\lambda^c}$-module, and its singular  
support  $\subset Spec(gr D^0_{\lambda^c})$  is $0$ section since $M$ is lisse; 
hence  $\Delta_{C_S}(M)$  is ${\Cal O}_S$-coherent. 
 
Let  $e_i$  be a basis of  $L_{h{\Bbb C}((t))}$	 that consists of $L_0$-eigenvectors, 
so  $L_0e_i = (h-n_i)e_i$  for $n_i \in {\Bbb Z} \ge 0$; let  $e^*_i$  be the  
dual basis in  $L_{h{\Bbb C}((t))} = *L_{h{\Bbb C}((t))}$.  It is easy to see  
that   $\Delta_{C^\vee_\eta}(M \otimes L_h \otimes L_h)_h \subset  
\Delta_{C^\vee_\eta}(M \otimes L_h \otimes L_h)$  is ${\Cal O}_S$-submodule  
generated by images of elements  $q^{-n_i}_m \otimes e_i \otimes e^*_j$, where  
$m \in M_{A,C_S}, e_i \in L_{h({\Bbb C}((t_1)),q^{-1}t)}, e^*_j \in  
L_{h({\Bbb C}((t_2)),t_2)}$ (see 6.1.4 for notations). 
 
To prove 8.2.5 it suffices to construct a morphism of  $D^0_{\lambda^c}$-modules  
$\Delta_{C_S}(M) \to \oplus \Delta_{C^\vee_\eta}(\quad )_h$  which induces  
isomorphism mod $q$ (since both are coherent ${\Cal O}_S$-modules, and the one  
on the right hand has no $q$-torsion, this morphism will be isomorphism).   
 
The $h$-component of this morphism just maps the image of   
$m \in M_{A,C_S} = M_{A,C^\vee_S}$  in  $\Delta_{C_S}(M)$  to the image of  
${\dsize\sum_i} m \otimes e_i \otimes e^*_i$  in  $\Delta_{C^\vee_\eta}(M \otimes  
L_h \otimes L_h)$.  It is easy to see that this formula defines a correctly  
defined morphism of  $D^0_{\lambda^c}$-modules (cf. 6.1.5).  It induces  
isomorphism modulo $q$ by 8.1.7 (since $\Delta_{C_S}(M)/q = M_{N_{(0)}/q N_{(0)}}  
= M_{{\Cal T}(C^\vee_0\setminus A,B)'}$, see 8.2.2).  \hfill $\square$ 
\enddemo 
 
\bigpagebreak 
 
\subheading{8.3 Definition of minimal theories}  Now we may define the minimal  
theory.  Pick central charge  $c = c_{p,q}$  from the list 7.3.7(i). 
	    
The fusion category ${\Cal A} = {\Cal A}_{p,q}$  is category of finitely generated  
lisse higher weight modules for Virasoro algebra  $\widetilde{\Cal T}_c$  of  
central charge $c$.  By 7.3.12 it satisfies the conditions listed in the beginning  
of 4.5.1.  The data from 4.5.1 ar the following ones: 
 
The duality functor $*: {\Cal A}^0 \to {\Cal A}$  is contravariant duality  
(see 7.3.1). 
 
The vacuum module $\Bbb 1$  is $L_{c,o}$; the isomorphism  $*\Bbb 1 = \Bbb 1$   
is canonical one (that identifies the vacuum vectors). 
 
The Dehn automorphism $d$ is equal to the monodromy automorphism $T = exp 2\pi i L_o$ 
from 7.3.2. 
 
We will define a canonical fusion structure on ${\Cal A}$  simultaneously with  
the structures 6.1 of algebraic field theory.  Namely, our realization functor  
$r: A \to (\widetilde{\Cal T}_c, v_1)$-mod is ``identity" embedding.  The  
vacuum vector $1 \in r(\Bbb 1) = L_0$  is  $v_0$.   
 
Let  $\pi: C_S \to S, A \subset C_S(S), \nu_A$,  be as in 6.1.2.  Assume that  
$A \ne \emptyset$.   For any $X \in {\Cal A}^{\otimes A}$ the $D_{\lambda^c}$-module  
$\Delta_{\psi_c}(X)$  is lisse holonomic with regular singularities at $\infty$.  
We put  $\langle X\rangle_{C_S} = \Delta_{\psi_c}(X)$ and $\gamma$ from 6.1.2 (iv)  
is identity map.   
 
Assume now that  $A = \emptyset$.  We should define a canonical lisse  
$D_{\lambda^c}$-module $\langle \Bbb 1\rangle_{C_S}$.  Let us make the base  
change and consider  $\pi_C: C_C = C_S \times_S C_S \to C_S$:  this is a family  
of curves with a canonical (diagonal) section a.  Consider the  $D_{\lambda^c}$-module  
$\langle \Bbb 1\rangle_{C_C}$; this is a lisse $D_{\lambda^c}$-module on  $C_S$  generated  
by the holomorphic section  $\langle 1\rangle_{C_C}$.  Note that  
$\langle 1\rangle_{C_C}$  is horizontal along the fibers of  $\pi: C_S \to S$.   
Hence there exists a (unique) $D_{\lambda^c}$-module $\langle \Bbb 1\rangle_{C_S}$  
on $S$ together with a holomorphic section $\langle 1\rangle_{C_S}$ such that  
$\langle \Bbb 1_a\rangle_{C_C} = \pi^*\langle \Bbb 1\rangle_{C_S}$,   
$\langle 1\rangle_{C_C} = \pi^*\langle 1\rangle_{C_S}$.   
 
Note that the axioms 4.5.4 (ii) and 6.1.2e hold by 8.1.5.  The axiom 6.1.3f  
holds automatically.  It remains to define the isomorphism 4.5.5 (ii) that  
will satisfy the axiom $g$ from 6.1.  This was done in 8.2.5 above (note that since  
$*L_h = L_h$, we have  $R = \oplus L_h \otimes L_h$). 
 
By the way, the covariant fusion functor  ${\Cal F}_C^{A,B}$  from 4.6 is  
$*{\Cal F}_C$  for contravariant ${\Cal F}_C$ from 8.1 (by 8.1.3 (iii)). 




\end
