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

§1. Tate’s linear algebra

1.1 Crossed modules and central extensions of Lie algebras. We will need Lie and associative algebra versions of crossed modules:

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^{#} ⟶^{∂} L of L-modules. For ℓ ∈ L we will denote the action of L on L^{#} as [ℓ, · ]; so one has ∂[ℓ, ℓ̃] = [ℓ, ∂ℓ̃], ℓ̃∈ L^{#}. (ii) Let R be an associative algebra. An R-crossed module is an R-bimodule R^{#} together with a morphism R^{#}⟶^{∂} R of R-bimodules. □

We have canonical pairings {, } : Sym^{2}L^{#} → L, ⟨, ⟩ : R^{#} ⊗_{R} R^{#} → R^{#} defined by formulas {m_{1}, m_{2}} := [∂m_{1}, m_{2}]+[∂m_{2}, m_{1}], ⟨s_{1}, s_{2}⟩ := (∂s_{1})s_{2}−s_{1}(∂s_{2}). These are morphisms of L-modules and R-bimodules respectively; one has ∂{, } = 0, ∂⟨, ⟩ = 0.

Crossed modules in both versions form categories in an obvious manner. For example, if R_{1}⟶^{f}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} → R_{2} of bimodules such that ∂f^{#} = f∂. 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 Rcrossed module, then it has also an R^{Lie}-crossed module structure R^{#Lie} with [r, r̃] = rr̃− r̃r. One has {s_{1}, s_{2}} = ⟨s_{1}, s_{2}⟩ + ⟨s_{2}, s_{1}⟩ for s_{i} ∈ R^{#} = R^{#} = R^{#Lie}.

Below “DG algebra” means “differential graded algebra”; so “Lie DG algebra” is the same as differential graded Lie superalgebra.

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}⟶^{d}L^{0} (resp. R^{−1}⟶^{d}R^{0}) is a Lie (resp. associative) algebra crossed module. For m_{1}, m_{2} ∈ L^{−1} (resp. s_{1}, s_{2} ∈ R^{−1}) one has {m_{1}, m_{2}} = d[m_{1}, m_{2}] (resp. ⟨s_{1}, s_{2}⟩ = d(s_{1}s_{2})). (ii) Conversely, let L^{#}⟶^{∂}L (resp. R^{#}⟶^{∂}R) be a crossed module, and i : N ⊂ L^{#} (resp. i : T ⊂ R^{#}) be an L-submodule (resp. R-sub-bimodule) such that

{L^{#}, L^{#}} ⊂ N ⊂ ker ∂ (resp. ⟨R^{#}, R^{#}⟩ ⊂ T ⊂ ker ∂). Then N⟶^{i}L^{#}⟶^{∂}L (resp.

T ⟶^{i}R^{#}⟶^{∂}R) is a dg Lie (resp. associative) dg algebra placed in degrees −2, −1, 0. □

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^{#}⟶^{∂}L; N), where L^{#}⟶^{∂}L is a crossed module and N ⊂ L^{#} is a submodule as in (ii) above. For example, one may take N = image of {, } (or image of ⟨, ⟩ in the associative algebra version); we will say that the corresponding DG algebra is defined by our crossed module. 1.1.3 The simplest example of a Lie algebra crossed module is a central extension L̃ → L of a Lie algebra L (the bracket on L̃ 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 [ℓ_{1}, ℓ_{2}] := [dℓ_{1}, ℓ_{2}]^{0,−1} is a Lie algebra, and d : L^{−1}/dL^{−2} → L^{0} is a morphism of Lie algebras such that (H^{−1} → L^{−1}/dL^{−2} → d(L^{−1})) is a central extension of dL^{−1} by H^{−1}. Hence if L^{#}⟶^{∂}L is an L-crossed module such that ∂ is surjective, then ker ∂/{L^{#}, L^{#}} → L^{#}/{L^{#}, L^{#}} → L is a central extension of L. If tr : ker ∂/{L^{#}, L^{#}} → ℂ is any linear functional, then it defines, by push-out, a central ℂ-extension L_{tr}^{#} of L. 1.1.4 The following example of a crossed module will be used below. Let L be a Lie algebra, and let L_{+}, L_{−} ⊂ L be ideals. Then we have an L-crossed module L_{+} ⊕ L_{−}⟶^{∂}L, ∂(ℓ_{+}, ℓ_{−}) = ℓ_{+} + ℓ_{−}. We have isomorphism i : L_{+} ∩ L_{−} \overrightarrow{∼} ker ∂, i(ℓ) = (ℓ, −ℓ) ∈ L_{+} ⊕ L_{−}. Or we may take an associative algebra R equipped with 2-sided ideals R_{+}, R_{−}, and get an R-crossed module R_{+} ⊕ R_{−}⟶^{∂}R. Note that {, } vanishes on L_{+} and L_{−} (and ⟨, ⟩ vanishes on R_{+} and R_{−}) and one has {ℓ_{+}, ℓ_{−}} = i([ℓ_{−}, ℓ_{+}]), ⟨r_{+}, r_{−}⟩ := −i(r_{+}r_{−}), ⟨r_{−}, r_{+}⟩ = i(r_{−}r_{+}).

If L_{+} +L_{−} = L, then we get a central extension L_{+} ∩L_{−}/[L_{+}, L_{−}] ⟶^{i} L̃ → L of L, where L̃ = L_{+}⊕L_{−}/i([L_{+}, L_{−}]). This central extension is equipped with obvious splittings s_{±} : L_{±} → L̃ such that s_{±}(L_{±}) are ideals in L̃; it is easy to see that L̃ is universal among all central extensions of L equipped with such splittings. Note also that the embedding s_{+} : L_{−} ↪ L̃ yields an isomorphism L_{+}/[L_{+}, L_{−}] \overrightarrow{∼} L̃/s_{−}(L_{−}) and we have the Cartesian square

L̃ ⟶ L̃/s_{−}(L_{−}) \overleftarrow{∼} L_{+}/[L_{+}, L_{−}]
↓ ↓ ↓
L ⟶ L/L_{−}  \overleftarrow{∼} L_{+}/L_{+} ∩ L_{−}

and the same for ± interchanged. 1.1.5 Now let tr : L_{+} ∩ L_{−}/[L_{+}, L_{−}] → ℂ be any linear functional. According to 1.1.3 it defines a central ℂ-extension L̃_{tr} of L. One has the splittings s_{+} : L_{+} → L̃_{tr}, s_{−} : L_{−} → L̃_{tr} such that s_{±}(L_{±}) are ideals and (s_{+} − s_{−})|_{L_{+}∩L_{−}} = tr. Clearly L_{tr} is the unique extension equipped with this data. 1.1.6 The above constructions are functorial with respect to (L, L_{±}). Hence if L′_{±} ⊂ L are other ideals such that L_{±} ⊂ L′_{±}, then we get a canonical morphism L̃ → L̃′ between the corresponding central extensions of L. If tr : L_{+} ∩ L_{−}/[L_{+}, L_{−}] → ℂ extends to tr : L′_{+} ∩ L′_{−}/[L′_{+}, L′_{−}] → ℂ, then L̃_{tr} = L̃′_{tr}. In particular, assume that tr : L_{+} ∩ L_{−}/[L_{+}, L_{−}] → ℂ extends to tr : L_{−}/[L_{−}, L_{−}] → ℂ. Then we may take L′_{+} = L, L′_{−} = L_{−} to get the same extension L̃_{tr}, hence we get the splitting s̃_{+} : L → L̃_{tr} that extends our old s_{+} : L_{+} → L̃_{tr}. Explicitly, s̃_{+}(ℓ_{+} + ℓ_{−}) = s_{+}(ℓ_{+}) + s_{−}(ℓ_{−}) + trℓ_{−}; clearly s̃_{+} − s_{−} = tr : L_{−} → ℂ. In the same way, an extension of tr : L_{+} ∩ L_{−} → ℂ to L_{+} determines the splitting s̃_{−} : L → L̃_{tr} that extends the old s_{−} : L_{−} → L̃_{tr}. If we have the trace functional on the whole L, i.e. tr : L/[L, L] → ℂ, then s̃_{+} − s̃_{−} = tr : L → ℂ. 1.1.7 We will often use the following notation. If 𝔤 is a Lie algebra, V is a vector space, and 0 → V → 𝔤̃ → 𝔤 → 0 is a central V -extension of 𝔤, then for any c ∈ ℂ we will denote by 𝔤̃_{c} a V -extension of 𝔤 which is the c-multiple of 𝔤̃. So we have a canonical morphism 𝔤̃ → 𝔤̃_{c} of central extensions of 𝔤 that restricted to V ’s is multiplication by c. For example, in situation 1.1.3 one has (L_{tr}^{#})_{c} = L_{ctr}^{#}.

1.2 Tate’s vector spaces. For subspaces V_{0}, V_{1} of a vector space V we will write V_{0} ≺ V_{1} if V_{0}/V_{0} ∩ V_{1} is of finite dimension, and V_{0} ∼ V_{1} (V_{i} are commensurable) if V_{0} ≺ V_{1} and V_{1} ≺ V_{0}. Clearly ≺ is partial order on a set of commensurability classes of subspaces. 1.2.1 A Tate’s topological vector space (or, simply, Tate’s space) V is a linearly topologized complete separated vector space V that admits a basis {V_{α}} of neighbourhoods of 0 with V_{α} mutually commensurable. Equivalently, V is the projective limit of a family of epimorphisms of usual vector spaces with finite dimensional kernels: V = lim_{\overleftarrow{α}}V/V_{a}.

Let L ⊂ V be a vector subspace. We will say that L is bounded if for any open U ⊂ V one has L ≺ U, and L is discrete if for some open U one has U ∩ L = 0. Clearly simultaneously bounded and discrete subspaces are just finite dimensional ones.

A lattice V_{+} ⊂ V is a bounded open subspace; equivalently, this is a maximal (with respect to ≺) bounded closed subspace. The lattices form a maximal basis of neighbourhoods of 0 that consists of mutually commensurable subspaces.

A colattice V_{−} ⊂ V is a maximal discrete subspace. Equivalently, this means that for (any) lattice V_{+} one has V_{+} ∩ V_{−} ∼ 0, V_{+} + V_{−} ∼ V (or for some lattice

V_{+} one has V_{+} ⊕ V_{−}⟶^{∼}V).

Tate’s vector spaces form an additive category T V with kernels and cokernels. The category T V is self-dual: Namely, for a Tate’s space V its dual V^{∗} is Hom(V, 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_{+} ⟼ V_{+}^{⊥} is 1-1 correspondence between lattices in V and V^{∗}; and the same for colattices. 1.2.2 Let V be a Tate’s vector space. One has a canonical ℤ-torsor Dim_{V} together with a map dim : { Set of all lattices in V } → Dim_{V} such that for a pair V_{+1}, V_{+2} of lattices one has dimV_{+1} −dimV_{+2} := dim(V_{+1}/V_{+1} ∩V_{+2})−dim(V_{+2}/V_{+1} ∩V_{+2}) ∈ ℤ. One has a natural map codim : { Set of all colattices in V } → Dim_{V} defined by formula codimV_{−} = dimV_{+} + dim(V/V_{+} + V_{−}) − dim(V_{+} ∩ V_{−}), where V_{+} is any lattice. The ℤ-torsor Dim_{V^{∗}} coincides with the opposite torsor to Dim_{V}: one has dimV_{+}^{⊥} = −dimV_{+}. The group Aut V acts on Dim_{V}; if V is neither bounded nor discrete, then the action is non-trivial. 1.2.3 Let V_{1}, V_{2} be Tate’s vector spaces. We will say that a linear operator f ∈ Hom(V_{1}, V_{2}) is bounded if Imf is bounded, is discrete if ker f is open, and is finite if Imf is finite dimensional. Denote by Hom_{+}, Hom_{−} and Hom_{00} respectively the corresponding spaces of operators; put Hom_{0} := Hom_{+} ∩ Hom_{−}. Clearly Hom_{+} + Hom_{−} = Hom, Hom_{?} (where ? = +, −, 0, 00) is a 2-sided ideal in Hom (i.e., if for V_{1}⟶^{f_{1}}V_{2}⟶^{f_{2}}V_{3} either f_{1} or f_{2} is in Hom_{?}, then f_{2}f_{1} is in Hom_{?}), and Hom_{−}Hom_{+} ⊂ Hom_{00}.

Remark. Let T V_{+}, T V_{−} ⊂ T V be full subcategories of bounded, resp. discrete, spaces. Then T V_{−} coincides with the category of usual vector spaces, and ∗ identifies T V_{+} with the dual category T V_{−}^{∘}; in particular these are abelian categories. Consider the quotient categories T V/+, T V/−, T V/0, whose objects are Tate’s vector spaces, and Hom’s are the corresponding quotients Hom/± := Hom/Hom_{±}, Hom/0 := Hom/Hom_{0} (clearly T V/± are just the quotient categories T V/T V_{±}). These quotient categories are abelian. In fact, the projection T V/0 → T V/ + ⊕T V− is an equivalence of categories, and embeddings T V_{±} ↪ T V composed with projections define equivalences T V_{+}/V ect \overrightarrow{∼} T V/−, T V_{−}/V ect \overrightarrow{∼} T V/+ (here V ect = T V_{+} ∩ T V_{−} is the category of finite dimensional vector spaces).

1.2.4 For V ∈ T V consider the algebra EndV equipped with 2-sided ideals End_{±} ⊃ End_{0} ⊃ End_{00}. We will write 𝔤ℓ = 𝔤ℓV for EndV^{Lie} = EndV considered as Lie algebra. Since End_{0}^{2} ⊂ End_{00}, we have a canonical trace functional tr : 𝔤ℓ_{0} → ℂ which vanishes on [𝔤ℓ_{+}, 𝔤ℓ_{−}]

According to 1.1.4, we get an End-crossed module End_{+} ⊕ End_{−} → End. By 1.1.5, tr defines a central ℂ-extension 𝔤̃ℓ → 𝔤ℓ of 𝔤ℓ, together with canonical Lie algebra splittings s_{±} : 𝔤ℓ_{±} → 𝔤̃ℓ such that s_{+} − s_{−} = tr on 𝔤ℓ_{0}. 1.2.5 Let T ⊂ V be a Tate’s subspace (= a closed subspace with induced Tate struc-

ture), and V/T be the quotient. Denote by P_{T} ↪^{i} 𝔤ℓV the parabolic subalgebra of endomorphisms that preserve T; let π = (π_{T} , π_{V/T}) : P_{T} → 𝔤ℓT ×𝔤ℓV/T be the ob-

vious projection. Let { \atop 𝔤ℓT × 𝔤ℓV/T} be a central ℂ-extension of 𝔤ℓT × 𝔤ℓV/T which

is the Baer sum of 𝔤̃ℓT and 𝔤̃ℓV/T; one has { \atop 𝔤ℓT × 𝔤ℓV/T} = 𝔤̃ℓT ×𝔤̃ℓV/T/\{(a_{1}, a_{2}) ∈

ℂ × ℂ : a_{1} + a_{2} = 0\}. Clearly { \atop 𝔤ℓT × 𝔤ℓV/T} coincides with the ℂ-extension constructed by the recipe of 1.1.4, 1.1.5 using the ideals 𝔤ℓ_{+}T ×𝔤ℓ_{+}V/T, 𝔤ℓ_{−}T ×𝔤ℓ_{−}V/T and the trace functional tr = tr_{T} + tr_{V/T}.

Let P̃_{T} = i^{∗}𝔤̃ℓV be the ℂ-extension of P_{T} induced by 𝔤̃ℓV. Since P_{T} = P_{T +} + P_{T −}, where P_{T ±} = P_{T} ∩ 𝔤ℓ_{±}V, this ℂ-extension coincides with the one constructed by means of ideals P_{T ±} and the trace functional tr_{V} |_{P_{T}}. Note that π(P_{T ±}) = 𝔤ℓ_{±}T ×𝔤ℓ_{±}V/T and tr_{V} |_{P_{T}} = tr ∘π. By 1.1.6 this defines a canonical morphism π̃ :

P̃_{T} → { \atop 𝔤ℓT × 𝔤ℓV/T} of ℂ-extensions that lifts π. In other words, P̃_{T} is canonically isomorphic to the Baer sum of ℂ-extensions induced by projections π_{T}, π_{V/T} from 𝔤̃ℓT, 𝔤̃ℓV/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_{+} : 𝔤ℓV_{+} = 𝔤ℓ_{+}V_{+} → 𝔤̃ℓV_{+}, s_{−} : 𝔤ℓV/V_{+} = 𝔤ℓ_{−}V/V_{+} → 𝔤̃ℓV/V_{+}, hence a canonical splitting s_{V_{+}} = s_{+}π_{V_{+}} + s_{−}π_{V/V_{+}} : P_{V_{+}} → 𝔤̃ℓ_{V}. Note that s_{V_{+}} actually depends on V_{+}: if V′_{+} is another lattice, then s_{V_{+}} −s_{V′_{+}} : P_{V_{+}} ∩P_{V′_{+}} → ℂ is given by formula (s_{V_{+}} −s_{V′_{+}})(a) = tr_{V_{+}/V_{+}∩V′_{+}}(a) − tr_{V′_{+}/V_{+}∩V′_{+}}(a).

Similarly, if T = V_{−} is a colattice, then we have the splittings s_{−} : 𝔤ℓV_{−} = 𝔤ℓ_{−}V_{−} → 𝔤̃ℓV_{−}, s_{+} : 𝔤ℓV/V_{−} = 𝔤ℓ_{+}V/V_{−} → 𝔤̃ℓV/V_{−}, hence the splitting s_{V_{−}} = s_{−}π_{V_{−}}+s_{+}π_{V/V_{−}} : P_{V_{−}} → 𝔤̃ℓ_{V}. On P_{V_{−}}∩P_{V_{+}} the difference s_{V_{+}}−s_{V_{−}} : P_{V_{−}}∩P_{V_{+}} → ℂ is given by formula

(s_{V_{+}} − s_{V_{−}})(a) = tr_{V_{−}∩V_{+}}(a) − tr_{V/V_{−}+V_{+}}(a).

The following subsection 1.3 could be omitted on first reading.

1.3 Elliptic complexes. Let (V^{·}, d) be a finite complex of Tate’s vector spaces. We will call it elliptic, if for some (or any) subcomplex (V_{+}^{·}, d) ⊂ (V^{·}, d) formed by lattices in V^{·} both V_{+}^{·} and V^{·}/V_{+}^{·} have finite dimensional cohomology spaces.

Clearly, elliptic complexes have finite dimensional cohomology.

Remark. V^{·} is elliptic iff its image in the abelian category T V/0 (see 3.2.2) is acyclic.

1.3.1 Let (U^{·}, d), (V^{·}, d) be elliptic complexes. Then Hom = Hom(U^{·}, V^{·}) := ∏ Hom(U^{i}, V^{i}) carries a bunch of subspaces. First, one has the subspaces Hom_{±} := ∏ Hom_{±}(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} := {f ∈ Hom : [f, d] ∈ Hom_{±}(U^{·}, V^{·+1})}, Hom_{0}^{d} := Hom_{+}^{d} ∩ Hom_{−}^{d}, Hom_{d} := {f ∈ Hom : [f, d] = 0} (= usual morphisms of complexes). Clearly Hom_{±} ⊂ Hom_{±}^{d}, Hom_{0} ⊂ Hom_{0}^{d}, and all Hom_{?}^{d} are compatible with ± decomposition: one has Hom_{?}^{d} = (Hom_{?}^{d} ∩ Hom_{+}) + (Hom_{?}^{d} ∩ Hom_{−}).

The following easy technical lemma is quite useful. Assume that we picked subcomplexes U_{+}^{′·} ⊂ U_{+}^{·} ⊂ U, V′_{+} ⊂ V_{+} ⊂ V formed by lattices. Put P := {f ∈ Hom(U^{·}, V^{·}) : f(U_{+}^{′·}) ⊂ V_{+}^{′·}, f(U_{+}) ⊂ V_{+}}, P_{+d} := {f ∈ P : [f, d](U^{·}) ⊂ V_{+}^{·+1}}, P_{−d} := {f ∈ P : [f, d](U_{+}^{′·}) = 0}, P_{od} = P_{+d} ∩ P_{−d}.

1.3.2 Lemma. One has Hom_{±}^{d} = P_{±d} + Hom_{00}, Hom_{0}^{d} = P_{0d} + Hom_{00}.

Proof. Consider, e.g., the case of Hom_{+}^{d}. One has Hom_{+}^{d} = (P ∩ Hom_{+}^{d}) + Hom_{0}. An element f ∈ P ∩ Hom_{+}^{d} induces the linear map f̄: U^{·}/U_{+}^{·} → V^{·}/V_{+}^{·} such that α = [f̄, d] is of finite rank. One may find ḡ of finite rank such that [ḡ, d] = α. Lift ḡ to an element g ∈ P ∩ Hom_{0}; then f − g ∈ P_{+d}, and we are done. □

Now let us define the traces. Consider a single elliptic complex (V^{·}, d). We have a bunch of Lie subalgebras in 𝔤ℓ = 𝔤ℓV^{·} = Π𝔤ℓV^{i}. Pick subcomplexes V_{+}^{′·} ⊂ V_{+}^{·} ⊂ V^{·} formed by lattices; we get the corresponding parabolic subalgebra P ⊂ 𝔤ℓ and its standard subalgebras. Define the trace functional tr: P_{0d} → ℂ by formula trf := Σ(−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 = Σ(−1)^{i}tr_{V_{+}^{i}/V_{+}^{i′}}. The algebra 𝔤ℓ_{00} also carries the trace tr = Σ(−1)^{i}tr_{V^{i}}. Clearly on P_{0d} ∩ 𝔤ℓ_{00} these traces coincide, so, by 1.3.2, they define tr : 𝔤ℓ_{0}^{d} → ℂ.

1.3.3 Lemma. The trace functional tr : 𝔤ℓ_{0}^{d} → ℂ does not depend on the choice of V_{+}^{·}, V_{+}^{′·} and vanishes on [𝔤ℓ_{0}^{d}, 𝔤ℓ_{0}^{d}]. □

Let 𝔤̃ℓ be the central extension of 𝔤ℓ by ℂ which is the alternating Baer sum of 𝔤̃ℓV^{i}. Equivalently, to get 𝔤̃ℓ take the ideals 𝔤ℓ_{±} ⊂ 𝔤ℓ and the trace functional tr = Σ(−1)^{i}tr_{V^{i}} on 𝔤ℓ_{0}, and apply constructions 1.1.4, 1.1.5. We have canonical splittings s_{±} : 𝔤ℓ_{±} → 𝔤̃ℓ.

1.3.4 Lemma. These splittings extend to canonical splittings s_{±} : 𝔤ℓ_{±}^{d} → 𝔤̃ℓ; one has s_{+} − s_{−} = tr : 𝔤ℓ_{0}^{d} → ℂ.

Proof. Consider, say, the case of s_{+}. Let 𝔤̃ℓ_{+}^{d} be 𝔤̃ℓ restricted to 𝔤ℓ_{+}^{d}. Note that

𝔤ℓ_{+}^{d} = 𝔤ℓ_{+} + (𝔤ℓ_{−} ∩ 𝔤ℓ_{+}^{d}), so 𝔤̃ℓ_{+}^{d} comes from constructions 1.1.4, 1.1.5 applied to 𝔤ℓ_{+}^{d}, its ideals 𝔤ℓ_{+} and 𝔤ℓ_{−} ∩ 𝔤ℓ_{+}^{d} and the trace functional tr. We may even replace 𝔤ℓ_{−} ∩ 𝔤ℓ_{+}^{d} by the larger ideal 𝔤ℓ_{0}^{d} and, since tr extends to 𝔤ℓ_{0}^{d} by 1.3.3, according to 1.1.6 we get the desired section s_{+} : 𝔤ℓ_{+}^{d} → 𝔤̃ℓ. One treats s_{−} in a similar way; the formula s_{+} − s_{−} = tr results from 1.1.6. □

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 \overrightarrow{∼} W^{∗}). 1.4.1 For a lattice W_{+} ⊂ W let W_{+}^{⊥} be the orthogonal complement with respect to ( , ). This is also a lattice, and the parity of dimW_{+}^{⊥} − dimW_{+} ∈ ℤ does not depend on W_{+} (and depends on (W, ( , )) only). We will say that W is even or odd dimensional if dimW_{+}^{⊥} − dimW_{+} is even or odd, respectively. 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 ∈ M there is a lattice W_{+} such that W_{+}m = 0. Denote by CM_{W} the category of Clifford modules.

Let W_{+} ⊂ W be a lattice such that ( , )|_{W_{+}} = 0. Then the finite-dimensional vector space W_{+}^{⊥}/W_{+} carries an induced non-degenerate form. If M is a Clifford module, then M^{W_{+}} := {m ∈ M : W_{+}m = 0} is a W_{+}^{⊥}-invariant subspace of M, hence a Cliff(W_{+}^{⊥}/W_{+}, ( , ))-module.

1.4.3 Lemma. The functor CM_{W} → CM_{W_{+}^{⊥}/W_{+}}, M ⟼ M^{W_{+}}, is an equivalence of categories. The inverse functor is given by formula N ⟼ Cliff(W) ⨂_{Cliff(W_{+}^{⊥})}N. □

In particular, we see that CM_{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 CℓW the completion lim_{←} Cliff(W)/Cliff(W) · 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 CℓW is an associative algebra. Clearly, it acts on any Clifford module. 1.4.4 Let L_{+} ⊂ W be a maximal ( , )-isotropic lattice (so either L_{+}^{⊥} = L_{+} or dimL_{+}^{⊥}/L_{+} = 1 depending on parity of dimension of W). If L′_{+} is another such lattice, put λ(L_{+} : L′_{+}) := det(L_{+}/L_{+} ∩ L′_{+}). One has a canonical embedding i : λ(L_{+} : L′_{+}) ↪ CℓW/CℓW · L′_{+}, given by the formula v_{1} ∧ ⋯ ∧ v_{n} ⟼ ṽ_{1} ⋯ ṽ_{n} mod CℓW · L′_{+}. Here {v_{i}} is a basis of L_{+}/L_{+} ∩ L′_{+}, ṽ_{i} are any liftings of v_{i} to elements of L_{+}. For a Clifford module M one has a canonical isomorphism λ(L_{+} : L′_{+}) ⊗ M^{L′_{+}} \overrightarrow{∼} M^{L_{+}}, v ⊗ m ⟼ i(v)m.

Now let L_{−} ⊂ L be a maximal isotropic colattice (so codimL_{−} = dimL_{+} in case dimW is even, or codimL_{−} = dimL_{+} + 1 if dimW is odd). Put λ(L_{+}, L_{−}) = det(L_{+}∩L_{−}). For a Clifford module M put M_{L_{−}} := M/L_{−}M. One has a canonical isomorphism λ(L_{+}, L_{−}) ⊗ M_{L_{−}} \overrightarrow{∼} M^{L_{+}}, defined by formula v ⊗ m ⟼ vm̃, where v ∈ λ(L_{+}, L_{−}) ⊂ Cliff(W), m ∈ M_{L_{−}}, and m̃ ∈ M_{L_{−}}, and m̃ ∈ M is any element such that m̃ mod L_{−}M = m and vm̃ ∈ M^{L_{+}}. If M is irreducible, then dimM^{L_{+}} = dimM_{L_{−}} = 1, and we may rewrite the above isomorphisms as

λ(L_{+} : L′_{+}) = M^{L_{+}}/M^{L′_{+}}, λ(L_{+}, L_{−}) = M^{L_{+}}/M_{L_{−}}.

1.4.5 The algebra CℓW carries a natural ℤ/2-grading such that W lies in the degree 1 component. Denote by CM_{W}^{ℤ/2} the corresponding category of ℤ/2-graded Clifford modules. This is a semisimple category. If dimW is odd, then it has a single irreducible object; if dimW is even, then there are two irreducible objects that differ by a shift of ℤ/2-grading.

If dimW is even, then each M ∈ CM_{W} carries a natural 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 dimL_{+}/L_{+} ∩ L′_{+} is even. Denote the two element set of these components by ℤ/2_{W}; we will consider it as ℤ/2-torsor. Then any M ∈ CM_{W} carries a canonical ℤ/2_{W} -grading determined by the property that M^{L_{+}} ⊂ M^{α} for L_{+} ∈ α ∈ ℤ/2_{W}.

1.4.6 Let Cℓ^{Lie}W denote the Clifford algebra considered as Lie (super)algebra (with the above ℤ/2-grading; the (super)commutator is defined by the usual formula [a, b] = ab − (−1)^{αβ}ba for a ∈ Cℓ^{Lie}W^{α}, b ∈ Cℓ^{Lie}W^{β}). Denote by 𝔞W the normalizer of W ⊂ Cℓ^{Lie}W^{1} in Cℓ^{Lie}W. This is a Lie subalgebra of Cℓ^{Lie}W. As a vector space 𝔞W is the completion in CℓW of the subspace of all degree ≤ 2 polynomials of elements of W. One has 𝔞W^{1} = W. The Lie algebra ÕW := 𝔞W^{0} is called the spinor algebra of W. The subspace ℂ ⊂ CℓW coincides with center of 𝔞W. One has a canonical isomorphism 𝔞W/ℂ = OW ⋊ W. Here OW is the orthogonal Lie algebra of all ( , )-skew symmetric elements in 𝔤ℓW; the projection π : ÕW → ÕW /ℂ = OW is given by the adjoint action on W = 𝔞W^{1}.

The Lie superalgebra 𝔞W acts on any M^{·} ∈ CM_{W}^{ℤ/2} in an obvious manner. If M^{·} is irreducible, this action identifies 𝔞W with the normalizer of W in the Lie superalgebra End_{ℂ}M^{·}. Similarly, ÕW acts on any M ∈ CM_{W}, and, in case M is irreducible, ÕW coincides with the normalizer of W in End_{ℂ}M. 1.4.7 Here is another construction of ÕW. For a ∈ 𝔤ℓW denote by ^{t}a ∈ 𝔤ℓW the adjoint operator with respect to ( , ); for a ∈ 𝔤ℓ_{−}W one has ^{t}a ∈ 𝔤ℓ_{+}W. Consider now the ideal 𝔤ℓ_{−}W ⊂ 𝔤ℓW as an OW -module with respect to Ad-action. Then 𝔤ℓ_{−}W together with the surjective morphism 𝔤ℓ_{−}W ⟶^{∂} OW, a ⟼ a −^{t}a, is an OW -crossed module. The pairing {, } : 𝔤ℓ_{−}W × 𝔤ℓ_{−}W → ker ∂ (see 1.1.1) is given by formula {a_{1}, a_{2}} = [a_{1},^{t}a_{2}] + [a_{2},^{t}a_{1}]. Clearly ker ∂ ⊂ 𝔤ℓ_{0}W. The usual trace tr(1.2.4) vanishes on {ker ∂, ker ∂}; put o tr = 1/2tr. By 1.1.3 we get a central ℂ-extension ÕW′ = (𝔤ℓ_{−}W)_{o tr} of OW.

We define a canonical isomorphism α : ÕW′ \overrightarrow{∼} ÕW of central ℂ-extensions of OW as follows. One has a canonical identification W ⊗ W ≃ 𝔤ℓ_{00}W, w_{1} ⊗ w_{2} corresponds to a linear operator w ⟼ (w_{2}, w)w_{1}. This isomorphism extends by continuity to the isomorphism of completions lim_{\overleftarrow{W_{+}}} W ⊗(W/W_{+}) ≃ 𝔤ℓ_{−}W. Hence the

map 𝔤ℓ_{00}W = W ⊗ W → Cliff(W, (,)), a_{1} ⊗ a_{2} ⟼ a_{1}a_{2}, extends by continuity to the map α^{#} : 𝔤ℓ_{−}W → CℓW. Clearly α^{#} maps 𝔤ℓ_{−}W to 𝔞W^{0} = ÕW. For a_{1}, a_{2} ∈ 𝔤ℓ_{−}W, w ∈ W one has [α^{#}(a), w] = ∂(a)(w), [α^{#}(a_{1}), α^{#}(a_{2})] = α^{#}([∂a_{1}, a_{2}]). For b ∈ ker ∂ ∩ 𝔤ℓ_{00}W one has b = 1/2(b + ^{t}b) = Σ(w_{i} ⊗ w′_{i} + w′_{i} ⊗ w_{i}), hence α^{#}(b) = Σ(w_{i}, w′_{i}) = o tr b; by continuity this holds for any b ∈ ker ∂. This implies that α^{#} yields a map α : 𝔤ℓ_{−}W/ ker tr = ÕW′ → ÕW, which is the desired isomorphism of ℂ-extensions of OW. 1.4.8 Let L_{+} ⊂ W be a maximal isotropic lattice; denote by P_{L_{+}}O ⊂ OW the “parabolic” subalgebra of operators that preserve L_{+}. One has a canonical Lie algebra splitting s_{L_{+}} : P_{L_{+}}O → ÕW defined by formula s_{L_{+}}(a) = α^{#}(b), where b ∈ 𝔤ℓ_{−}W is any operator such that ∂(b) = a, b(L_{+}) = 0, (a − b)(W) ⊂ 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 ÕW with this property).

Similarly, let L_{−} ⊂ W be a maximal isotropic colattice. The corresponding parabolic subalgebra P_{L_{−}}O ⊂ OW also has a canonical Lie algebra splitting s_{L_{−}} : P_{L_{−}}O → ÕW defined by formula s_{L_{−}}(a) = α^{#}(b), where b ∈ 𝔤ℓ_{−}W is an operator such that ∂(b) = a, b|_{L_{−}} = a|_{L_{−}}, b(W) ⊂ L_{−}. For a Clifford module one has s_{L_{−}}(a)(M_{L_{−}}) = 0 (i.e., s_{L_{−}}(a)(M) ⊂ L_{−}M).

According to 1.4.4 for a ∈ P_{L_{+}}O∩P_{L_{−}}O one has (s_{L_{−}} −s_{L_{+}})(a) = tr_{L_{−}∩L_{+}}(a) ∈ ℂ ⊂ ÕW. If L′_{+} is another maximal isotropic lattice, then for a ∈ P_{L_{+}}O ∩ P_{L′_{+}}O one has (s_{L′_{+}} − s_{L_{+}})(a) = tr_{L_{+}/L_{+}∩L′_{+}}(a). 1.4.9 Let V be any Tate’s vector space. Then W := V ⊕V^{∗}, equipped with the form ((v, v^{∗}), (v′, v^{∗′})) := v^{∗}(v′) + v^{∗′}(v), is an even-dimensional space. For any lattice V_{+} ⊂ V and a colattice V_{−} ⊂ V a lattice L(V_{+}) = V_{+} ⊕ V_{+}^{⊥} ⊂ W and a colattice L(V_{−}) = V_{−} ⊕ V_{−}^{⊥} ⊂ W are maximal isotropic ones; clearly one has a canonical isomorphisms

λ(L(V_{+}) : L(V′_{+})) = det(V_{+}/V_{+} ∩ V′_{+})/ det(V′_{+}/V_{+} ∩ V′_{+})
λ(L(V_{+}), L(V_{−})) = det(V_{+} ∩ V_{−})/ det(V/V_{+} + V_{−}).

The algebra CℓW gets a natural ℤ-grading such that the subspaces V, V^{∗} (⊂ W ⊂ CℓW) lie in degrees 1, -1, respectively. Any Clifford module M has a canonical Dim_{V} -grading such that M^{L(V_{+})} lies in degree dimV_{+}.

The embedding i : 𝔤ℓV ↪ OW, ℓ ⟼ ℓ⊕(−^{t}ℓ), lifts canonically to a morphism of ℂ-extensions ĩ: 𝔤̃ℓV ⟶ ÕW constructed as follows. For ℓ_{+} ∈ 𝔤ℓ_{+}V choose a lattice V_{+} ⊃ Imℓ_{+}. Then i(ℓ_{+}) ∈ P_{L(V_{+})}O. Put ĩ_{+}(ℓ_{+}) = s_{L(V_{+})}i(ℓ_{+}) ∈ ÕW; by 1.4.8 this element is independent of a choice of V_{+}. Similarly, for ℓ_{−} ∈ 𝔤ℓ_{−}V choose a lattice V′_{+} ⊂ Ker ℓ_{−}; then i(ℓ_{−}) ∈ P_{L(V′_{+})}O, and ĩ_{−}(ℓ_{−}) := s_{L(V′_{+})}i(ℓ_{−}) ∈ ÕW depends on ℓ_{−} only. For ℓ_{0} ∈ 𝔤ℓ_{0}V one has (ĩ_{−} − ĩ_{+})(ℓ_{0}) = tr_{L(V_{+})/L(V_{+})∩L(V′_{+})}(iℓ_{0}) = trℓ_{0}

by 1.4.8. According to 1.2.3 we get a canonical morphism ĩ : 𝔤̃ℓ_{−1}V ⟶ ÕW of ℂ-extensions such that ĩs_{±} = ĩ_{±} : 𝔤ℓ_{±}V ⟶ ÕW (here 𝔤̃ℓ_{−1}V = (𝔤̃ℓV)_{−1}, see 1.1.7).

The action of 𝔤̃ℓV on M preserves the Dim_{V} -grading. If M is irreducible, then it is natural to denote the 𝔤̃ℓ_{−1}V -module M^{a}, a ∈ DimV, as Λ^{a}V (“semi-infinite wedge power”). Note that ∧^{a}V (as well as M itself) is defined up to tensorization with 1-dimensional ℂ-vector space. 1.4.10 We will need a version “with formal parameter” of the above constructions. Namely, let O = ℂ[[q]] be our base ring. Consider a flat complete O-module V (so lim_{←}V/q^{n}V). A Tate structure on V is given by Tate’s ℂ-vector space structure on

each V/q^{n}V such that each short exact sequence 0 → V/q^{m}V ⟶^{q^{n}} V/q^{m+n}V → V/q^{n}V → 0 is strongly compatible with the Tate structures (i.e., V/q^{m}V is a Tate’s subspace of V/q^{m+n}V and V/q^{n}V is the quotient space). A lattice V_{+} ⊂ V is an O-submodule such that V/V_{+} is O-flat, V_{+} = lim_{←}V_{+}/q^{n}V_{+} and V_{+}/q^{n}V_{+} is a lattice in V/q^{n}V for each n. One defines a colattice V_{−} ⊂ V in a similar way. For a Tate Omodule V one defines its dual V^{∗} in an obvious way; one has V^{∗}/q^{n}V^{∗} = (V/q^{n}V)^{∗}, V^{∗∗} = V.

Let W be Tate’s O-module and (,) : W ×W → O be a non-degenerate symmetric form (i.e., a symmetric isomorphism W \overrightarrow{∼} W^{∗}). Let Cliff(W) be the Clifford Oalgebra of (,). A Clifford module M is a Cliff(W)-module such that M is flat as O-module, M = lim_{←}M/q^{n}M, and W/q^{n}W acts on each M/q^{n}M in a continuous way (in discrete topology of M/q^{n}M). Such M carries the action of completed Clifford algebra

CℓW = lim_{\overleftarrow{n}} lim_{\overleftarrow{W_{+}^{(n)}}} Cliff(W)/q^{n}Cliff(W) + Cliff(W)W_{+}^{(n)}

(where W_{+}^{(n)} is a lattice in W/q^{n}W). Clearly M_{0}; = M/qM is Clifford module for (W_{0}, (,)_{0}) := (W/qW, (,)modq); if M′ is another Clifford module, then Hom(M, M′) is a flat O-module and Hom(M, M′)/qHom(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 EndM = O. All the facts 1.4.3-1.4.9 have an obvious ℂ[[q]]-version.

§2. Tate’s residues and Virasoro-type extensions

2.1 Tate’s construction of the local extension. Let F be a 1-dimensional local field, and O_{F} ⊂ F be the corresponding local ring. A choice of uniformization parameter t ∈ O_{F} identifies O_{F} with ℂ[[t]], and F with ℂ((t)). Let E be an F - vector space of dimension n < ∞. Denote by DE the algebra of F -differential operators acting on E. A choice of a basis of E identifies DE with the algebra of matrix differential operators a_{N}∂_{t}^{N} + ⋯ + a_{1}∂_{t} + a_{0}, a_{i} ∈ Mat_{n}(F).

2.1.1 The space E, considered as ℂ-vector space, is actually a Tate’s vector space in a canonical way. A basis of neighbourhoods of 0 is formed by O_{F} -submodules of E that generate E as F -module. We will denote by EndE, 𝔤ℓ_{±}E, etc., the corresponding algebras of endomorphisms of E, considered as Tate’s ℂ-vector space.

Clearly DE ⊂ EndE. We may restrict the central extension 𝔤̃ℓE of 𝔤ℓE to DE^{Lie} to get a central extension 0 → ℂ → D̃E → DE^{Lie} → 0 of the Lie algebra DE^{Lie}.

It is easy to compute a 2-cocycle of this extension explicitly. Namely, let us choose a parameter t ∈ O_{F} and an F -basis {v_{i}} in E. Put E_{+} = ∑_{i}O_{F} v_{i}, E_{−} =

∑_{i}t^{−1}ℂ[t^{−1}]v_{i}: this is a lattice and a colattice in E and E = E_{+} ⊕ E_{−}. For

ℓ ∈ 𝔤ℓE define the operator ℓ_{+} ∈ 𝔤ℓ_{+}E by formula ℓ_{+}|_{E_{+}} = ℓ|_{E_{+}}, ℓ_{+}|_{E_{−}} = 0. Clearly this map 𝔤ℓE → 𝔤ℓ_{+}E, ℓ ⟼ ℓ_{+}, lifts the canonical projection 𝔤ℓE → 𝔤ℓE/𝔤ℓ_{−}E = 𝔤ℓ_{+}E/𝔤ℓ_{0}E. Hence by 1.1.4 it defines a section σ : 𝔤ℓE → 𝔤̃ℓE; the corresponding 2-cocycle is given by formula ℓ_{1}, ℓ_{2} ⟼ α(ℓ_{1}, ℓ_{2}) = [σ(ℓ_{1}), σ(ℓ_{2})] −

σ([ℓ_{1}, ℓ_{2}]) = tr([ℓ_{1+}, ℓ_{2+}] − [ℓ_{1}, ℓ_{2}]_{+}). Take now ℓ_{1} = At^{a} \frac{∂_{t}^{b}}{b!} , ℓ_{2} = A′t^{a′} ^{\frac{∂_{t}^{b′}}{b′!}} , where A, A′ ∈ Mat_{n}(ℂ), a, a′ ∈ ℤ, b, b′ ∈ ℤ_{≥0}. Clearly α(ℓ_{1}, ℓ_{2}) = 0 if a − b ≠ b′ − a′. Assume that a − b = b′ − a′; since α is skew-symmetric we may assume that n = a − b ≥ 0. Then one has

α(ℓ_{1}, ℓ_{2}) = −T r(AA′) ∑_{i=0}^{n−1} \binom{i}{b′}\binom{i − n}{b}.

2.1.2 Let AE ⊂ DE^{Lie} be a Lie subalgebra that consists of operators of order ≤ 1 with scalar symbol (i.e., the operators of type a_{0} + a_{1}∂_{t}, a_{0} ∈ End_{F} E, a_{1} ∈ F). Denote by T_{F} the Lie algebra of vector fields on F. One has a canonical short exact sequence of Lie algebras 0 → End_{F} E^{Lie} → AE ⟶^{σ} T_{F} → 0, σ(a_{0} + a_{1}∂_{t}) = a_{1}∂_{t}. Let ÃE be the ℂ-extension of AE induced from D̃E. The above formulas reduce to the following ones:

α(At^{a}, Bt^{b}) = bδ_{a}^{−b}trAB, α(At^{a}, t^{b+1}∂_{t}) = \frac{a − a^{2}}{2} δ_{a}^{−b}trA, α(t^{a+1}∂_{t}, t^{b+1}∂_{t}) = \frac{n}{6}(a^{3}−a)δ_{a}^{−b}.

This is the Kac-Moody-Virasoro cocycle.

2.1.3 Consider the case E = F. One has an obvious embedding T_{F} ⊂ AF which defines the ℂ-extension T̃_{F} of T_{F} with cocycle α_{V ir}(t^{a+1}∂_{t}, t^{b+1}∂_{t}) = \frac{1}{6}(a^{3} − a)δ_{a}^{−b}. This T̃_{F} is called (a local) Virasoro algebra. For any c ∈ ℂ consider the ℂ-extension T̃_{F c} (see 1.1.7). Since T_{F} is perfect, T̃_{F c} has no automorphisms. One knows that any central ℂ-extension of T_{F} is isomorphic (canonically) to a unique T̃_{F c} (one has H^{2}(T_{F} , ℂ) ≃ ℂ).

2.1.4 Now consider for j ∈ ℤ a 1-dimensional F -vector space ω_{F}^{⊗j} of j-differentials (the elements of ω_{F}^{⊗j} are tensors fdt^{⊗j}, f ∈ F). The Lie algebra T_{F} acts canonically on ω_{F}^{⊗j} by Lie derivatives, i.e., we have a canonical embedding T_{F} ↪ Aω_{F}^{⊗j}. Denote

by T̃_{F}^{(j)} the corresponding ℂ-extensions of T_{F} induced from Ãω_{F}^{⊗j}. The explicit formula for this action is φ∂_{t}(fdt^{⊗j}) = (φ∂_{t}(f) + jf∂_{t}(φ))dt^{⊗j}, i.e., with respect to the basis dt^{⊗j} a field t^{a+1}∂_{t} acts as t^{a+1}∂_{t} + j(a + 1)t^{a}. The formulas 2.1.2 immediately show that a 2-cocycle for T̃_{F}^{(j)} coincides with (6j^{2} −6j +1)α_{V ir}. Hence T̃_{F}^{(j)} coincides with T̃_{F (6j^{2}−6j+1)}.

2.2 A geometric construction of a global extension. Let us describe the above extensions in geometric language.

2.2.1 Let C be a smooth algebraic curve (not necessary compact). Denote by ω = Ω_{C}^{1} the sheaf of 1-forms, and by H = H_{DR}^{1} = Ω_{C}^{1}/dO_{C} the de Rham cohomology sheaf (in the Zariski topology of C). For a vector bundle E on C let D = DE denote the sheaf of differential operators on E, and E^{∘} := ωE^{∗}. Then E is a left D-module, E^{∘} is a right D-module (so one has a canonical anti-isomorphism t : DE → DE^{0},

see, e.g., [B]), and the pairing E^{0}⊗E ⟶^{⟨ ⟩} ω quotients to the pairing E^{0} ⨂_{DE} E → H.

Let Δ : C → C × C be the diagonal; we will identify the sheaves on C with ones on C × C supported on the image of Δ. Consider the sheaf E ⊠ E^{0} := p_{1}^{∗}E ⊗ p_{2}^{∗}E^{0} on C × C. Recall that one has a canonical isomorphism δ : E ⊠ E^{0}(∞Δ)/E ⊠ E^{0} \overrightarrow{∼} D. Explicitly, for a “kernel” k(t_{1}, t_{2}) = e(t_{1})e^{0}(t_{2})f(t_{1}, t_{2}), e ∈ E, e^{0} ∈ E^{0}, f(t_{1}, t_{2}) ∈ O_{C×C}(∞Δ), the corresponding differential operator δ(k) acts on sections of E according to formula (δ(k)ℓ)(t_{1}) = Res_{t_{2}=t_{1}}⟨k(t_{1}, t_{2})ℓ(t_{2})⟩ = e(t_{1})Res_{t_{2}=t_{1}}f(t_{1}, t_{2})⟨e^{0}(t_{2})ℓ(t_{2})⟩. Here ℓ ∈ E, ⟨e^{0}(t_{2})ℓ(t_{2})⟩ ∈ ω, ⟨k(t_{1}, t_{2})ℓ(t_{2})⟩ ∈ E ⊠ ω(∞Δ); we take the residue along the t_{2} variable. The right action of δ(k) on sections of E^{0} is given by formula (mδ(k))(t_{2}) = Res_{t_{1}=t_{2}}f(x, t_{2})⟨m(t_{1})e(t_{1})⟩)e^{0}(t_{2}).

2.2.2 Put PE_{n} := lim\overleftarrow{i}E ⊠ E^{0}((n + 1)Δ)/E ⊠ E^{0}(−iΔ), PE = ∪PE_{n}, so we

have an isomorphism δ : PE/PE_{−1} \overrightarrow{∼} DE. Clearly PE is a DE-bimodule (the left and right actions are the obvious actions along the first, resp. the second variable), and δ is a morphism of bimodules, i.e., PE is a DE-crossed module (see 1.1). Let t : PE → PE^{0} be minus the isomorphism “transposition of coordinates” (here minus comes since E, E^{0} have “odd” nature). Then for k ∈ PE one has tδ(k) = δ(^{t}k), and ^{t} is an “anti-isomorphism” between crossed modules.

The pairing ⟨ ⟩ : PE⨂_{DE}PE → PE_{−1} from 1.1, ⟨k_{1}, k_{2}⟩ = δ(k_{1})k_{2} − k_{1}δ(k_{2}), is

given by formula

⟨k_{1}k_{2}⟩(t_{1}, t_{2}) = (Res_{z=t_{1}} + Res_{z=t_{2}})⟨k_{1}(t_{1}, z)k_{2}(z, t_{2})⟩ = ∫_{γ_{t_{1},t_{2}}} ⟨k_{1}(t_{1}, z)k_{2}(z, t_{2})⟩.

Here ⟨k_{1}(t_{1}, z)k_{2}(z, t_{2})⟩ is the 1-form of variable z (with values in E_{t_{1}} ⊗ E_{t_{2}}^{0}), and γ_{t_{1},t_{2}} is a loop round z = t_{1} and z = t_{2}. The corresponding Lie algebra pairing { } : S^{2}PE → PE_{−1} is {k_{1}, k_{2}} := ⟨k_{1}, k_{2}⟩ + ⟨k_{2}, k_{1}⟩. Let tr : PE_{−1} → ω be the composition PE_{−1} → PE_{−1}/PE_{−2} = E ⊗ E^{0} → ω. We have

tr{k_{1}, k_{2}} = (Res_{1} − Res_{2})⟨k_{1}(t_{1}, t_{2})k_{2}(t_{2}, t_{1})⟩.

Here k_{2}(t_{2}, t_{1}) = ^{t}k_{2} ∈ PE^{0} is k_{2} with coodinates transposed, ⟨k_{1}(t_{1}, t_{2})k_{2}(t_{2}, t_{1})⟩ is a 2-form with poles along the diagonal and Res_{1}, Res_{2} : Ω_{C×C}^{2}(∞Δ) → ω_{C} are residues around the diagonal along the first and second coordinates, respectively. Clearly, Res_{1} − Res_{2} vanishes on Ω_{C×C}^{2}(Δ) and has image in exact forms. In fact, there is a canonical map R̃es : Ω_{C×C}^{2}(∞Δ)/Ω_{C×C}^{2}(Δ) → O_{C} such that dR̃es = Res_{1} − Res_{2} (see [B Sch] (2.11)). An explicit formula for R̃es is

R̃es(f(t_{1}, t_{2})(t_{1} − t_{2})^{−i−1}dt_{1} ∧ dt_{2}) = i!^{−1} ∑_{a+b=i−1} ∂_{t_{1}}^{a}∂_{t_{2}}^{b}f(t_{1}, t_{2})|_{t_{1}=t_{2}=t}.

Here f(t_{1}, t_{2}) ∈ O_{C×C}. Hence one has tr{k_{1}, k} = dR̃es⟨k_{1},^{t}k_{2}⟩. Note that the symmetric pairing PE ⊗ PE → O_{C}, k_{1}, k_{2} ⟼ {k_{1}, k_{2}}^{∼} := R̃es⟨k_{1},^{t}k_{2}⟩ vanishes on ∑_{a+b=−1}PE_{a} ⊗ PE_{b}; in particular, it induces the pairing on PE_{1}/PE_{−2}.

According to 1.1.2, 1.1.3 we get a central extension D̃E of the Lie algebra DE^{Lie} by H defined by a following commutative diagram:

0 ⟶ PE_{−1} ⟶ PE ⟶^{δ} DE ⟶ 0
↓tr  ↓  ∥
ω ↓
0 ⟶ H ⟶ D̃E ⟶ DE^{Lie} ⟶ 0.

2.2.3 Denote by AE ⊂ DE^{Lie} the Lie subalgebra of differential operators of order ≤ 1 with scalar symbol. In other words, AE is the Lie algebra of infinitesimal symmetries of (C, E): the elements of AE are pairs (τ, τ̃), where τ ∈ P_{C} is a vector field, and τ̃ is an action of τ on E (so τ̃ is an order 1 differential operator with symbol equal to τ).

The constructions of 2.2.2 give rise to a differential graded Lie algebra AE defined as follows. One has A^{0}E = AE, A^{−1}E is pre-image of AE ⊂ DE by the projection PE/ ker tr ⟶^{δ} D_{E} (so we have short exact sequence 0 → ω → A^{−1}E ⟶^{δ} AE → 0), and finally A^{−2}E = O_{C}; all the other components of A^{·}E are zero ones. The differential d : A^{−2}E = O_{C} → ω ⊂ A^{−1}E is the de Rham differential, and A^{−1}E → AE is δ. The bracket components [ ]^{ij} : A^{i}E × A^{j}E → A^{i+j}E are the following. [ ]^{00} is the usual bracket [ ]^{0−1} comes from D^{Lie}-action on PE, [ ]^{0,−2} is the action of AE on O_{C} via σ : AE → T_{C}, and [ ]^{−1−1} is { , }^{∼} defined above. So A^{·}E contains de Rham complex Ω_{C}^{·}[2] as an ideal, A^{·}E/Ω_{C}^{·}[2] is acyclic and the central extension ÃE = A^{−1}E/dA^{−2}E of AE by H (see 1.13) coincides with restriction of D̃E to AE ⊂ DE^{Lie}.

2.2.4 Consider the case E = O_{C}. An obvious embedding P_{C} ↪ AO_{C} defines the central H-extension P̃_{C} called a global Virasoro algebra. As in 2.1.3 for c ∈ ℂ we will denote by P̃_{Cc} the H-extension of P_{C} which is c-multiple of P̃_{C}. Since P_{C} is perfect (see 2.5 below), the extensions P̃_{Cc} have no automorphisms.

2.2.5 Consider for j ∈ ℤ the sheaf ω^{⊗j}. The natural action of P_{C} on ω^{⊗j} by Lie derivatives defines a canonical embedding of Lie algebras P_{C} ↪ Aω^{⊗j}. Denote by P̃_{C}^{(j)} the induced H-extension Ãω^{⊗j}|_{P_{C}}. Given a local coordinate t, one may

consider elements of P̃_{C}^{(j)} as expressions

φ_{(f,g)}^{(j)} = [\frac{f(t_{1})}{(t_{2} − t_{1})^{2}} + j \frac{∂_{t_{1}}f(t_{1})}{t_{2} − t_{1}} + g(t_{1})] dt_{1}^{⊗j}dt_{2}^{⊗1−j},

where f, g ∈ O_{C}, modulo the ones of type φ_{(0,∂_{t}h)}. The map P̃_{C} = P̃_{C}^{(0)} → P̃_{C}^{(j)} defined by formula φ_{(f,g)}^{(0)} ⟼ φ_{(f,(6j^{2}−6j+1)g)}^{(j)} is a morphism of Lie algebras, and does not depend on a choice of a local coordinate t. Hence it defines a canonical isomorphism P̃_{C(6j^{2}−6j+1)} \overrightarrow{∼} P̃_{C}^{(j)} of H-extensions of C (see [B Sch]). Unfortunately, we do not know any “coordinate-free” explanation of this isomorphism.

2.3 Compatibility with Tate’s construction. Let x ∈ C be a point. We may consider the constructions of 2.2 locally at x. Namely, let O_{x}^{∧} be the completed local ring of C at x, O_{(x,x)}^{∧} be the completed local ring of C × C at (x, x), F_{x} ⊃ O_{x}^{∧} the local field at x, so if t is a parameter at x then O_{(x,x)}^{∧} = ℂ[[t_{1}, t_{2}]]. Denote by R the localization of O_{(x,x)}^{∧} with respect to t_{1}^{−1}, t_{2}^{−1}, (t_{1} − t_{2})^{−1}. Put ω_{(x)} := F_{x} ⊗_{O} ω, E_{(x)} := F_{x} ⊗_{O} E, D_{(x)} = DE_{(x)} := F_{x} ⊗_{O} DE_{(x)}, P_{(x)} = PE_{(x)} = E ⊗_{O} R⊗_{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 D̃_{(x)} of D̃_{(x)}^{Lie} by

H_{(x)} = ω_{(x)}/dF_{x} \overrightarrow{∼}^{Res} ℂ.

2.3.1 By 2.1, E_{(x)} is a Tate’s vector space, and we have the embedding i_{x} : D_{(x)} ↪ EndE_{(x)}. For k = k(t_{1}, t_{2}) ∈ P_{(x)} let k_{−}, k_{+} ∈ EndE_{(x)} be the linear operator defined by formulas

[k_{−}(e)](t) = −Res_{t_{2}=0}⟨k(t, t_{2})e(t_{2})⟩, [k_{+}(e)](t) = (Res_{t_{2}=t_{1}}+Res_{t_{2}=0})⟨k(t, t_{2})e(t_{2})⟩.

Here e(t) ∈ E_{(x)}, ⟨k(t, t_{2})e(t_{2})⟩ ∈ E ⊗ R ⊗ ω, and the residues are taken along the second variable. According to 2.2.1 one has i_{x}δ(k) = k_{−} + k_{+}. Denote by i_{x±}^{#} : P_{(x)} → EndE_{(x)} the maps i_{x±}^{#}(k) = k_{±}.

2.3.2 Lemma. (i) For k ∈ P_{(x)} one has k_{±} ∈ End_{±}E_{(x)}. (ii) The commutative diagram

P_{(x)} ⟶^{i_{x}^{#}=(i_{x+}^{#},i_{x−}^{#})} End_{+}E_{(x)} ⊕ End_{−}E_{(x)}
↓δ  ↓
D  ⟶^{i_{x}}  End E_{(x)}

is an i_{x}-morphism of crossed modules (see 1.1). (iii) For k ∈ ker δ ⊂ P_{(x)} one has Res_{x}tr(k) = tri_{x}^{#}(k)(= trk_{+} = −trk_{−}). (iv) Let us identify E_{(x)}^{0} with E_{(x)}^{∗} via the pairing ( , ) : E × E^{0} → ℂ, (e, e^{0}) =

Res⟨e, e^{0}⟩; this gives the anti-isomorphism t : EndE_{(x)} → EndE_{(x)}^{0}. Then the diagram

PE_{(x)} ⟶^{i_{+}^{#}} End_{+}E_{(x)}
t↓≀  t↓≀
PE_{(x)}^{0} ⟶^{i_{−}^{#}} End_{−}E_{(x)}^{0}

commutes.

Proof. Assume for simplicity of notation that E = O_{C}, so E_{(x)} = F_{x}. The statement k_{−} ∈ End_{−}F_{x} from (i) is clear, since k_{−} vanishes on the lattice t^{N}O_{x}^{∧} ⊂ 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_{+} ∈ 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 γ_{±}(t) be the following loops in the t_{2}-complex plane t_{1} = t:

Then for any function f ∈ F_{x} one has [k_{±}(f)](t) = \frac{1}{2πi} ∫_{γ_{±}(t)} k(t, t_{2})f(t_{2}).

Denote by U a small neighbourhood of zero in ℂ × ℂ with coordinate cross and diagonal removed. One has the following 2-dimensional cycles C_{±} in U. Fix a small real numbers 0 < ϵ < r ≪ 1. Then C_{+} = {(z_{1}, z_{2}) ∈ ℂ × ℂ : |z_{1}| = ϵ, |z_{2}| = r}, C_{−} = {(z_{1}, z_{2}) ∈ ℂ × ℂ : |z_{1}| = r, |z_{2}| = ϵ}; the orientation of C_{+} is a standard orientation of S^{1} × S^{1}, and the one of C_{−} is minus the standard orientation.

The above formula for the action of k_{±} implies that for a 1-form g ∈ F_{x}^{0} = ω_{(x)} one has (g, k_{±}(f)) = ∫_{C_{±}} 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)) = ((^{t}k)_{−}(g), f). □

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 ℂ-extensions ĩ_{x} : D̃_{(x)} → 𝔤̃ℓE_{(x)}, ĩ_{x}(k) = s_{+}(k_{+}) + s_{−}(k_{−}), or, equivalently, the isomorphism of ℂ-extensions D̃_{(x)} \overrightarrow{∼} D̃E_{(x)} (see 2.1.1).

2.3.4 Assume now that our curve C is compact. Let X = {x_{i}} ⊂ C be a finite non empty set of points, and E be a vector bundle on U = C \ X. Put E_{(X)} = ΠE_{(x_{i})}, D_{(X)} = ΠD_{(x_{i})}. Denote by D̃_{(X)}, a ℂ-extension of D_{(X)}^{Lie} which is the Baer

sum of ℂ-extensions D_{(x_{i})}, so D̃_{(X)} = ΠD̃_{(x_{i})}/{(a_{i}) ∈ ℂ^{X} : ∑ a_{i} = 0}. Clearly D̃_{(X)} coincides with the ℂ-extension D̃E_{(X)} induced from 𝔤̃ℓE_{(X)} via the embedding D_{(X)} ↪ ∏ End E_{(x_{i})} ↪ End E_{(X)}.

Put D_{U} := H^{0}(U, DE_{U}) and consider the central extension 0 → H_{DR}^{1}(U) → D̃_{U} → D_{U} → 0 constructed in 2.2.2. One has the ”localization around x_{i}” maps D_{U} ↪ ∏ D_{(x_{i})}, D̃_{U} → ∏ D̃_{(x_{i})}. The composition D̃_{U} → ∏ D̃_{(x_{i})} → D̃_{(X)} vanishes on H_{DR}^{1}(U) (since ∑_{X}Res_{x_{i}} = 0). Hence it defines a canonical morphism s_{X} :

D_{U}^{Lie} → D̃_{(X)} that lifts the embedding D_{U} ↪ D_{(X)}.

This morphism could be constructed by pure linear algebra means. Namely, consider the colattice E_{U} = H^{0}(U, E) ⊂ E_{(X)}. Clearly D_{U}^{Lie} ⊂ P_{E_{U}} ⊂ 𝔤ℓE_{(X)}, hence we have the splitting s_{E_{U} |D_{U}} : D_{U}^{Lie} → D̃E_{(X)} = D̃_{(X)} (see 1.2.5).

2.3.5 Lemma. This splitting coincides with the above s_{X}.

Proof. Let ∂ ∈ D_{U} be a differential operator. Choose a section k ∈ H^{0}(U × U, E ⊠ E^{0}(∞Δ)) such that δ(k) = ∂. Denote by k_{−} = (k_{−}^{x_{i}}) ∈ Hom(E_{(X)}, E_{U}) the morphism given by formula k_{−}(e_{x_{i}}) = Σk_{−}^{x_{i}}(e_{x_{i}}), k_{−}^{x_{i}}(e_{x_{i}}) = −Res_{x_{i}}⟨k · e_{x_{i}}⟩ ∈ E_{U}. Here e_{x_{i}} ∈ E_{x_{i}}, ⟨k · e_{x_{i}}⟩ ∈ H^{0}(U × SpecF_{x_{i}}, E ⊠ ω(∞Δ)) is a section obtained by convolution of k and e_{x_{i}} (where e_{x_{i}} is considered as a section of O_{U} ⊠ 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).

Let j = (j_{x_{i}}) : E_{U} ↪ E_{(X)} be the embedding. The residue formula implies that for e ∈ E_{U} one has k_{−}(j(e)) = ∂(e). Hence j ∘ k_{−} ∈ P_{E_{U}} ⊂ 𝔤ℓE_{(X)}, one has j ∘ k_{−} ∈ 𝔤ℓ_{−}E_{(X)}, ∂ − j ∘ k_{−} ∈ 𝔤ℓ_{+}E_{(X)}, and, according to 1.2.5, s_{E_{U}} (∂) coincides with s_{−}(j ∘ k_{−}) + s_{+}(∂ − j ∘ k_{−}).

Now consider j ∘ k_{−} as a matrix (j ∘ k_{−})_{x_{j}}^{x_{i}} ∈ Hom(E_{(x_{i})}, E_{(x_{j})}). Let j ∘ k_{−}^{diag} = Σ(j ∘ k_{−})_{x_{i}}^{x_{i}} ∈ End E_{(X)} be the diagonal part of j ∘ k_{−}. According to 2.3.2, one has s_{X}(∂) = s_{−}(j∘k_{−}^{diag})+s_{+}(∂−j∘k_{−}^{diag}). Hence s_{X}(∂)−s_{E_{U}} (∂) = tr(j∘k_{−}−j∘k_{−}^{diag}). This is a trace of a matrix in 𝔤ℓ_{0}E_{(X)} with zero diagonal component which is zero, q.e.d. □

2.3.6 We will often use the morphism s_{X} for appropriate subalgebras of D_{U}^{Lie}, say, for AE_{U}.

2.4 Spinors and theta-characteristics. Let W be a vector bundle on our curve C equipped with a symmetric non-degenerate pairing ( , ) : W × W → ω.

2.4.1 One may consider ( , ) as an isomorphism W ≃ W^{0}, hence we have the involution ^{t} : DW → DW such that ^{t}(∂_{1}∂_{2}) = ^{t}∂_{2}^{t}∂_{1}, and ^{t} acts on degree n symbols as multiplication by (−1)^{n}. Denote by ODW the anti-invariants of ^{t}; this is a Lie subalgebra of DW^{Lie}.

The isomorphism W ≃ W^{0} also defines an involution ^{t} : PW → PW (see 2.2.2) such that ^{t}δ = δ^{t}. Let OPW be the anti-invariants of ^{t} in PW; put oδ = δ|_{OPW}. The action of DW on PW defines the ODW -action on OPW, and oδ : OPW → ODW is an ODW -crossed module. The trace otr which is −\frac{1}{2} of the composition

ker oδ → W ⊗ W^{0} ⟶^{( , )} ω ⟶ H defines by 1.1.3, a canonical central H-extension { \atop ODW} of ODW. In ODW we have a Lie subalgebra OAW = AW ∩ ODW of infinitesimal symmetries of (C, W, ( , )): this is an extension of P_{C} by an orthogonal Lie algebra OW ⊂ End W. Denote by { \atop OAW} the central extension { \atop ODW} |_{OAW}. Note that if rkW = 1, i.e., if W = ω^{⊗1/2} is a theta-characteristic, then Oω^{⊗1/2} = 0, hence OAω^{⊗1/2} = T_{C}. The formula from 2.2.5 applied to j = 1/2 gives a canonical

isomorphism { \atop OAω^{1/2}} = T̃_{C−1/2}.

2.4.2 If E is any vector bundle, and W = E ⊕ E^{0} with obvious ( , ), then the Lie algebras embedding j : DE → ODW, ∂ ⟼ (∂, −^{t}∂), lifts to a morphism of crossed modules j^{#} : PE → OPW, k ⟼ (k, −^{t}k). For k ∈ ker δ one has otr(j^{#}k) = −trk. So we get a canonical morphism j̃ : D̃E_{−1} → ODW of H-extensions (see 1.1.7 for -1 index).

2.4.3 Let us consider a local version of the above construction. Now our curve is a punctured disc SpecF_{x}, so one has the identification Res_{x} : H(F_{x}) \overrightarrow{∼} ℂ. The Tate ℂ-vector space W_{(x)} carries a non-degenerate symmetric form ( , )_{•} defined by formula (w_{1}, w_{2})_{•} = Res_{x}(w_{1}, w_{2}). The action of DW_{(x)} on W_{(x)} gives the embedding oi_{X} : ODW_{(x)} ↪ OW_{(x)}. It lifts to an oi_{x}-morphism oi_{x}^{#} : OPW_{(x)} ⟶ 𝔤ℓ_{−}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 ∈ ker δ one has otr(k) = \frac{1}{2}trk_{−} = otr(k_{−}) by 2.3.2 (iii), 1.4.7. Hence oi_{x}^{#} defines a canonical morphism of ℂ-extensions õi_{x} :

{ \atop ODW_{(x)}} ↪ { \atop OW_{(x)}}.

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 ⊂ C, and our bundle (W, ( , )) on U = C \ X. We get a Tate vector space W_{(X)} = ∏ W_{(x_{i})} with the form ( , )_{(X)} = ∑( , )_{(x_{i})}, a central ℂ-

extension { \atop ODW_{(X)}} ⊂ { \atop OW_{(X)}} of ODW_{(X)} = ∏ ODW_{(x_{i})} ⊂ OW_{(X)}. Just as in

2.3.4 a localization at X morphism ODW_{U} := H^{0}(U, ODW) ⟶ ODW_{(X)} lifts

canonically to a morphism s_{X} : ODW_{U} ⟶ { \atop ODW_{(X)}}; as in 2.3.5 this s_{X} coincides with the lifting s_{W_{U}} |_{ODW_{U}} from 1.4.8. Certainly s_{X} extends in an obvious manner to a morphism of Lie superalgebras ODW_{U} ⋉ W_{U} → 𝔞W_{(X)} (here W_{U} has odd degree, for 𝔞W_{(X)}, see 1.4.6).

2.4.5 By Serre’s duality W_{U} is a maximal isotropic colattice in W_{(X)}.

2.5 Simplicity of Lie algebra of vector fields. The following lemma will be of use:

2.5.1 Lemma. Let C be a smooth curve. Then the Lie algebra T = H^{0}(C, T_{C}) of vector fields on C is simple.

Proof. The case of compact C is clear, so we will assume that C is affine. Let I ⊂ T be a non-zero ideal; we have to show that I = T. Let τ ∈ I be a non-zero vector field. Note that if g ∈ O(C) is a function such that gτ ∈ I and f ∈ O(C) is any function, then τ(f)gτ = \frac{1}{2}([gτ, fτ] + [τ, fgτ]) ∈ I. Let A_{τ} ⊂ O(C) be the subalgebra of functions generated by all functions τ(f), f ∈ O(C). The previous remark implies (by induction) that A_{τ} τ ⊂ I. One may describe A_{τ} explicitly, namely A_{τ} consists precisely of those f ∈ O(C) that take equal values at zeros of τ and ord_{x}(f − f(x)) ≥ ord_{x}(τ) for any x ∈ C; this condition is non empty only for x = zero of τ. (To see this, consider the morphism π : C → C′ = SpecA_{τ}. Clearly A_{τ} is a curve. An easy local analysis at points at ∞ of C shows that π is finite. If x, y ∈ C, x ≠ y, are not zeros of τ, then a finite jet at x, y of the functions τ(f), f ∈ O(C), could be arbitrary ones, hence π is isomorphism on the complement of zeros of τ. An easy local analysis at zeros of τ finishes the proof). In particular, any function that vanishes at zeros of τ with large order of zero lies in A_{τ}. Hence I contains any vector field that vanishes at zeros of τ with sufficiently large order of zero (namely, twice that of τ). A trivial local analysis at zeros of τ (take brackets of elements of I with vector fields non-vanishing at zeros of τ) shows that I = T. □

2.5.2 Corollary. If C is an affine curve, then T has no non-trivial finite dimensional representations. □

§3. Localization of representations

3.1 Harish-Chandra modules. Recall some definitions.

3.1.1 Let K be a pro-algebraic group. A K-module M is a comodule over the coalgebra O(K). Equivalently, M is a vector space with an algebraic K(ℂ)-action. Here “algebraic” means that M is a union of finite dimensional K(ℂ)-invariant subspaces M_{α} such that K(ℂ) acts on M_{α} via an algebraic action of a factor group K/K_{α} of finite type. Any K-module is a Lie K-module in a natural way.

3.1.2 A Harish-Chandra pair (𝔤, K) consists of a Lie algebra 𝔤 and a pro-algebraic group K together with an “adjoint” action Ad of K(ℂ) on 𝔤 and a Lie algebra embedding i : LieK ↪ 𝔤 that satisfy the compatibilities:

(i) The embedding i commutes with adjoint actions of K. (ii) The action Ad is “pro-algebraic”: for any normal subgroup K′ ⊂ K such that

K/K′ has finite type the action of K(ℂ) on 𝔤/i(LieK′) is algebraic. (iii) The ad ∘ i-action of Lie K on 𝔤 coincides with the differential of the Ad-action.

3.1.3 Let (𝔤, K) be a Harish-Chandra pair. A (𝔤, K)-module, or a Harish-Chandra module, is a ℂ-vector space equipped with 𝔤- and K-module structures such that (i) For k ∈ K, h ∈ 𝔤, m ∈ M one has Ad_{k}(h)m = khk^{−1}(m). (ii) The two Lie K-actions on M (the one that comes from 𝔤-action via i, and the

differential of K-action) coincide.

We denote by (𝔤, K)-mod the category of (𝔤, K)-modules.

3.1.4 Let T be any K-torsor. Denote (𝔤, K)_{T} = (𝔤_{T} , K_{T}) the T -twist of (𝔤, K) with respect to adjoint action; this is a Harish-Chandra pair. If M is a (𝔤, K)- module, then the T -twist M_{T} is a (𝔤_{T} , K_{T})-module, and M ⟼ M_{T} is equivalence of categories (𝔤, K)-mod \overrightarrow{∼} (𝔤_{T} , K_{T})-mod.

3.1.5 The following version of the above definitions is quite convenient.

A pro-algebraic groupoid V is a groupoid such that for any object X the group AutX carries a pro-algebraic structure and for any f : X → Y the map Ad_{f} : AutY \overrightarrow{∼} AutX preserves the pro-algebraic structures (the objects of V form a usual set with no algebraic structure). A V-module is a functor M : V → V ect_{ℂ} such that for any X ∈ V the AutX-action on M_{X} is algebraic.

A Harish-Chandra groupoid (𝔤, V) is a pro-algebraic groupoid V together with a functor X ⟼ (𝔤_{X}, K_{X}) from V to the category of Harish-Chandra pairs equipped with a canonical identification of “group part” K_{X} of the functor with AutX; we assume that for g ∈ AutX = K_{X} the “functorial” action of g on 𝔤_{X} coincides with the Ad-action from 3.1.3.

One defines a representation of our Harish-Chandra groupoid (or simply a (𝔤, V)- module) in the obvious manner. For any X ∈ V one has a canonical “fiber” functor (𝔤, V)-mod → (𝔤_{X}, K_{X})-mod, M ⟼ M_{X}. If V is connected, this functor is an equivalence of categories. Note that if T is a K_{X}-torsor, and X_{T} ∈ V is T -twist of X (i.e., X_{T} is an object of V equipped with isomorphism of K_{X}- torsors T \overrightarrow{∼} Hom(X, X_{T}), then one has a canonical isomorphism (𝔤_{X_{T}} , K_{X_{T}}) = (𝔤_{X}, K_{X})_{T}, M_{X_{T}} = (M_{X})_{T} (see 3.1.4).

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 étale topology of S; here K′ ⊂ K is any normal subgroup such that K/K′ has finite type.

Let V be a pro-algebraic groupoid. An S-object Y_{S} of V is a rule that assigns to each object X ∈ V on AutX-torsor Y_{S}(X) = \underline{Hom}(X, Y_{S}) on S together with canonical identifications of AutX-torsors Y_{S}(X) = Y_{S}(X′)_{Hom(X,X′)} (= the twist of Y_{S}(X′) by AutX′-torsor Hom(X, X′)) for each X, X′ ∈ V; these identifications should satisfy an obvious compatibility condition for three objects X, X′, X′′ ∈ V. In other words, Y_{S} is a functor from V to schemes over S such that the AutX-action defines on Y_{S}(X) the structure of AutX-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 V-module, then an S-object Y_{S} of V defines a locally free O_{S}-module M_{Y_{S}} on S. If Y_{S}(X) for X ∈ V is non-empty then M_{Y_{S}} coincides with Y_{S}(X)-twist of M_{X} ⊗ O_{S}.

Let now (𝔤, V) be a Harish-Chandra groupoid, and Y_{S} be an S-object of V (considered as pro-algebraic groupoid). We get a sheaf 𝔤_{Y_{S}} of O_{S}-Lie algebras; 𝔤_{Y_{S}} is a projective limit of locally free O_{S}-modules. For any (𝔤, V)-module M the O_{S}-module M_{Y_{S}} is a 𝔤_{Y_{S}}-module.

3.2 Lie algebroids. Let S be a scheme.

3.2.1 A Lie algebroid on S (which is an infinitesimal version of Lie groupoid) is a sheaf A of Lie algebras on S together with an O_{S}-module structure on A and an O_{S}-linear map σ : A → T_{S} such that σ is a morphism of Lie algebras, and the formula [a, fb] = σ(a)(f)b + f[a, b] holds for a, b ∈ A, f ∈ O_{S}. Clearly A_{(0)} = ker σ is a sheaf of O_{S}-Lie algebras. In the case when S is smooth we will say that A is transitive if σ is surjective.

The Lie algebroids form a category Lie(S) with final object T_{S}. This category has products: for A, B ∈ Lie(S) we have A × B = A ×_{T} B in the obvious notations. The categories Lie(S) form a fibered category over the category of schemes. For a morphism f : S′ → S of schemes and A ∈ Lie(S) the inverse image f^{∗}A ∈ Lie(S′) is defined by the formula f^{∗}A = T_{S′} ×f^{∗}(A). Here f^{∗}(A), f^{∗}(T_{S}) are inverse images in the categories of O-modules, and the fibered product is f^{∗}(T_{S}) taken with respect

to projections T_{S′} ⟶^{df} f^{∗}(T_{S}) ⟵^{f^{∗}(σ)} f^{∗}(A).

3.2.2 Let A be a Lie algebroid. An A-module is a sheaf F of A-modules on S together with an O_{S}-module structure such that for a ∈ A, f ∈ O_{S}, m ∈ F one has a(fm) = σ(a)(f)m + f(am). We will also call such a structure an action of A on O_{S}-module F. If A, B are Lie algebroids, F is an A-module, G is a Bmodule, then F ⊗_{O_{S}} G is A × B-module: for (a, b) ∈ A × B, m ∈ F, n ∈ G one has (a, b)(m ⊗ n) = (am) ⊗ n + m ⊗ (bn).

3.2.3 Let A be a Lie algebroid, and 𝔤 an O_{S}-Lie algebra equipped with an Aaction. An A-morphism ψ : A_{(0)} → 𝔤 is a morphism of O_{S}-Lie algebras that commutes with A-action (here the A-action on A_{(0)} is adjoint one). Note that if φ : A → B is a morphism of Lie algebroids, then A acts on B_{(0)} by ad ∘ φ, and φ_{(0)} : A_{(0)} → B_{(0)} is an A-morphism. Conversely, for an A-morphism ψ : A_{(0)} → 𝔤 let A_{ψ} be the quotient of the semi-direct product A ⋉ 𝔤 by the ideal A_{(0)} ↪ A ⋉ 𝔤, a ⟼ (a, −ψ(a)). Then A_{ψ} is a Lie algebroid, A_{ψ_{(0)}} = 𝔤, and we have a canonical morphism ψ : A → A_{ψ} with ψ_{(0)} = old ψ. These constructions are mutually inverse: if 𝔤 = B_{(0)}, φ : A → B is a morphism of Lie algebroids, and ψ = φ_{(0)}, then we have a canonical morphism i : A_{ψ} → B which is an isomorphism if A is transitive.

3.2.4 Let A be a Lie algebroid. A central extension of A by O_{S} is a Lie algebroid Ã together with a surjective morphism π : Ã → A and a central element 1 ∈ ker π such that the map O_{S} \overrightarrow{∼} ker π, f ⟼ f · 1, is isomorphism. Note that the adjoint action of Ã on Ã_{(0)} quotients to an A-action. We will call a central extension L of T_{S} by O_{S} an invertible Lie algebroid (so L_{(0)} = O_{S}).

3.2.5 Remarks. (i) Let B be any Lie algebroid, and let tr : B_{(0)} → O_{S} be a Bmorphism (we will call such tr a trace on B). If B is transitive, then B_{tr} is an invertible algebroid. (ii) Let Ã ⟶^{π} A be a central extension of A by O_{S}, and γ : A_{(0)} → Ã be an O-linear section of π such that γ commutes with adjoint action of A. Then γ(A_{(0)}) is ideal in Ã, and Ã/γ(A_{(0)}) is invertible algebroid.

3.2.6 The invertible Lie algebroids form a category PLie(S) which is a Picard category, and, more generally, a “ℂ-vector space” in categories. This means that for α, β, ∈ ℂ, A, B ∈ PLie(S) we may form the linear combination C = αA + βB ∈ PLie(S): by definition C = (A × B)_{tr_{α,β}}, where tr_{α,β}(f, g) = αf + βg. For A ∈ PLie(S) we have AutA = Ω_{S}^{1cℓ}: for a closed 1 form ω the corresponding automorphism of A is a ⟼ a + ⟨ωσ(a)⟩ · 1. The trivial invertible algebroid is T_{SO} = T_{S} ⋉ O_{S} (where O : T_{S(0)} = 0 → O_{S} is the trivial trace map). The locally trivial invertible Lie algebroids form a full ℂ-linear subcategory canonically equivalent to the one of Ω^{1cℓ}-torsors.

3.2.7 For A ∈ PLie(S) define D_{A} to be the sheaf of associative ℂ-algebras on S together with a morphism of ℂ Lie algebras i : A → D_{A} such that i|_{O_{S}} is a morphism of associative algebras (in particular, i(1) is 1 in D_{A}) and one has i(f)i(a) = i(fa) for f ∈ O_{S}, a ∈ A, and universal with respect to these data. For example, if A is trivial, then D_{A} is the usual algebra of differential operators on S. For arbitrary A this is a twisted differential operators ring, see, e.g. Appendix to [BK] for details. Clearly a D_{A}-module F is the same as an A-module such that 1 ∈ A acts on F as the identity operator. Since D_{A} carries an obvious filtration with grD_{A} = SymT_{S}, for a coherent D_{A}-module F we have its singular support SSF which is a closed conical subset in the cotangent bundle of S. A D_{A}-module F is called lisse if SSF = (0): this condition is equivalent to the fact that F is a vector bundle (as O_{S}-module).

3.2.8 The standard example of a Lie algebroid is the current (or Atiyah) algebra A(E) of a vector bundle E. This is the Lie algebra of infinitesimal symmetries of E. The sections of A(E) are pairs (σ(τ), τ), where σ(τ) ∈ T_{S} and τ is an action of σ(τ) on E, or, equivalently, a first order differential operator on E with symbol σ(τ) · id_{E}. Clearly A(E) is transitive and A(E)_{(0)} = 𝔤ℓ(E). If L is a line bundle, then A(L) is invertible algebroid; one has A(L_{1} ⊗ L_{2}) = A(L_{1}) + A(L_{2}), i.e., A : Pic(S) → PLie(S) is a morphism of Picard categories. The ring D_{A(L)} coincides with the algebra D_{L} of differential operators on L. If E is any vector bundle, then tr: 𝔤ℓ(E) → O_{S} is a trace on A(E), and A(E)_{tr} = A(det E): this canonical isomorphism comes from a natural action of A(E) on det E given explicitly by the Leibnitz rule a(e_{1} ∧ … ∧ e_{n}) = ae_{1} ∧ e_{2} ∧ … ∧ e_{n} + ⋯ + e_{1} ∧ … ∧ ae_{n}.

3.3 Localization of (𝔤, K)-modules. Below we will explain the general pattern how to transform representations to D-modules. We will start with some notations.

3.3.1 Let (𝔤̃, V) be a Harish-Chandra groupoid. We will say that it is centered if for any X ∈ V there is a fixed central element 1 ∈ 𝔤̃_{X}, 1 ∉ LieAutX, that depends on X in a natural way. Put 𝔤_{X} = 𝔤̃_{C}/ℂ1, so 𝔤̃_{X} is a central ℂ-extension of 𝔤_{X}.

Our (𝔤̃, V) defines several Harish-Chandra groupoids with the same underlying proalgebraic groupoid V. Namely, we have the groupoid (𝔤, V) that corresponds to 𝔤_{X}; for any c ∈ ℂ we have the centered groupoid (𝔤̃_{c}, V) with 𝔤̃_{cX} equal to c-multiple of the central extension 𝔤̃_{X} of 𝔤_{X}. Denote by (𝔤̃, V)_{c}-mod the category of (𝔤̃_{c}, V)-modules on which 1 ∈ ℂ ⊂ 𝔤̃_{c} acts as identity.

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 AY_{S} the Lie algebroid of infinitesimal symmetries of (S, Y_{S}). Its sections are pairs (τ, τ_{Y_{S}}), where τ ∈ τ_{Y_{S}} and T_{Y_{S}} is a lifting of τ to Y_{S} that commutes with K-action. Clearly AY_{S(0)} = LieK_{Y_{S}} (= Y_{S}-twist of Lie K ˆ⊗O_{S} with respect to the adjoint action of K); AY_{S} is a transitive groupoid. If (𝔤, K) is a Harish-Chandra pair, then we have the O_{S}-Lie algebra 𝔤_{Y_{S}} (= Y_{S}-twist of 𝔤ˆ⊗O_{S} with respect to the adjoint action). The Lie algebroid AY_{S} acts on 𝔤_{Y_{S}} in an obvious manner, and the canonical embedding i : AY_{S(0)} = LieK_{Y_{S}} ↪ 𝔤_{Y_{S}} is an AY_{S}- morphism. According to 3.2.3 we get the transitive Lie algebroid A𝔤_{Y_{S}} = AY_{Si} with A𝔤_{Y_{S}(0)} = 𝔤_{Y_{S}}. If M is a (𝔤, K)-module, then M_{Y_{S}} (= Y_{S}-twist of M ⊗ O_{S}) is an A𝔤_{Y_{S}}-module.

Now let (𝔤, V) be a Harish-Chandra groupoid, and let Y_{S} be an S-object of V. The above construction defines a transitive Lie algebroid A𝔤_{Y_{S}} on S with A𝔤_{Y_{S}(0)} = 𝔤_{Y_{S}}. If M is a (𝔤, V)-module, then M_{Y_{S}} is an A𝔤_{Y_{S}}-module in a natural way. Note that if (𝔤̃, V) is a centered groupoid, then A𝔤̃_{Y_{S}} is a central O_{S}-extension of A𝔤_{Y_{S}}.

3.3.3 Definition. Let S be a smooth scheme and (𝔤̃, V) be a centered Harish-Chandra groupoid. An S-localization data ψ for (𝔤̃, V) is a collection (Y_{S}, N, φ, φ̃_{(0)}) where

(i) Y_{S} is an S-object of V. (ii) N is a transitive Lie algebroid on S. (iii) φ : N → A𝔤_{Y_{S}} is a morphism of Lie algebroids. (iv) φ̃_{(0)} : N_{(0)} → 𝔤̃_{Y_{S}} is a lifting of φ_{(0)} such that for n ∈ N, m ∈ N_{(0)} one has

φ̃_{(0)}([n, m]) = [φ(n), φ_{(0)}(m)]. □

3.3.4 A localization data ψ defines an invertible Lie algebroid A_{ψ} on S as follows. Consider a fiber product A𝔤̃_{Y_{S}}N = A𝔤̃_{Y_{S}} ×_{A𝔤_{Y_{S}}} N: this is a central O_{S}-extension of N. This central extension splits over N_{(0)} by means of the section s : N_{(0)} → A𝔤_{Y_{S}}N_{(0)}, s(m) = (φ̃_{(0)}(m), m). Put A_{ψ} := A𝔤̃_{Y_{S}}N/s(N_{(0)}). Let D_{ψ} = D_{A_{ψ}} be the corresponding algebra of twisted differential operators.

3.3.5 Let M ∈ (𝔤̃, V)_{1}-mod be a Harish-Chandra module such that 1 acts as id_{M}. Then M_{Y_{S}} is an A𝔤̃_{Y_{S}}N-module (via the projection A𝔤̃_{Y_{S}}N → A𝔤̃_{Y_{S}}), and Δ_{ψ}M = M_{Y_{S}}/s(N_{(0)})M_{Y_{S}} is A_{ψ}-module on which 1 ∈ A_{ψ} acts as identity. Hence Δ_{ψ}M is a D_{ψ}-module. Clearly Δ_{ψ} : (𝔤̃, V)_{1}-mod → D_{ψ}-mod is a right exact functor; we call it the S-localization functor that corresponds to ψ. Note that for a point s ∈ S we have a Lie algebra map N_{(0)s} → 𝔤̃_{Y_{S}} (where N_{(0)s} = N_{(0)}/m_{s}N_{(0)}), hence the fiber Δ_{ψ}(M)/m_{s}Δ_{ψ}(M) coincides with coinvariants M_{Y_{s}}/N_{(0)s}M_{Y_{S}}.

3.3.6 The above constructions are functorial with respect to morphisms of localization data. Precisely, let (𝔤̃′, V′) be another centered Harish-Chandra groupoid, and r : (𝔤̃, V) → (𝔤̃′, V′) is a morphism of centered groupoids. One defines an r-morphism of S-localization data r^{#} : ψ → ψ′ in an obvious manner. Such r^{#} defines the isomorphisms r_{A}^{#} : A_{ψ} \overrightarrow{∼} A_{ψ′}, r_{D}^{#} : D_{ψ} \overrightarrow{∼} D_{ψ′}. For M ∈ (𝔤̃, V)_{1}- mod, M ∈ (𝔤̃′, V′)_{1}-mod and an r-morphism ℓ : M → M′ we have r_{D}^{#}-morphism r_{Δ}^{#} : Δ_{ψ}(M) → Δ_{ψ′}(M′).

One has also functoriality with respect to base change. If f : S′ → S is a morphism of smooth schemes, and ψ is an S-localization data for (𝔤̃, V), then one gets an S′-localization data f^{∗}ψ for (𝔤̃, V). One has A_{f^{∗}ψ} = f^{∗}A_{ψ}, and for M ∈ (𝔤̃, V)_{1}- mod one has a natural isomorphism f^{∗}Δ_{ψ}(M) = Δ_{f^{∗}ψ}(M) of D_{f^{∗}ψ}-modules.

3.3.7 An S-localization data ψ for (𝔤̃, V) defines in an obvious way for each c ∈ ℂ an S-localization data ψ_{c} for (𝔤̃_{c}, V). One has A_{ψ_{c}} = cA_{ψ} (see 3.2.6).

3.4 Localization along moduli of curves. This section collects some basic examples of the above localization constructions.

3.4.1 Let us describe a centered Harish-Chandra groupoid (T̃, V) called the Virasoro groupoid. The underlying connected proalgebraic groupoid V is the groupoid of one-dimensional local fields with residue field equal ℂ (the morphisms are isomorphisms of the local fields). Precisely, let F ∈ V be a local field, O_{F} ⊂ F the corresponding local ring, and m_{F} ⊂ O_{F} the maximal ideal. A choice of uniformizing parameter t identifies F with ℂ((t)) and O_{F} with ℂ[[t]]. The group AutF = AutO_{F} is the projective limit of groups AutO_{F} /m_{F}^{n} = AutF/Aut_{n}F. These groups are obviously algebraic groups, our AutF is a proalgebraic group, and V is a proalgebraic groupoid. Note that AutF/Aut_{1}F = ℂ^{∗}, and Aut_{i}F/Aut_{i+1}F is isomorphic to ℂ for i ≥ 1; in particular Aut_{1}F is the pro-unipotent radical of AutF. Explicitly, Autℂ((t)) coincides with the group of power series a_{1}t + a_{2}t^{2} + ⋯, a_{1} ≠ 0, with multiplication law equal to composition of series.

Now for F ∈ V let T_{F} be the Lie algebra of vector fields on F and T̃_{F} be the Virasoro ℂ-extension of T_{F} defined in 2.1.3. The Lie algebra T_{F} carries a canonical filtration T_{iF}; for F = ℂ((t)) one has T_{iF} = t^{i+1}ℂ[[t]]∂_{t}. The subalgebra T_{−1F} preserves the lattice O_{F} ⊂ F, hence we have a canonical splitting s_{O_{F}} : T_{−1F} → T̃_{F}. Clearly LieAutF = T_{0F}, and the embedding s_{O_{F}} : LieAutF ↪ T̃_{F} together with the natural AutF -action on T̃_{F} define the Harish-Chandra pair (T̃_{F} , AutF). This defines our centered Virasoro groupoid (T̃, V).

3.4.2 Let S be a scheme. It is easy to see that an S-object Y_{S} of V is the same as a “family of formal discs” over S or, equivalently, a formal O_{S}-algebra O_{Y} locally isomorphic to O_{S}[[t]]. The corresponding Lie algebroid AY_{S} consists of pairs (τ, τ_{Y_{S}}) where τ ∈ T_{S} and τ_{Y_{S}} ∈ DerO_{Y_{S}} is a τ-derivation of O_{Y_{S}}.

3.4.3 Now let π : C_{S} → S be a family of smooth projective curves and a : S → C_{S} be a section of π. These define an S-localization data ψ = ψ(C_{S}, a) for (T̃, V) as follows. Our Y_{S} is the formal completion of C_{S} along a(S), and N is the Lie algebroid of pairs (τ, τ_{U}) where τ ∈ T_{S} and τ_{U} is a lifting of τ to U = C_{S} \ a(S). Clearly AT_{Y_{S}} is the Lie algebroid of pairs (τ, τ_{Y_{S\(a)}}), where τ ∈ T_{S} and τ_{Y_{S\(a)}} is a lifting of τ to a meromorphic vector field on Y_{S} with possible pole at a(S). Our φ : N → AT_{Y_{S}} is just the restriction of a vector field τ_{U} on Y_{S} \ {a} = punctured neighbourhood of a. Now the lifting φ̃_{(0)} : N_{(0)} = π_{∗}T_{U/S} → T̃_{Y_{S}} is the restriction to T_{U/S} ⊂ D_{U/S} of the morphism s_{a} : π_{∗}D_{U/S} → D̃_{(a)} (here D = D_{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, φ, φ̃_{(0)}) is our localization data ψ(C_{S}, a). According to 3.3.4, 3.3.5, 3.3.7 for any c ∈ ℂ we have the localization functor Δ_{ψ_{c}(C_{S},a)} : (T̃, V)_{c}-mod → D_{ψ_{c}(C_{S},a)}-mod.

3.4.4 Here is an explicit description of A_{ψ(C_{S},a)} and Δ_{ψ(C_{S},a)}. Choose (locally on S) a formal parameter t at a, so O_{Y_{S}} = O_{S}[[t]]. Consider the space B of triples (τ, τ_{U}, τ_{U}^{v}), where τ ∈ T_{S}, τ_{U} is a lifting of τ to U, and τ_{U}^{v} : S → T̃_{ℂ((t))} is a lifting of a vertical component of τ_{U}, τ_{U}^{v} = τ_{U}(t)∂_{t} : S → T_{ℂ((t))}. This B is a Lie algebroid on S in an obvious manner. We have a canonical morphism T_{U/S} → B_{(0)}, ν ⟼ (o, ν, s_{a}(ν)), see 2.3.4. One has A_{ψ(C_{S},a)} = B/T_{U/S}. Now let M be a (T̃, V)_{c}- module. One has M_{Y_{S}} = M_{ℂ((t))} ⊗ O_{S}. The algebroid B acts on M_{Y_{S}} by formula (τ, τ_{U}, τ_{U}^{v})(m⊗f) = m⊗τ(f)+τ_{U}^{v}(m⊗f). One has Δ_{ψ(C_{S},a)}(M) = M_{Y_{S}}/T_{U/S}M_{Y_{S}}.

3.4.5 Variant. For any non empty finite set A we may consider the centered groupoid (T̃^{A}, V^{A}). Here V^{A} is the A-th power of V and T̃_{{F_{a}}}^{A} is the Baer sum of

ℂ-extension T̃_{F_{a}}, a ∈ A (so T̃_{{F_{a}}}^{A} is a ℂ-extension of ∏_{a∈A}T_{F_{a}}). A family π : C_{S} → S

of curves together with a disjoint set A of sections (where “disjoint” means that for a_{i} ≠ a_{j} ∈ A and any s ∈ S one has a_{i}(s) ≠ a_{j}(s) ∈ C_{S}) defines an Slocalization data ψ(C_{S}, A) for (T̃^{A}, V^{A}) in a way similar to 3.4.2. For example, the corresponding Lie algebroid N consists of pairs (τ, τ_{U}), where τ ∈ T_{S} and τ_{U} is a lifting of τ to U = C_{S} \ ∐_{a∈A}a_{i}(S).

3.4.6 Remark. Let B ⊂ A be a non-empty subset. The groupoids (T̃^{B}, V^{B}) and (T̃^{A}, V^{A}) are related by an obvious correspondence (T̃^{B}, V^{B}) ⟵^{π_{B}} (T̃^{A,B}, V^{A}) ⟶^{i_{A}} (T̃^{A}, V^{A}),

where T̃_{{F_{a}}}^{A,B} = T̃_{{F_{b}}_{b∈B}}^{B} × ∏_{a∈A\B}T_{−1F_{a}} ↪ T̃_{{F_{a}}}^{A}. Any family of curves π : C_{S} → S

and a set A of disjoint sections defines an S-localization data ψ(C_{S}, A, B) for (T̃^{A,B}, V^{A}) in an obvious manner together with corresponding morphisms ψ(C_{S}, B)

⟵^{π_{B}^{#}} ψ(C_{C}, A, B) ⟶^{i_{A}} ψ(C_{S}, A). These define the corresponding isomorphisms D_{ψ_{c}(C_{S},B)} \overrightarrow{∼} D_{ψ_{c}(C_{S},A,B)} \overrightarrow{∼} D_{ψ_{c}(C_{S},A)}. For M_{B} ∈ (T^{B}, V^{B})_{c}-mod, M_{A} ∈ (T^{A}, V^{A})- mod a morphism f : M_{B} → M_{A} is, by definition, an i_{A}-morphism from M_{B}, considered as (T̃^{A,B}, V^{A})-module via π_{B}, to M_{A}. Since Δ_{ψ_{c}(C_{S},B)}M_{B} = Δ_{ψ_{c}(C_{S},A,B)}M_{B}, such an f defines a morphism Δ(f) : Δ_{ψ_{c}(C_{S},B)}M_{B} → Δ_{ψ_{c}(C_{S},A)}M_{A}. For example, if M_{A} = IndT̃ ^{A} ; T̃ ^{A,B}(M_{B}) and f is the canonical embedding, then Δ(f) is isomorphism.

Note that the above canonical identification D_{ψ_{c}(C_{S},A)} = D_{ψ_{c}(C_{S},B)} for B ⊂ A actually provides a canonical algebra D_{ψ_{c}(C_{S})} that depends on C_{S} only together with canonical isomorphisms D_{ψ_{c}(C_{S})} = D_{ψ_{c}(C_{S},A)} for any set A of disjoint sections. To construct D_{ψ_{c}(C_{S})} we may assume, working locally in étale topology of S, that C_{S} has many sections. To construct D_{ψ_{c}(C_{S})} it suffices to define for any two sets A, A′ of disjoint sections a canonical isomorphism D_{ψ_{c}(C_{S},A)} = D_{ψ_{c}(C_{S},A′)}. Choose a non-empty set B of sections such that both A ⊔ B, A′ ⊔ B are sets of disjoint sections. Our isomorphism is D_{ψ_{c}(C_{S},A)} = D_{ψ_{c}(C_{S},A⊔B)} = D_{ψ_{c}(C_{S},B)} = D_{ψ_{c}(C_{S},A′⊔B)} = D_{ψ_{c}(C_{S},A′)}. One verifies easily that this does not depends on the choice of B. We will compute D_{ψ_{c}(C_{S})} explicitly in 3.5.6.

3.4.7 Variant. Often the Virasoro modules are integrable only with respect to the subgroup Aut_{1}F (see 3.4.1). To localize them one needs to consider the groupoid (T̃, V_{1}). The objects of V_{1} are pairs (F, ν), where F is a local field and ν ∈ m_{F} /m_{F}^{2} , ν ≠ 0, is a 1-jet of a parameter. One has Aut(F, ν) = Aut_{1}F. The Lie algebra T̃_{(F,ν)} is T̃_{F}. If π : C_{S} → S is a family of curves, a : S → C_{S} a section, and ν ∈ a^{∗}Ω_{C_{S}/S}^{1} a

1-jet of parameters at a, then we get an S-localization data ψ(C_{S}, a, ν) for (T̃, V_{1}). We may also consider many points, as in 3.4.5.

We have a “forgetting of ν” morphism r : (T̃, V_{1}) → (T̃, V) and a corresponding r-morphism of localization data ψ_{c}(C_{S}, a, ν) → ψ_{c}(C_{s}, a). This defines a canonical isomorphism r_{D} : D_{ψ_{c}(C_{S},a,ν)} \overleftarrow{∼} D_{ψ_{c}(C_{S},a)} and for any M ∈ (T , V)_{c}-mod the r_{D}- isomorphism r_{M} : Δ_{ψ_{c}(C_{S},a,ν)}M \overleftarrow{∼} Δ_{ψ_{c}(C_{S},a)}M.

3.4.7.1 Let C be a fixed curve, a ∈ C, and ν a 1-jet of parameter at a. Consider a constant ℂ^{∗}-family C_{ℂ^{∗}} = C × ℂ^{∗} with constant point a, and put ν^{∨}(u) = uν for u ∈ ℂ^{∗}. We get the corresponding ℂ^{∗}-localization data ψ = ψ(C_{ℂ^{∗}}, a, ν^{∨}). One has D_{ψ} = D_{ψ(C_{ℂ^{∗}} ,a,ν^{∨})} = D_{ψ(C_{ℂ^{∗}} ,a)} = D_{ℂ^{∗}} – the usual ring of differential operators. In particular, we have λ∂_{λ} ∈ D_{ψ_{c}}. Let us compute the action of u∂_{u} on Δ_{ψ_{c}}(M) for M ∈ (T , V_{1})_{c}-mod. Choose a parameter t_{a} at a on C such that dt(a) = ν. Then t_{au} = ut is a ℂ^{∗}-family of parameters which identifies our O_{Y_{ℂ^{∗}}} with O_{ℂ^{∗}}[[t]]. We have M_{Y_{ℂ^{∗}}} = M_{ℂ((t))} ⊗ O_{ℂ^{∗}}, and Δ_{ψ_{c}}(M) is a quotient of M_{Y_{ℂ^{∗}}}. For m ∈ M_{ℂ((t))} denote by \overline{m} its image in Δ_{ψ_{c}}(M). Put L_{0} = s_{ℂ[[t]]}(t∂_{t}) ∈ T̃_{ℂ((t))}. One has u∂_{u}(\overline{m}) = \overline{L_{0}m}. In particular, if M is a higher weight module (see 7.3.1), then Δ_{ψ}M is smooth along ℂ^{∗} with monodromy equal to the action of T = exp(2πiL_{0}) (see 7.3.2).

3.4.8 Now consider the case “vector symmetries”. Our “Virasoro-Kac-Moody” centered Harish-Chandra groupoid (Ã, VV) defined as follows. The objects of VV are pairs (F, E_{O}) where F is a local field, and E_{O} is a free O_{F} -module of finite rank; we put E_{F} = F ⊗ E_{O}. The morphisms are defined in an obvious manner. Clearly Aut(F, E_{O}) is extension of Aut F by GL(E_{O}) = Aut_{O_{F}} (E_{O}); this is a proalgebraic group. We put Ã(F, E_{O}) = ÃE_{F}, see 2.1.2. The canonical embedding s_{E_{O}}: Lie Aut (F, E_{O}) → ÃE_{F} defines the Harish-Chandra pair (ÃE_{F} , Aut(F, E_{O})). This defines our centered groupoid (Ã, VV).

Let S be a scheme. An S-object of VV is a pair (Y_{S}, E_{Y_{S}}), where Y_{S} is an S-object of V (see 3.4.2) and E_{Y_{S}} is a locally free O_{Y_{S}}-module of finite rank.

Assume that S is smooth. Let π : C_{S} → S be a famly of smooth projective curves, a : S → C_{S} a section, and let E be a vector bundle on C_{S}. These define an S-localization data ψ(C_{S}, E, a). Namely, the corresponding S-object of VV is the completion of C_{S}, E along a. The Lie algebroid N consists of triples (τ, τ_{U}, τ_{E_{U}}), where τ ∈ T_{S}, τ_{U} is a lifting of τ to U = C_{S} \ a(S), and τ_{E_{U}} is an action of τ_{E_{U}} on E_{U}. The morphisms φ, φ̃_{(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.

3.4.9 Let us consider now the spinor or “fermionic” version. The corresponding centered Harish-Chandra groupoid (ÕA, OV) is defined as follows. Its objects are triples Q = (F, W_{O}, ( , )), where F is a local field, W is a free O_{F} -module of finite rank, and ( , ): W_{O} × W_{O} → ω_{O_{F}} is a symmetric bilinear form with values in 1-forms of O_{F}. We assume that ( , ) is maximally non-degenerate, i.e., the cokernel of the corresponding map W_{O} → W_{O}^{0} = Hom_{O_{F}} (W_{O}, ω_{O_{F}}) is either trivial (such Q is called even) or a 1-dimensional ℂ-vector space (such Q is called odd). The morphisms in OV are the obvious ones. For Q as above, put W_{F} = F ⊗ W_{O}; our ( , ) extends to non-degenerate form ( , ) : W_{F} × W_{F} → ω_{F}. Note that our condition means that W_{O} is a maximal isotropic lattice in W_{F}. We may consider W_{F} as a Tate’s ℂ-vector space with form ( , )_{•} = Res( , ) (see 2.4.3); then W_{O} is also a maximal isotropic ( , )_{•}-lattice so Q is even iff W_{F} is even-dimensional, see 1.4.1. We put ÕA(Q) = { \atop OAW}_{F} (see 2.4.1). The Lie algebra Lie Aut Q ⊂ OAW_{F} preserves W_{O}, hence we have a canonical embedding s_{W_{O}} : Lie Aut Q ↪ ÕA(Q). This defines the Harish-Chandra pair (ÕA(Q), Aut Q), and we get the groupoid (ÕA, OV).

Remark. Clearly Q is even (resp. odd) iff (W_{F} , ( , )_{•}) 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.

Now let S be a smooth scheme. Let π : C_{S} → S be a family of smooth projective curves, a : S → C_{S} a section, W a vector bundle on C_{S}, and ( , ) : W ×W → ω_{C_{S}/S} a symmetric bilinear pairing. Assume that the cokernel of the corresponding map W → W^{0} = Hom(W, ω_{C_{S}/S}) is either trivial or supported on a(S) and is an O_{S}- module of rank 1. This collection (C_{S}, a, W, ( , )) defines an S-localization data ψ for (ÕA, OV) in a way similar to 3.4.3, 3.4.8. Namely, the formal completion of W along a defines an S-object of OV. The Lie algebroid N consists of triples (τ, τ_{U}, τ_{W_{U}}), where τ ∈ T_{S}, τ_{U} ∈ T_{U} is a lifting of τ to U = C_{S} \ a(S), and τ_{W_{U}} is an action of τ_{U} on W_{U} that preserves ( , ). The corresponding map φ is obvious, and φ̃_{(0)} comes from 2.4.4.

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).

3.4.10 Note that we have a canonical morphism r : (Ã_{−1}^{.}, VV) → (ÕA, OV) of centered Harish-Chandra groupoids. It assigns to (F, E_{O}) ∈ VV the triple (F, E_{O} ⊕ E_{O}^{0}, ( , )) where ( , ) is the obvious pairing. The morphism ÃE_{F} → ÕA(E_{F} ⊕ E_{F}^{0} ) 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 ⊕ E^{0}, ( , )) from 3.4.9. We have an obvious r-morphism of corresponding localization data r^{#} : ψ_{c}(C_{S}, a, E) → ψ_{−c}(C_{S}, a, E ⊕ E^{0}, ( , )) (see 2.4), hence the isomorphism r_{D} : D_{ψ_{c}(C_{S},a,E)} \overrightarrow{∼} D_{ψ_{c}(C_{S},a,E⊕E^{0},( , ))}.

3.5 Fermions and determinant bundles. In this section the rings of twisted differential operators D_{ψ} that appeared in 3.4 will be canonically identified with the rings D_{L} for some natural line bundles L (see 3.2.8). Equivalently, we will construct a D_{ψ}-module L which is a line bundle (as O-module). This will be done by means of Clifford modules.

3.5.1 Let us start with the situation in 3.4.9. For Q = (F, W_{O}, ( , )) ∈ OV denote by M_{Q} the Clifford module (for Clifford algebra Cℓ(Q) = Cℓ (W_{F} .( , )_{•}), see 1.4) generated by a single fixed vector v with the only relation W_{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 ÕAW_{F} = ÕA_{Q}-module. Clearly these actions are compatible, hence M_{Q} is an (ÕA_{Q}, Aut Q)-module. This way we get the (ÕA, OV)-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 ψ for (OV, ÕA). Let Q_{S} = (F_{S}, W_{O_{F_{S}}} , ( , )) be the corresponding S-object of OV (= the completion of our data along a), and M_{Q_{S}} be the corresponding O_{S}-module with ÕA_{Q_{S}}- action. Certainly, M_{Q_{S}} is a Clifford module for the O_{S}-Clifford algebra Cℓ(W_{F_{S}}, ( , )_{•}) generated by the section v with the only relation W_{O_{F_{S}}} v = 0. Note that π_{∗}W_{U} = π|_{U^{∗}}(W |_{U}) is an S-family of maximal isotropic colattices in W_{F_{S}} (see 2.4.5). Put L_{ψ} = M_{Q_{S}}/π_{∗}W_{U}M_{Q_{S}}. This is a line bundle on S if Q_{S} is even (which means that ( , ) : W × W → W_{C_{S}/S} is non-degenerate). If Q_{S} is odd, then L_{ψ} is a twodimensional 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 γ ∈ W_{O_{F_{S}}}^{⊥} /W_{O_{F_{S}}} with (γ, γ)_{•} = 1.

3.5.2 Lemma. L_{ψ} is naturally a D_{ψ}-module: it is a D_{ψ}-module quotient of Δ_{ψ}M .

Proof. Consider the action of Lie algebroid AÕA_{Q_{S}}N (see 3.3.4) on M_{Q_{S}}. Since

for (a, n) ∈ AÕA_{Q_{S}}N = AÕA_{Q_{S}} AOA_{Q_{S}}^{×}π_{∗}OAW_{U} and w ∈ π_{∗}W_{U} one has

[(a, n), w] = n(w) (as operators on M_{Q_{S}}), we see that this action ”quotients down” to L_{ψ}. It remains to show that L_{ψ} is actually an A_{ψ}-module. We need to prove that the O_{S}-Lie subalgebra s(N_{(0)}) ⊂ { \atop AOA}_{Q_{S}}N acts trivially on L_{ψ}. Note that s(N_{(0)}) = π_{∗}OAW_{U/S} acts on L_{ψ} O_{S}-linearly, hence it suffices to consider the case when S is a point. Then N_{(0)} = OAW_{U} is an extension of T_{U} by the orthogonal Lie algebra OW_{U}. Since both OW_{U} and T_{U} are perfect ℂ-Lie algebras, we see that N_{(0)} is perfect, hence every 1-dimensional representation of N_{(0)} is trivial. Since L_{ψ} is either 1-dimensional or a sum of two 1-dimensional N_{(0)}-invariant subspaces, we are done. □

Actually we have proven that L_{ψ} is a quotient of the D_{ψ}-module Δ_{ψ}(M). Certainly, 3.5.2 implies

3.5.3 Proposition. One has a canonical isomorphism of twisted differential operators algebras D_{ψ} = D_{L_{ψ}} if Q_{S} is even, and D_{ψ_{2}} = D_{det L_{ψ}} if Q_{S} is odd. □

3.5.4 Remarks. (i) According to 1.4.4 the fibers L_{ψ_{s}}, s ∈ 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_{ψ_{s}} = det^{⊗2} H^{0}(C_{s}, W_{s})). Hence the automorphism - id_{W} of our data acts on L_{ψ} as ±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. (ii) Of course we may consider the situation with several points a_{1}, …, a_{n} ∈ C. By a reason similar to 3.4.6 one may see that the corresponding line bundle L_{ψ} actually does not depend on these points; certainly, we may delete only “even” points where ( , ) is non-degenerate. □

Now let us consider the situation 3.4.8 of vector symmetries. By 3.4.10 we have a canonical isomorphism D_{ψ_{c}(C_{S},a,E)} = D_{ψ_{−c}(C_{S},a,E⊕E^{0},( , ))}. By 3.5.4(i) the fibers of the line bundle L_{ψ} = L_{ψ}(C_{S}, a, E ⊕ E^{0}, ( , )) coincide with det H^{0}(C_{s}, E) ⊗ det H^{0}(C_{s}, E_{s}^{0}) = det H^{0}(C_{s}, E)/ det H^{1}(C_{s}, E) = det RΓ(C_{s}, E). It is easy to see that L_{ψ} = det Rπ_{∗}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

3.5.5 Corollary. One has a canonical isomorphism D_{ψ_{c}(C_{S},E)} = D_{det^{⊗−c} Rπ_{∗}E}.□

Consider finally the pure Virasoro situation. We have an obvious embedding of Harish-Chandra groupoids r : (V, T̃) → (VV, Ã), F ⟼ (F, O_{F}), T̃ ↪ ÃF (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 ψ_{(C_{S},a)} ⟶ ψ_{(C_{S},a,O_{C_{S}})} which identifies D_{ψ_{c}(C_{S},a)} with D_{ψ_{c}(C_{S},a,O_{C_{S}})}. Now 3.5.5 implies

3.5.6 Corollary. One has a canonical isomorphism D_{ψ_{c}(C_{S})} = D_{det^{⊗−c} Rπ_{∗}O_{C_{S}}}.□

3.6 Quadratic degeneration. In this section we will describe the determinant bundle of a family of curves that degenerates quadratically. Below S = Spec ℂ[[q]] is a formal disc, 0 ∈ S is the special point q = 0, η = Spec ℂ((q)) is the generic point.

3.6.1 Lemma. There is a canonical 1-1 correspondence between the following data (i) and (ii):

(i) A proper S-family of curves, C_{S} such that C_{η} 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_{1}t_{2}.

(ii) A proper smooth S-family of curves C_{S}^{∨} together with two disjoint points a_{1}, a_{2} ∈ C_{S}(S) and formal coordinates t_{i} at a_{i}.

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 B̂_{S} = the completion of B_{S} along B_{0}.

(i) ⟼ (ii). Let C_{S}, t_{1}, t_{2} be a (i)-data. The corresponding C_{S}^{∨}, a_{i}, t_{i} are the following ones. One has C_{0}^{∨} = normalization of C_{0}, so t_{i} define formal coordinates at points a_{1}(0), a_{2}(0) ∈ C_{S}^{∨}. To define C_{S}^{∨} as a formal scheme, we have to construct the corresponding sheaf Ô_{C_{S}^{∨}} of functions on C_{0}^{∨}. We demand that on U = C_{S}^{∨} \ {a_{1}, a_{2}} = C_{0} \ {a} our Ô_{C_{S}^{∨}} coincides with Ô_{C_{S}}. Note that any function φ ∈ Ô_{C_{S}}(V), where V ⊂ U, has Laurent series expansions φ_{i}(t_{i}, q) ∈ ℂ((t_{i}))[[q]] at a_{i}(0). We say that φ is regular at a_{i}(0) if φ_{i}(t_{i}, q) ∈ ℂ[[t_{i}, q]]. This defines Ô_{C_{S}^{∨}}. The points a_{i} are defined by equations t_{i} = 0. (ii) ⟼ (i). Let C_{S}^{∨}, a_{i}, t_{i} be (ii)-data. The zero fiber C_{0} of our curve C_{S} is C_{0}^{∨} with points a_{1}, a_{2} glued together. The sheaf Ô_{C_{S}} coincides with Ô_{C_{S}^{∨}} on U = C_{0} \ {0} = C_{0}^{∨}\{a_{1}, a_{2}}. For a Zariski open V ⊂ U a function φ ∈ Ô_{C_{S}}(V) is regular at a if the t_{i}-Laurent series expansions φ_{i} ∈ ℂ((t_{i}))[[q]] of φ at a_{i} lie in ℂ[[t_{1}, t_{2}]] ⊂ ℂ((t_{i}))[[q]] and φ_{1} = φ_{2} ∈ ℂ[[t_{1}, t_{2}]]. Here the embedding ℂ[[t_{1}, t_{2}]] ↪ ℂ((t_{1}))[[q]] is t_{1} ⟼ t_{1}, t_{2} ⟼ q/t_{1}, and the one ℂ[[t_{1}, t_{2}]] ↪ ℂ((t_{2}))[[q]] is t_{1} ⟼ q/t_{2}, t_{2} ⟼ t_{2}. This defines Ô_{C_{S}}. □

Below we will say that a vector bundle E on a scheme X is stratified at x ∈ X if we are given an isomorphism E ≃ A ⊗_{ℂ} O_{X} on a formal neighbourhood of x (here A is a vector space; A = E_{x}).

3.6.2 Lemma. Let C_{S} and C_{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^{∨} on C_{S}^{∨} together with a stratifications of E^{∨} at a_{1}, a_{2} and an isomorphism of fibers E_{a_{1}}^{∨} ≃ E_{a_{2}}^{∨}. □

3.6.3 Proposition. Let (C_{S}, E), (C_{S}^{∨}, E^{∨}) be the related objects from 3.6.1, 3.6.2. Then there is a canonical stratification of the line bundle L = det Rπ_{∗}E/ det Rπ_{∗}^{∨}E^{∨} on S.

Remark. Here “stratification” = “stratification at 0” = (isomorphism L ≃ L_{0} ⊗ O_{S}). Note that L_{0} = det RΓ(C_{0}, E_{0})/ det RΓ(C_{0}^{∨}, E_{0}^{∨}) is naturally isomorphic to det^{−1} E_{a}, so 3.6.3 is canonical isomorphism det Rπ_{∗}^{∨}(C^{∨}, E^{∨}) = det E_{a} det Rπ_{∗}(C, E).

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_{a_{1}}^{∨} = E_{a_{2}}^{∨}. Our data identifies the formal completion Eâ of E at a with A ⊗ ℂ[[t_{1}, t_{2}]], and the formal completion of E∨ ; â_{i} of E^{∨} at a_{i} with A ⊗ ℂ[[t_{i}, q]]. The restrictions of E and E^{∨} to the formal scheme Û = (U, Ô_{U}) coincide; put P = H^{0}(U, E|Û) = lim_{←}H^{0}(U, E/q^{n}E). Also put V = A ⊗ {ℂ((t_{1}))[[q]] ⊕ ℂ((t_{2}))[[q]]}, V_{+0} = A ⊗ {ℂ[[t_{1}, q]] ⊕ ℂ[[t_{2}, q]]}, V_{+1} = A ⊗ {ℂ[[t_{1}, t_{2}]]. We may compute Rπ_{∗}E, Rπ_{∗}^{∨}E^{∨} by means of “adelic” complexes for our formal schemes. Namely, Rπ_{∗}^{∨}E^{∨} = Cone(P ⊕ V_{+0} → V)[−1], Rπ_{∗}E = Cone(P ⊕V_{+1} → V)[−1]; here the map P → V is minus the Laurent series expansion map, the map V_{+1} → V is given by formula a ⊗ t_{1}^{m}t_{2}^{n} ⟼ a ⊗ {q^{n}t_{1}^{m−n} + q^{m}t_{2}^{n−m}} (see the proof of 3.6.1), and V_{+0} → V is the obvious embedding.

Note that V is a flat complete 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 ⊕ 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 ⊕ P^{⊥}, L(V_{i+}) = V_{i+} ⊕ V_{i+}^{⊥} are maximal isotropic colattice and lattices respectively. A ℂ[[q]]-version of 1.4.9 shows that coinvariants M_{L(P)} and invariants M^{L(V_{i+})} are free ℂ[[q]]-modules of rank one, and there are canonical isomorphisms

det Rπ_{∗}^{∨}E^{∨} = M^{L(V_{0+})}/M_{L(P)}, det Rπ_{∗}E = M^{L(V_{1+})}/M_{L(P)}.

Hence det Rπ_{∗}E/ det Rπ_{∗}^{∨}E^{∨} = 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 γ : M^{L(V_{0+})} → M^{L(V_{1+})} ⊗ det A; the desired stratification of the ratio of determinants then will be γ(v)/v for a non-zero generator v (clearly it does not depend on M). Let a_{1}, …, a_{ℓ} be a basis of A. Consider the vectors e_{α1}^{k} = a_{α}⊗t_{1}^{k}, e_{α2}^{k} = a_{α}⊗t_{2}^{k}, k ∈ ℤ, α = 1, …, ℓ. This is a basis (in an obvious sense) of V; denote by e_{αi}^{k∗} ∈ V^{∗} the dual basis. The vectors {e_{αi}^{k}}, k ≥ 0, form a basis of V_{0+}, and the vectors f_{α1}^{k} := e_{α1}^{k} + q^{k}e_{α2}^{−k}, f_{α2}^{k} := q^{k}e_{α1}^{−k} + e_{α2}^{k}, e_{α1}^{0} + e_{α2}^{0}, k ≥ 1, form a basis of V_{1+}. In a slightly informal way our γ could be defined as follows. A generator of M^{L(V_{0+})} is an infinite wedge product ⋀_{\begin{matrix}k≥0 \\ α,i \end{matrix}}e_{αi}^{k}, a generator of

M^{L(V_{1+})} ⊗ det A is ⋀_{\begin{matrix}k≥1 \\ α,i \end{matrix}}f_{αi}^{k} ∧ ⋀_{α}(e_{α1}^{0} + e_{α2}^{0}) ⊗ ⋀_{α}a_{α}, and γ just identifies these gener-

ators. To be precise, consider the elements γ_{n} = ∏_{\begin{matrix}1≤k≤n \\ α \end{matrix}}(f_{α1}^{k}f_{α2}^{k}e_{α2}^{k∗}e_{α1}^{k∗}) ∈ Cliff(W).

These γ_{n} do not depend on a choice of basis {a_{α}} in A, and it is easy to see that γ_{∞} = lim_{n}γ_{n} ∈ CℓW is correctly defined. Let V_{0++} ⊂ V_{0+}, V_{1++} ⊂ V_{1+} be sublat-

tices with bases {e_{αi}^{k}}, k ≥ 1, and {f_{αi}^{k}}, k ≥ 1, respectively. It is easy to see that γ_{∞}(M^{L(V_{0++})}) = M^{L(V_{1++})} (more precisely, γ_{n}(M^{L(V_{0+})}) ≡ M^{L(V_{1+})}modq^{n+1}M).

Since M^{L(V_{0+})} = ⋀_{α,i}e_{αi}^{0} · M^{L(V_{0++})}, M^{L(V_{1+})} = ⋀_{α}(e_{α1}^{0} + e_{α2}^{0}) · M^{L(V_{1++})}, we have

⋀_{α}(e_{1}^{0∗} − e_{2}^{0∗}) · γ_{∞}M^{L(V_{0+})} = M^{L(V_{1+})}. Put ⋀_{α}(e_{1}^{0∗} − e_{2}^{0∗}) · γ_{∞} ⊗ ⋀_{α}a_{α} ∈ CℓW ⊗

det A. This γ does not depend on a choice of basis {a_{α}} of A, and the desired M^{L(V_{0+})} \overrightarrow{∼} M^{L(V_{1+})} ⊗ det A is just multiplication by γ. □

3.6.4 Let C^{∨} be a curve, a_{1}, a_{2} ∈ C^{∨}, a_{1} ≠ a_{2}, a pair of points, and t_{i} a formal parameter at a_{i}. Consider the constant S-family C_{S}^{∨} := C^{∨} × S; let a_{i} ∈ C_{S}^{∨}(S), t_{i} be the “constant” points and parameters. According to 3.6.1 these define an Scurve C_{S} with quadratic singularities along zero fiber and smooth generic fiber. Consider the trivial vector bundles O_{C_{S}}, O_{C_{S}^{∨}}; they correspond to each other by 3.6.2 correspondence. Note that det Rπ_{∗}^{∨}O_{C_{S}^{∨}} = det RΓ(C^{∨}, O_{C^{∨}})⊗O_{S} is obviously stratified, hence 3.6.3 defines the stratification of det Rπ_{∗}O_{C_{S}} which is a natural generator γ of the ℂ[[q]]-module det^{−1} RΓ(C^{∨}, O_{C^{∨}}) ⊗_{ℂ[[q]]} det Rπ_{∗}O_{C_{S}}. Let us compute γ in a couple of simple situations.

3.6.5 Assume that C^{∨} is a disjoint union of two copies of ℙ^{1}’s, C^{∨} = ℙ_{1}^{1} ∐ ℙ_{2}^{1}, a_{1} ∈ ℙ_{1}^{1}, a_{2} ∈ ℙ_{2}^{1} 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 𝔸^{2} → S, q = t_{2}t_{2}. This is a genus 0 curve, hence Rπ_{∗}O_{C_{S}} = O_{S}, so we have a canonical trivialization α of det Rπ_{∗}O_{C_{S}} of “global” origin. In fact, it coincides with our γ. To see this, note that (in the notations of proof of 3.6.3) in our case P is colattice with basis {e_{1}^{k}, e_{2}^{k}}, k ≤ 0, so one has P ⊕V_{1++} = V = P ⊕V_{0++}. The operator (e_{1}^{0}+e_{2}^{0})· identifies M^{L(V_{1++})} with M^{L(V_{1+})}, hence det Rπ_{∗}O_{C_{S}} = M^{L(V_{1++})}/M_{L(P)}. The “global” trivialization α comes from the isomorphism M^{L(V_{1++})} \overrightarrow{∼} M_{L(P)}, m ⟼ m mod L(P)M. The trivialization γ comes from composition M^{L(V_{1++})} \overrightarrow{∼} M^{L(V_{0++})} \overrightarrow{∼} M_{L(P)} where the first arrow is inverse to multiplication by γ_{∞} and the second one is projection m ⟼ m mod L(P)M. Since f_{i}^{k} = e_{i}^{k} mod P for k ≥ 1, the formula for γ_{∞} shows that this composition coincides with projection M^{L(V_{1++})} → M_{L(P)}, hence α = γ.

3.6.6 Assume now that C^{∨} = ℙ^{1}, a_{1} = 0, a_{2} = ∞ 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 ∞. The Tate curve carries a canonical relative 1-form ν (that corresponds to the standard invariant form on G_{m} via Tate’s uniformization). One has R^{0}π_{∗}O_{C_{S}} = O_{S}, R^{1}π_{∗}O_{C_{S}} = (R^{0}π_{∗}ω_{C_{S}})^{∗} by Serre duality (here ω_{C_{S}} is relative dualizing sheaf), hence det Rπ_{∗}O_{C_{S}} = R^{0}π_{∗}ω_{C_{S}} and ν is a canonical trivialization of det Rπ_{∗}O_{C_{S}}. Let us calculate γ. The colattice P has basis {e_{1}^{k} + e_{2}^{k}}, k ∈ ℤ. One has O_{S} = R^{0}π_{∗}O_{C_{S}} = O_{S}(e_{1}^{0} + e_{2}^{0}) = P ∩ V_{1+}, R^{1}π_{∗}O_{C_{S}} = V/P + V_{1+} = V/P + V_{1++}. The relative differential ν 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 ν ∈ (R^{1}π_{∗}O_{C_{S}})^{∗} = (V/P + V_{1+})^{∗} ⊂ V^{∗} is e_{1}^{0∗} − e_{2}^{0∗}. As above, multiplication by e_{1}^{0} + e_{2}^{0} identifies M^{L(V_{1++})} with M^{L(V_{1})}, hence det Rπ_{∗}O_{C_{S}} = M^{L(V_{1++})}/M_{L(P)}. The trivialization ν comes from the isomorphism M^{L(V_{1++})} → M_{L(P)}, m ⟼ (e_{1}^{0}m) mod L(P)M. The trivialization γ comes from composition M^{L(V_{1++})} \overrightarrow{∼} M^{L(V_{0++})} \overrightarrow{∼} M_{L(P)} where the first arrow is inverse to multiplication by γ_{∞} isomorphism M^{L(V_{0++})} \overrightarrow{∼} M^{L(V_{1++})} and the second arrow is m ⟼ (e_{1}^{0}m) mod L(P)M. Since f_{1}^{k} = (1 − q^{k})e_{1}^{k} mod P, f_{2}^{k} = (1 − q^{k})e_{2}^{k} mod P we see that γ = [∏_{k≥1}(1 − q^{k})^{2}]ν, or, in terms of Dedekind’s

η-function η(q) = q^{1/24} ∏_{k≥1}(1 − q^{k}), one has

γ = q^{−1/12}η(q)^{2}ν.

One may reformulate this as follows. Recall that the line bundle λ = det Rπ_{∗}O_{C} = π_{∗}ω_{C} on moduli space of elliptic curves carries a canonical global integrable connection ∇ such that the discriminant Δ is a global horizontal section of λ^{⊗12} (with respect to the corresponding connection on λ^{⊗12}). We see that our γ is a horizontal section of a connection ∇ + \frac{1}{12} \frac{dq}{q}.

§4. Fusion Categories

4.1 Recollections from symplectic linear algebra. Let V be a symplectic ℝ-vector space of dimension 2g with symplectic form ⟨ , ⟩. To (V, ⟨ , ⟩) there corresponds a canonical transitive groupoid T_{V}. In 1.1-1.3 below we give three different constructions of T_{V}. Assume first that V ≠ 0.

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 ⊂ V_{ℂ} := V ⊗ ℂ such that i⟨x, \overline{x}⟩ > 0 for x ≠ 0 ∈ L. Equivalently, one may consider a point of H as a complex structure ℓ on V such that the form ⟨·, i_{ℓ}·⟩ is symmetric and positive definite; here i_{ℓ} ∈ End V is multiplication by i ∈ ℂ with respect to ℓ (the 1-1 correspondence ℓ ⟷ L is ℓ ⟼ L_{ℓ} := the i-eigenspace of i_{ℓ}, L ⟼ ℓ_{L} := the complex structure that comes from the isomorphism V \overrightarrow{∼} V_{ℂ}/L). The space H is a complex variety, and the L’s form a rank g holomorphic bundle L on H. Put λ := det L : this is a holomorphic line bundle on H. Denote by H̃ the space of λ^{⊗2} \ { zero section }; the projection H̃ → H is a ℂ^{∗}-fibration. Let H be the space of C^{∞}-sections H → H̃. One has obvious maps

(4.1.1.1)  H ⟵ H × H ⟶ H̃, φ ← (φ, h) ↦ φ(h).

Since H is contractible, these are homotopy equivalences. Note that for any a ∈ H̃ the map i_{a} : S^{1} ↪ H̃, i_{a}(e^{iθ}) := e^{iθ}a, is a homotopy equivalence which defines a canonical identification

(4.1.1.2)  π_{1}(H̃, a) = ℤ.

For a topological space X let T (X) be the fundamental groupoid of X: its objects are points of X, and its morphisms are homotopy classes of paths. Put T′_{V} := T (H̃).

4.1.2 Denote by Λ = Λ_{V} the grassmannian of real non-oriented Lagrangian subspaces of V; the planes form a canonical Lagrangian sub-bundle L_{ℝ} of V_{Λ} := V ×Λ. Put λ_{ℝ} := det L_{ℝ}: this is a real line sub-bundle of Λ^{g}V_{Λ}. Let Λ′ be the space λ_{ℝ} \ {zero section}/ ± 1: the map x ⟼ x^{2} identifies Λ′ with the “positive ray” of λ_{ℝ}^{⊗2}. The obvious projection Λ′ ⟶ Λ is an ℝ_{+}^{∗}-torsor, hence a homotopy equivalence. One has a canonical map

(4.1.2.1)  v : Λ′ ⟶ H

defined by the formula v(x^{2})(h) = λ^{2}, where λ ∈ det L_{h} ⊂ ∧^{g}V_{ℂ} is the unique vector such that vol(x ∧ λ) = 1 (here vol= \frac{⟨ , ⟩}{g!} ∈ Λ^{2g}V^{∗} is the canonical volume). The map v induces an isomorphism of fundamental groupoids. Put T′′_{V} := T (Λ). According to (1.1.1), 1.2.1) we have a canonical equivalence of groupoids

(4.1.2.2)  α : T′′_{V}⟶^{∼} T′_{V} .

4.1.3 Here is the third construction of T_{V}. For 3 Lagrangian planes one defines, according to Kashiwara, their Maslov index τ(L_{1}, L_{2}, L_{3}) as the signature of the quadratic form B on L_{1} ⊕ L_{2} ⊕ L_{3} given by the formula B(x_{1}, x_{2}, x_{3}) = ⟨x_{1}, x_{2}⟩ + ⟨x_{2}, x_{3}⟩ + ⟨x_{3}, x_{1}⟩ (see [LV] ( )). Let T′′′_{V} be the following groupoid. Its set of objects is Λ. For L_{1}, L_{2} ∈ Λ we put Hom_{T′′′_{V}}(L_{1}, L_{2}) = ℤ, and the composition of morphisms L_{1}⟶^{n}L_{2}⟶^{m}L_{3} is given by the formula m ∘ n := m + n + τ(L_{1}, L_{2}, L_{3}). Since τ satisfies a cocycle formula [LV] ( ), the composition is associative.

Let us define a canonical isomorphism

(4.1.3.1)  β : T′′′_{V}\overrightarrow{∼} T′′_{V}

which is the identity on objects. To construct β we need to choose for each pair L_{1}, L_{2} ∈ Λ a canonical path γ_{L_{1},L_{2}} ∈ Hom_{T′′_{V}}(L_{2}, L_{1}) such that

(4.1.3.2)  γ_{L_{3}L_{2}} ∘ γ_{L_{2}L_{1}} = γ_{L_{3}L_{1}} + τ(L_{1}, L_{2}, L_{3}).

Then one defines β by the formula β(n) = n + γ_{L_{1},L_{2}} for n ∈ Hom_{T_{V} ′′′}(L_{2}, L_{1}) = ℤ (recall that Hom_{T′′_{V}}(L_{2}, L_{1}) is a ℤ-torsor by 1.1.2).

To define γ_{L_{1}L_{2}} consider the subset U_{L_{1}L_{2}} ⊂ Λ that consists of L’s such that L_{1} + L_{2} ⊃ L ⊃ L_{1} ∩ L_{2} = L ∩ L_{1} = L ∩ L_{2}. A plane L ∈ U_{L_{1},L_{2}} defines a quadratic form φ_{L} on L_{1}/L_{1} ∩ L_{2} by the formula φ_{L}(a) = ⟨b, a⟩ where b ∈ L_{2} is a vector such that b + a ∈ L. In this way one gets a 1-1 correspondence between U_{L_{1}L_{2}} and the set of all non-degenerate forms on L_{1}/L_{1} ∩ L_{2}. Let U_{L_{1}L_{2}}^{+} ⊂ U_{L_{1}L_{2}} be the subspace that corresponds to positive-definite forms, so U_{L_{1}L_{2}}^{+} is contractible. Now γ_{L_{1},L_{2}} is the unique homotopy path from L_{2} to L_{1} which, apart from its ends, lies in U_{L_{1}L_{2}}^{+}. One verifies (4.1.3.2) immediately.

4.1.4 Below we will denote by T_{V} either of the groupoids T′_{V} , T′′_{V}, T′′′_{V} identified via (4.1.2.2), (4.1.3.1). In case V = 0, the groupoid T_{V}, by definition, has a single object 0 with End 0 = ℤ. For any V and y ∈ T_{V} we will denote by γ_{0} the generator 1 ∈ ℤ = Aut y.

4.1.5 The groupoid T_{V} has the following functorial properties. Let V be a symplectic space, N ⊂ V a vector subspace such that ⟨ ⟩|_{N} = 0, and let N^{⊥} be the ⟨ ⟩- orthogonal complement to N. Then N^{⊥}/N has an obvious symplectic structure. Since the pre-image of a Lagrangian plane in N^{⊥}/N is a Lagrangian plane in V, we have an embedding Λ_{N^{⊥}/N} ↪ Λ_{V}, which defines a canonical equivalence of groupoids T′′_{N^{⊥}/N} \overrightarrow{∼} T_{V}.

4.1.6 Now let V_{1}, V_{2} be symplectic spaces. One has an obvious map Λ_{V_{1}} × Λ_{V_{2}} ⟶ Λ_{V_{1}⊕V_{2}}, (L_{1}, L_{2}) ⟼ L_{1} ⊕ L_{2}, and a similar map H̃_{V_{1}} × H̃_{V_{2}} ⟶ H̃_{V_{1}⊕V_{2}}, which comes from multiplication det^{⊗2} L_{1}×det^{⊗2} L_{2} ⟶ det^{⊗2} L_{1}⊗det^{⊗2} L_{2} = det^{⊗2}(L_{1}⊕ L_{2}). These define morphisms between corresponding fundamental groupoids, compatible with the canonical equivalences (4.1.2.2). Hence we have a canonical morphism T_{V_{1}} × T_{V_{2}} ⟶ T_{V_{1}⊕V_{2}}.

4.2 The Teichmüller groupoid. This groupoid appears in two equivalent versions: a “combinatorial” or “topological” version, and a “holomorphic” version. 4.2.1 An object of the “topological” Teichmüller groupoid T eich′ is an oriented surface S (possibly non-connected and with boundary) together with a set of points P_{S} = {x_{α}} of the boundary ∂S such that each connected component of ∂S contains exactly one x_{α} (we will denote this component ∂S_{x_{α}}). The morphisms are isotopy classes of diffeomorphisms.

Let us define an “enhanced” groupoid { \atop T eich}′. For a surface S denote by H(S) the image of the canonical map H_{c}^{1}(S, ℝ) ⟶ H^{1}(S, 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 { \atop T eich}′ is a triple (S, P_{S}, y), where (S, P_{S}) ∈ T eich′ and y ∈ T_{H(S)}. A morphism (S, P_{S}, y) ⟶ (S′, P_{S′}, y′) is a pair (φ, γ), where φ : (S, P_{S}) ⟶ (S′, P_{S′}) is a morphism in T eich′, and γ : φ_{∗}(y) ⟶ y′ is a morphism in T_{H(S′)}; the composition of morphisms is obvious.

The projection { \atop T eich}′ → T eich′, (S, P_{S}, y) ⟼ (S, P_{S}), is surjective. For

any (S, P_{S}, y) ∈ { \atop T eich}′ the group Aut{ \atop T eich}′(S, P_{S}, y) is a central extension of

Aut_{T eich′}(S, P_{S}) by ℤ(= Aut_{T_{H(S)}}(y)). So we may say that { \atop T eich}′ is a central extension of T eich′ by ℤ. We will denote the generator of this ℤ by γ_{0}.

Consider the functor T eich′ ⟶ Sets, (S, P_{S}) ⟼ P_{S} = set of boundary components of S. Clearly T eich′ is a fibered category over the groupoid of finite sets. For a finite set A denote by T eich′_{A} the fiber over A (the objects of this groupoid are pairs ((S, P_{S}), ν), where (S, P_{S}) ∈ T eich′, and ν : P_{S} \overrightarrow{∼} A is a bijection). For a bijection f : A \overrightarrow{∼} B, X ∈ T eich′_{A}, Y ∈ T eich′_{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}) = Aut_{id_{P_{S}}} (S, P_{S}). We will use the same notations

for { \atop T eich}′.

For (S, P_{S}) ∈ T eich′ and x_{α} ∈ P_{S} we denote by d_{x_{α}} ∈ Aut^{0}(S, P_{S}) the Dehn twist around ∂S_{x_{α}}. Since d_{x_{α}} acts as the identity on H(S) it lifts to the element (d_{x_{α}}, id_{y}) ∈ Aut_{{ \atop T eich}′}^{0}(S, P_{S}, y), which we will also denote by d_{x_{α}}. These d_{x_{α}} lie

in the center. In particular, we have a canonical morphism ℤ^{P_{S}} ⟶ Aut^{0}(S, P_{S}), (n_{x_{α}}) ⟼ ∏ d_{x_{α}}^{n_{xα}}; ℤ × ℤ^{P_{S}} ⟶ Aut^{0}(S, P_{S}, y), (n_{y}, n_{x_{α}}) ⟼ γ_{0}^{n_{y}} × ∏ d_{x_{α}}^{n_{xα}}.

4.2.2 Here is a “holomorphic” definition of the Teichmüller groupoid. An object of T eich′′ is a complex curve C (smooth, projective, possibly reducible) together with a finite set of points P_{C} = {y_{α}} ⊂ C equipped with non-zero co-tangent vectors ν_{α} ∈ Ω_{C,y_{α}}^{1}. The morphisms are 1-parameter C^{∞}-class families of such objects connecting two given ones, these families being considered up to homotopy. In other words, T eich′′ is the Poincaré groupoid of the modular stack M of the above

structures. In the same way, { \atop T eich}′′ is the Poincaré groupoid of the modular stack M̃ of the data (C, y_{α}, ν_{α}, y), where (C, y_{α}, ν_{α}) ∈ M, and y ∈ det^{⊗2}(H^{0}(C, Ω_{C}^{1})) \ {0}. Clearly, the second modular stack is a ℂ^{∗}-fibration over the first one, hence

{ \atop T eich}′′ is a ℤ(= π_{1}(ℂ^{∗}))-extension of T eich′′.

4.2.3 The groupoids T eich′ and T eich′′, are canonically equivalent, as are { \atop T eich}′

and { \atop T eich}′′. To define this equivalence, take (S, P_{S}) ∈ T eich′. Consider the data (μ; {r_{α}}), where μ is a complex structure on S, and r_{α} : S^{1} = {z ∈ ℂ : |z| = 1\} \overrightarrow{∼} ∂S_{x_{α}} is a parametrization such that r_{α}(1) = x_{α} and r_{α} extends μholomorphically to the ring {z ∈ ℂ : 1 ≤ |z| ≤ 1 + ϵ}. We may glue a collection of unit discs D_{α} = {z ∈ ℂ : |z| ≤ 1} (with their standard complex structure) to S using r_{α}. Denote the corresponding complex curve C = C(S, P_{S}; (μ, r_{α})). It is equipped with the set of points y_{α} = 0 ∈ D_{α}, and the cotangent vectors ν_{α} = dz_{0} ∈ Ω_{C,O}^{1}. Hence C(S, P_{S}; (μ, r_{α})) ∈ T eich′′. It is easy to see that for given (S, P_{S}) the data (μ; {r_{α}}) form a contractible space. So (S, P_{S}) ∈ T eich′ defines a canonical “homotopy point” in T eich′′. In this way we get a morphism of groupoids T eich′ ⟶ T eich′′ which is an equivalence of categories.

To lift this equivalence to { \atop T eich}′ ⟶ { \atop T eich}′′, note that H(S) = H^{1}(C, ℝ). The complex structure on C defines the Hodge subspace H^{0}(C, Ω_{C}^{1}) ⊂ H(S)_{ℂ}, which is a point h_{C} on the corresponding Siegel half plane (see 4.1.1). Now let us interpret T_{H(S)} as a fundamental groupoid of the space denoted by H in (4.1.1.1). For

y ∈ T_{H(S)} put y_{C} := y(h_{C}) ∈ det^{⊗2}(H^{0}(C, Ω_{C}^{1})) \ \{0\}. Our equivalence { \atop T eich}′ ⟶

{ \atop T eich}′′ is given by the formula (S, P_{S}, y) ⟼ (C, y_{α}, ν_{α}, y_{C}).

4.2.4 The above equivalence transforms γ_{y} to the loop θ ⟼ (C, y_{α}, ν_{α}, e^{iθ}y), and transforms the Dehn twist d_{x_{β}} to the loop θ ⟼ (C, y_{α}, e^{iθ}δ_{β}^{α}ν_{α}, y).

4.3 Operations in Teich. We will need the following ones:

(i) One has a functor “disjoint union” ∐ : T eich × T eich → T eich. According to 1.1.6 it lifts in a canonical way to a functor ∐ : { \atop T eich} × { \atop T eich} → { \atop T eich}. Clearly T eich, { \atop T eich} are strictly commutative monoidal categories, and the projection T eich → Sets, (S, P_{S}) ⟼ P_{S}, commutes with ∐. (ii) Deleting of a point. For a finite set A and α ∈ A one has a canonical functor

del_{α} : T eich_{A} → T eich_{A\\{α\}}, { \atop T eich}_{A} ⟶ { \atop T eich}_{A\\{α\}}. In “holomorphic” language

(4.2.2) this functor just deletes y_{α}, ν_{α}. In “topological” language (4.2.1) one should delete the component ∂S_{χ_{α}} by glueing a “cup” to ∂S_{χ_{α}}.

(iii) Sewing. Let A be a finite set, and α, β ∈ A, α ≠ β, two elements. One has a

canonical Sewing Functor S_{α,β} : T eich_{A} → T eich_{A\\{α,β\}}, { \atop T eich}_{A} → { \atop T eich}_{A\\{α,β\}}.

Let us define S_{α,β} in combinatorial language first. For a surface (S, A) ∈ T eich′ choose a diffeomorphism φ : ∂S_{x_{α}} \overrightarrow{∼} ∂S_{x_{β}}, φ(x_{α}) = x_{β}, reversing orientations. Our S_{α,β}(S, A) ∈ T eich′_{A\{α,β}} is S with two boundary components identified by means of φ. Since the φ’s form a contractible space, this surface does not depend on the choice of φ. Note that either H(S) = H(S_{α,β}(S, A)) (if α and β lie in different connected components of S), or H(S) coincides with a subquotient of H(S_{α,β}(S, A)) in a manner described in 4.1.5. In any case one has a canonical

equivalence T_{H(S)} \overrightarrow{∼} T_{H(S_{α,β}(S,A))}. This defines S_{α,β} : { \atop T eich}′_{A} → { \atop T eich}′_{A\\{α,β\}}.

4.3.1 To define S_{α,β} in holomorphic language, take (C, y_{γ}, ν_{γ}) ∈ T eich′′_{A}. Consider a curve C_{α,β} with a single quadratic singularity obtained from C by “clutching” y_{α} and y_{β} 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{M}_{A\\{α,β\}} of curves with at most quadratic singularities. The fiber of the normal bundle N to this divisor at C_{α,β} is canonically identified with T_{C,y_{α}} ⊗ T_{C,y_{β}}. Hence ν_{α}^{−1} · ν_{β}^{−1} is a non-zero vector of this normal bundle. It defines a “point at infinity” of the modular stack M_{A\{α,β}} of smooth curves (for a detailed account of “points at infinity” see [D]), which is a correctly defined (up to unique canonical isomorphism) object S_{α,β}(C, y_{γ}, ν_{γ}) ∈ T eich′′_{A\{α,β}}. To lift S_{α,β} to a functor between

{ \atop T eich}′′’s, notice that the line bundle λ over M with fibers λ_{C} := det H^{0}(C, Ω_{C}^{1}) extends canonically to a line bundle λ over \overline{M}: if C′ has quadratic singularities, one has λ_{C′} := det H^{0}(C, ω_{C′}), where ω_{C′} is the dualizing sheaf. Define the ℂ^{∗}-

bundle M^{≃} → \overline{M} to be λ^{⊗2} \ \{zero section\}. Recall that for any C′ ∈ \overline{M} one has a canonical isomorphism λ_{C′}^{⊗2} = λ⊗2 ; C̃′, where C̃′ is the normalization of C′ (recall that ω_{C′}/ωC̃′ is a skyscraper sheaf, supported at singular points, trivialized canonically up to sign using residues). Hence the fibers of M̃ over (C, y_{γ}, ν_{γ}) and S_{α,β}(C, y_{γ}, ν_{α}) are nearby fibers of the same ℂ^{∗}-fibration, and therefore one has a canonical identification of their fundamental groupoids. This defines the desired

lifting S_{α,β} : { \atop T eich}′′_{A} → { \atop T eich}′′_{A\\{α,β\}}. It is easy to verify that the equivalence 4.2.3 identifies the above “topological” and “holomorphic” constructions of S_{α,β}.

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 → B are pairs (i_{f}, ϕ_{f}), where i_{f} : B ↪ A is an embedding, and ϕ_{f} = {ϕ_{fδ}} is a collection of two-element mutually non-intersecting subsets ϕ_{fδ} of A \ i_{f}(B). The composition is obvious: if g : B → C is another morphism, then

g∘f = (i_{f} ∘i_{g}, ϕ_{f} ∪ϕ_{g}). For f as above we put A_{f}^{1} := ∐_{δ}ϕ_{fδ}, A_{f}^{0} = A\(i_{f}(B)∪A_{f}^{1}),

so A = i_{f}(B) ∐ A_{f}^{0} ∐ A_{f}^{1}.

Now for any morphism f : A → B we have a canonical functor f_{∗} : T eich_{A} → T eich_{B}, { \atop T eich}_{A} → { \atop T eich}_{B} that deletes points in A_{f}^{0} and sews pairwise points in all ϕ_{fδ}’s. One has (g ∘ f)_{∗} = g_{∗} ∘ 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 T eich^{#}, { \atop T eich}^{#} over Sets^{#} with old fibers T eich_{A}, { \atop T eich}_{A}, respectively.

Note that all these categories are strictly commutative monoidal categories with respect to “disjoint union” operation ∐; all the functors commute with ∐.

4.4 Representations of Teich; central charge. Let A be a finite set. Denote by R_{A} the category of finite dimensional ℂ-representations of T eich_{A} (i.e., the objects of R_{A} are functors L : T eich_{A} → V ect), and by R̃_{A} the same for { \atop T eich}_{A}. More generally, if Q is a component (i.e., a strictly full subcategory) of T eich_{A}, we denote by R_{A,Q} the category of representations of Q, identified with the full subcategory of R_{A} that consists of representations supported on Q (i.e., vanish off Q). For a representation V ∈ R̃_{A} and X ∈ { \atop T eich}_{A} we denote by V_{X} the value of V at X.

4.4.1 Definition. A representation V ∈ R̃_{A} has multiplicative central charge a ∈ ℂ^{∗} if for any X ∈ { \atop T eich} the canonical element γ_{0} ∈ AutX acts on V_{X} as multiplication by a. □

For any a ∈ ℂ^{∗} denote by R_{aA} ⊂ R̃_{A} the full subcategory of representations of central charge a. In particular, R_{1A} = R_{A}.

For any morphism f : A → B in Sets^{#} the functor f_{∗} : { \atop T eich}_{A} → { \atop T eich}_{B} defines the corresponding functor f^{∗} : R̃_{B} → R̃_{A}; one has f^{∗}(R_{aB}) ⊂ R_{aA}. The functors f^{∗} define a category R̃^{∗} fibered over Sets^{#} with fibers R̃_{A}, together with fibered subcategories R_{a}^{#} ⊂ R̃^{#} with fibers R_{aA}.

4.4.2 Here is an explicit description of representations. From a combinatorial point of view a representations V ∈ R̃_{A} assigns to each surface (S, A) ∈ T eich_{A} a local system V_{S} on the Lagrangian Grassmannian Λ_{H(S)} (see 4.1.2), and to each φ ∈ Hom((S, A), (S′, A)) a lifting of the corresponding diffeomorphism Λ_{H(S)} \overrightarrow{∼} Λ_{H(S′)} to V_{S} \overrightarrow{∼} V_{S′}. This V lies in R_{aA} if the monodromy matrix of the loop γ_{0} = 1 ∈ ℤ = π_{1}(Λ_{H(S)}) coincides with multiplication by a.

4.4.3 From a holomorphic point of view our V is a local system on the modular stack M̃_{A}; V lies in R_{aA} if the monodromy around the fiber of the projection π : M̃_{A} → M_{A} equals multiplication by a.

Recall that 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 M̃_{A} with regular singularities at ∞. Assume that V ∈ R_{aA}. Choose c ∈ ℤ (“additive central charge”) such that exp(2πic) = a. Let D_{λ^{c}} = D_{cA(λ)} be the ring of differential operators on the “line bundle” λ^{⊗c}. This is a twisted differential operator ring on M_{A} (see 3.2.6-3.2.8). Recall that D_{λ^{c}}-modules can be identified canonically with D-modules on M̃_{A}, monodromic along the fibers of π with monodromy a (see, e.g., [V]). In particular, V is a lisse D_{λ^{c}}-module on M_{A} having regular singularities at ∞.

4.5 Axioms of a fusion category. We will start with preliminary data. 4.5.1 Let A be an abelian ℂ-category (“category of modules”). We assume that A is semisimple, for X ∈ A the ℂ-vector space EndX is finite dimensional, and there are finitely many isomorphism classes of irreducibles. Denote by IrrA the set of isomorphism classes of irreducible objects in A.

We should also have the following data:

– a contravariant functor (“duality”) ∗ : A^{∘} → A together with a natural isomorphism ∗ ∗ \overrightarrow{∼} id_{A}

– a distinguished irreducible object (“vacuum module”) ⊮ together with an isomorphism ν : ⊮ \overrightarrow{∼} ∗ ⊮ such that ∗(ν) ∘ ν = id_{⊮}.

– an automorphism d of the identity functor id_{A}, called the Dehn automorphism, such that d∗ = ∗d and d_{⊮} = 1. Clearly to give d is the same as giving a collection of numbers d_{j} = d_{I_{j}} ∈ ℂ^{∗} for j ∈ IrrA (here I_{j} is an irreducible object of class j; recall that AutI_{j} = ℂ^{∗}).

4.5.2 For any finite set B we have a category A^{⊗B}: this is an abelian ℂ-category equipped with a polylinear functor ⊗ : A^{B} = ∏_{b∈B}A_{b} ⟶ A^{⊗B}, (X_{b})_{b∈B} ⟶

⨂_{b∈B}X_{b}, which is universal in an obvious sense (see [D] § for an extensive discussion

in a less trivial situation). The category A^{⊗B} is semisimple. Its irreducible objects are tensor products of irreducibles in A, so IrrA^{⊗B} = (IrrA)^{B}. Any isomorphism φ : B → B′ induces a canonical equivalence A^{⊗B} → A^{⊗B′}, ⊗X_{b} ⟼ ⊗X_{φ^{−1}(b′)}. 4.5.3 We put A^{⊗∅} = V ect. One may identify A^{⊗{1,2}} = A^{⊗2} with the category of ℂ-linear functors F = A^{0} → A. Namely, to an object X ⊗ Y ∈ A^{⊗2} there corresponds the functor F_{X⊗Y} defined by formula F_{X⊗Y} (Z) = Hom(Z, X) ⊗ Y. We define a canonical object (“regular representation”) R ∈ A^{⊗2} as an object that corresponds to the functor ∗ : A^{0} → A. Here is an explicit construction of R. For each j ∈ IrrA pick an irreducible object I_{j} of class j. Then one has a canonical isomorphism R = _{⨁j∈IrrA}I_{j} ⊗∗I_{j}. Note that R is symmetric: for the transposition σ = {1, 2} acting on A^{⊗2} one has a canonical isomorphism σ(R) = R. So for any two element set B we have a canonical object R_{B} ∈ A^{⊗B}.

4.5.4 For finite sets A, B and a morphism f : A → B in Sets^{#} (see 4.3.2) we define a ℂ-linear functor f^{∗} : A^{⊗B} → A^{⊗A} by the formula

f^{∗}(⨂_{b∈B} X_{b}) = [ ⨂_{a∈i_{f} (B)} X_{i_{f}^{−1}(a)}] ⊗ [ ⨂_{a∈A_{f}^{0}} ident_{a}] ⊗ [ ⨂_{ϕ_{fδ}∈ϕ_{f}} R_{ϕ_{fδ}} ] .

Clearly (g ∘ f)^{∗} = f^{∗} ∘ g^{∗}, so the f^{∗}’s define a fibered category A^{#} over Sets^{#} with fibers A_{A}^{#} = A^{⊗A}. The tensor product functor ⊗ : A^{⊗B_{1}} × A^{⊗B_{2}} ⟶ A^{⊗(B_{1} ∐ B_{2})} defines on A^{#} the structure of commutative monoidal category such that the projection A^{#} → Sets^{#} is a monoidal functor.

4.5.4 Definition. A fusion structure on A is a collection of functors ⟨ ⟩ : A^{⊗A} × { \atop T eich}_{A} ⟶ V ect, (X, S) ⟼ ⟨X⟩_{S} (here A is any finite set), together with natural isomomorphisms (i), (ii):

(i) ⟨X ⊗ Y ⟩_{S⊔T} = ⟨X⟩_{S} ⊗ ⟨Y ⟩_{T} for X ∈ A^{⊗A}, Y ∈ A^{⊗B}, S ∈ { \atop T eich}_{A}, T ∈ { \atop T eich}_{B}.

(ii) ⟨f^{∗}X⟩_{T} = ⟨X⟩_{f_{∗}T} for any morphism f : A → B in Sets^{#}, X ∈ A^{⊗B}, T ∈

{ \atop T eich}_{A}.

These isomorphisms should be compatible in an obvious sense. We also demand that: a. For fixed S ∈ { \atop T eich}_{A} the functor ⟨ ⟩_{S} : A^{⊗A} ⟶ V ect is additive. b. ⟨ ⟩ transforms Dehn automorphism to Dehn twist, i.e., for a finite set A, an

element α ∈ A and a collection of objects X_{γ} ∈ A, γ ∈ A, the automorphisms

of ⟨⊗X_{γ}⟩_{S} induced by ⨂_{γ≠α}id_{X_{γ}} ⊗ d_{X_{γ}} ∈ Aut ⊗ X_{γ} and by d_{α} ∈ AutS coincide.

c. ⟨ ⟩ is non degenerate in the sense that for any non-zero X ∈ A there exists Y ∈ A such that ⟨X ⊗ Y ⟩_{S_{0}} ≠ 0 where S_{0} is a 2-sphere with two punctures. We will say that (A, ⟨ ⟩) is a fusion category of multiplicative central charge

a ∈ ℂ^{∗} if for any X ∈ A^{⊗A} the representation ⟨X⟩ of { \atop T eich} lies in R_{aA}. □

4.5.5 Clearly (ii) just means that X ⟼ ⟨X⟩ is a cartesian functor A^{#} → R̃^{#} 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

(ii)′ ⟨X⟩_{del_{α}S} = ⟨X⊗ident_{α}⟩_{S} for any finite set A, α ∈ A, X ∈ A^{⊗A\\{α\}}, S ∈ { \atop T eich}_{A}.

(ii)′′ ⟨X⟩_{S_{α,β}S} = ⟨X ⊗ R_{αβ}⟩_{S} for any finite set A, a pair of elements α, β ∈ A, α ≠

β, X ∈ A^{⊗A\\{α,β\}}, S ∈ { \atop T eich}_{A}.

4.5.6 Here is a reformulation of 4.5.5(ii)′′ in “holomorphic” language 4.4.3. For X ∈ A^{⊗A\{α,β}} our ⟨X⟩ is a lisse D_{λ^{c}}-module with regular singularities at infinity. As was explained in 4.3.1 we have a canonical surjective smooth map π : M_{A} → N \ {zero section}, where N is the normal bundle to the (smooth part of) the divisor at infinity of M_{A\{α,β}}. We have the canonical specialization function Sp that assigns to a lisse D_{λ^{c}}-module with regular singularities at infinity on M_{A\{α,β}}, the one on N \ {zero section}. Hence we have the D_{λ^{c}}-module π^{∗}Sp⟨X⟩ on M_{A}, and 4.5.5 (ii)′ is an isomorphism π^{∗}Sp⟨X⟩ = ⟨X ⊗ R_{αβ}⟩.

4.6 Fusion functors. Let (A, ⟨ ⟩) be a fusion category. Let A, B be finite sets. Any object S ∈ { \atop T eich}_{A⊔B} defines a functor F_{S} = F_{S}^{A,B} : A^{⊗A} → A^{⊗B} by the formula Hom(F_{S}(X), Y) = ⟨X ⊗ ∗Y ⟩^{∗}, X ∈ A^{⊗A}, Y ∈ A^{⊗B}. We will call F_{S} the fusion functor along S. The automorphisms of S act as automorphisms of F_{S}. Note that if B = ∅ then A^{⊗B} = V ect and F_{S} = ⟨ ⟩_{S}. If A = ∅, then F is a functor { \atop T eich}_{B} → A^{⊗B}, i.e., an A^{⊗B}-valued representation of { \atop T eich}_{B}.

Let C be a third finite set, T ∈ { \atop T eich}_{B⊔C}. We define T ∘ S ∈ { \atop T eich}_{A⊔C} as the surface obtained from T ⊔ S by sewing the B-boundary components.

4.6.1 Lemma. There is a canonical isomorphism of functors F_{T ∘S} = F_{S} ∘ F_{T} : A^{⊗A} → A^{⊗C}.

Proof. For X ∈ A^{⊗A}, Z ∈ A^{⊗C} one has

Hom(F_{T ∘S}(X), Z) = ⟨X ⊗ ∗Z⟩_{T ∘S}^{∗} _{{= \atop 4.5.4(ii)}} ⟨X ⊗ R^{⊗B} ⊗ ∗Z⟩_{T ⊔S}^{∗}

{= \atop 4.5.4(i)} ⨁_{Ij⃗∈IrrA^{⊗B}}⟨X ⊗ ∗Ij⃗⟩_{S}^{∗} ⊗ ⟨Ij⃗ ⊗ ∗Z⟩_{T}^{∗}

= ⨁ Hom(F_{S}(X), Ij⃗) ⊗ Hom(F_{T} (Ij⃗), Z) = Hom(F_{T} ∘ F_{S}(X), Z).

The last equality comes since

F_{S}(X) = ⊕Hom(F_{S}(X), Ij⃗)^{∗} ⊗ Ij⃗.
□

Now assume that A = {0}, B = {∞} are one point sets. Let T eich_{{0,∞}}^{′0} ⊂ T eich′_{{0,∞}} be the full subcategory of “cylinders”. So T eich_{{0,∞}}^{′0} is a connected groupoid; for (S, 0, ∞) ∈ T eich_{{0,∞}}^{0} the group (of its automorphisms) is a free abelian group with generator d_{0} = d_{∞}^{−1}. Denote by S_{0} = (S_{0}, 0, ∞) the object of T eich_{{0,∞}}^{0} such that for any (S, 0, ∞) ∈ T eich_{{0,∞}}^{0} one has Hom(S_{0}, S) = { set of homotopy classes of paths in S connecting 0 and ∞}. This is a canonical object of T eich_{{0,∞}}^{′0}. Its “holomorphic” counterpart is (ℙ^{1}, 0, ∞, dt(0), dt^{−1}(∞)) ∈ T eich_{{0,∞}}^{′′0}, where t is a standard parameter on ℙ^{1}. One identifies this point of T eich′′ with S_{0} canonically by drawing the path ℝ_{≥0} from 0 to ∞. Note that since H(S) = 0 for S ∈ T eich_{{0,∞}}^{′0} we have an obvious embedding T eich_{{0,∞}}^{′0} ↪

{ \atop T eich}′_{\{0,∞\}}; the “holomorphic” counterpart of this section comes since the line bundle λ is canonically trivialized over the “moduli space” of genus zero curves. So we will consider S_{0} as a canonical object of { \atop T eich}_{\{0,∞\}}. Note that if A is any finite

set and T ∈ { \atop T eich}_{A⊔\{0\}}, then one has an obvious canonical isomorphism S_{0}∘T = T. According to 4.6.1 this gives a canonical isomorphism of functors F_{S_{0}} ∘ F_{T} = F_{T}. In fact, one has

4.6.2 Lemma. There is a canonical identification of the functor F_{S_{0}} : A → A with the identity functor id_{A} that generates the above isomorphisms F_{S_{0}} ∘F_{T} = F_{T}

for all T ∈ { \atop T eich}_{A⊔\{0\}}.

Proof. Assume that we know that F_{S_{0}} is an equivalence of categories. Then the desired isomorphism F_{S_{0}} = id_{A} would be F_{S_{0}}^{−1}(F_{S_{0}} ∘ F_{S_{0}} = F_{S_{0}}). Since A is semisimple, to see that F_{S_{0}} is an equivalence it suffices to prove that F_{S_{0}} induces the identity map of the Grothendieck group K(A). The irreducible I_{i} form a basis in K(A). Put F_{S_{0}}(I_{i}) = f_{i}^{j}I_{j}; we have to show that f_{i}^{j} = δ_{i}^{j}. We know that f_{i}^{j} ∈ ℤ_{≥0}. Since f_{i}^{j} = ⟨I_{j} ⊗ ∗I_{i}⟩_{S_{0}}^{∗} we see, by 4.5.4c, that any row or column of the matrix f_{i}^{j} is non-zero. Since F_{S_{0}}^{2} = F_{S_{0}}, these properties imply that F_{S_{0}} = id_{K(A)} (just note that F_{S_{0}}^{2}(I_{i}) = F_{S_{0}}(I_{i}) implies F_{S_{0}} induces a transposition of the set of those I_{j}’s that f_{i}^{j} ≠ 0; hence F_{S_{0}} is a surjective endomorphism of K(A), and hence it is the identity). □

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 F_{S} : A^{⊗A} ⟶ A, together with ∗ and d from 4.5.1, define on 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:

S_{3} :  O^{x_{1}}  O^{x_{2}}  O^{x_{3}}  x_{∞}

Put F_{S_{n}}(X_{1} ⊗ ⋯ ⊗ X_{n}) = X_{1} ˆ⊗ ⋯ ˆ⊗X_{n}. The axiom 1.5.4 (ii)′′ implies immediately that the operation ˆ⊗ : A^{n} → A is strictly associative: one has X_{1} ˆ⊗X_{2} ˆ⊗X_{3} = (X_{1} ˆ⊗X_{2})ˆ⊗X_{3} = X_{1} ˆ⊗(X_{2} ˆ⊗X_{3}). Consider the following diffeomorphism σ of S_{2} that fixes ∂S_{2x_{∞}} and interchanges ∂S_{2x_{1}} and ∂S_{2x_{2}} (we move the holes in a way that the marked point remain on the very right of the hole):

This diffeomorphism induces a natural isomorphism σ_{X_{1}X_{2}} : X_{1} ˆ⊗X_{2} \overrightarrow{∼} X_{2} ˆ⊗X_{1}. It is easy to see that σ satisfies the braid relations, and also one has a relation σ^{2} = d_{x_{∞}}d_{x_{1}}^{−1}d_{x_{2}}^{−1} in AutS_{2}. These imply the hexagon axiom for ˆ⊗, and the axiom σ_{X_{1},X_{2}}^{2} = d_{X_{1} ˆ⊗X_{2}} ∘ (d_{X_{1}} ˆ⊗d_{X_{2}})^{−1} of balanced tensor categories.

4.7 The fusion algebra. The above tensor structure on A defines a commutative ring structure on the Grothendieck group K(A). One calls K(A) the fusion algebra of A. Note that K(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(A).

Now 4.6.2 implies that (K(A), {I_{j}}) is a based ring in the sense of [L] 1.1. According to [L] 1.2, K(A) ⊗ ℚ is a semisimple algebra. Hence K(A) ⊗ 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 γ_{1}, γ_{2} in H_{1}(T, ℤ) compatible with the orientation, so that γ_{1}, γ_{2} are cycles on T that intersect at one point a. Consider the vector space ⟨⊮⟩_{T}. Note that if we cut T along γ_{1}, then γ_{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 ⟨⊮⟩_{T} numbered by irreducibles in A, i.e., we have the isomorphism i_{γ_{1},γ_{2}} : K(A) ⊗ ℂ → ⟨⊮⟩_{T} that transforms I_{j}’s to this basis. Interchanging γ_{1} and γ_{2} we get the isomorphism i_{γ_{2},−γ_{1}} : K(A) ⊗ ℂ \overrightarrow{∼} ⟨⊮⟩_{T}. The composition i_{γ_{2},−γ_{1}}^{−1} ∘ i_{γ_{1},γ_{2}} ∈ AutK(A) ⊗ ℂ is called the 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.

§6. Algebraic field theories

6.1 Axioms. Let c ∈ ℂ be any complex number. An algebraic rational field theory (in dimension 1) of central charge c consists of data (i) - (iv) subject to axioms a-g below: 6.1.1

(i) A fusion category A of multiplicative central charge exp(2πic) (see 4.5.4) (ii) An additive “realization” functor r : A → (T̃, V_{1})_{c}-mod (see 3.4.7).

We assume that for any X ∈ A

a. r(X) is a higher weight module, i.e., the “coordinate module” r(X)_{ℂ((t)),dt(o)} is a (direct) sum of generalized eigenspaces r(X)_{ℂ((t)),λ} = {m ∈ r(X)_{ℂ((t))} : (L_{0} − λ)^{N}m = 0 for N ≫ 0} for the operator L_{0} (see 3.4.7, 7.3.1). Each r(X)_{ℂ((t))λ}, λ ∈ ℂ, is a finite dimensional vector space.

b. r(d_{X}) = T_{r(X)}, where d_{X} is the Dehn automorphism (see 4.5.1) and T is the

monodromy automorphism (see 7.3.2). Note that these axioms imply that r(⊮) is actually a (T̃, V)_{c}-module (since

T_{r(⊮)} = id_{r(⊮)}. (iii) A fixed “vacuum” vector 1 ∈ Hom_{V}(ℂ, r(⊮)).

We assume that

c. 1 is a non-zero vector invariant with respect to the action of s_{O_{F}} (T_{−1F}) ⊂ T̃_{F} (see 3.4.1).

6.1.2. Now let S be a smooth scheme, π : C_{S} → S a family of smooth projective curves, A ⊂ C_{S}(S) a finite disjoint set of sections, and {ν_{a}}_{a∈A} 1-jets of parameters at points in A. This collection defines S-localization data ψ_{c} for (T̃_{c}^{A}, V_{1}^{A}) (see 3.4.7, 3.4.5). The corresponding algebra of twisted differential operators D_{ψ_{c}} coincides with D_{λ^{c}} (see 3.5.6). Hence, by 3.3.5, we have the S-localization functor Δ_{ψ_{c}} ∘r^{⊗A} : A^{⊗A} ⟶ D_{λ^{c}}-mod. On the other hand, by 4.5.4, 4.4.3, the fusion structure on A defines the functor ⟨ ⟩_{C_{S}} : A^{⊗A} ⟶ D_{λ^{c}}-mod such that for any ⊗X_{a} ∈ A^{⊗A} the corresponding D_{λ^{c}}-module ⟨⊗X_{a}⟩_{C_{S}} is lisse with regular singularities at infinity. Our next piece of data is

(iv) A morphism of functors γ : Δ_{ψ_{c}} ∘ r^{⊗A} ⟶ ⟨ ⟩_{C_{S}}.

For X ∈ A^{⊗A} denote by r(X)_{A,C_{S}} = r(X)_{A,ν_{A},C_{S}} the O_{S}-module that corresponds to the S-object “formal completion of C_{S} at A with 1-jet of parameters ν_{A}” of V_{1}^{A} (see 3.4.3, 3.4.6, 3.4.7). If X = ⊗X_{a}, then r(X)_{A,C_{S}} = ⊗_{O_{S}}r(X_{a})_{a,C_{S}}. Recall that Δ_{ψ_{c}} ∘ r^{⊗A}(X), considered as an O_{S}-module, is a quotient of r(X)_{A,C_{S}}. For any section φ of r(X)_{A,C_{S}} put ⟨φ⟩_{C_{S}} = γ(φ) ∈ ⟨X⟩_{C_{S}}. This is the “correlator of the field φ along C_{S}”.

The following axioms should hold:

d. γ commutes with base change, i.e., γ is a morphism of D_{λ^{c}}-modules on the

modular stack M_{A}. e. For a ∈ A, objects X ∈ A^{⊗A\{a}} and a section φ ∈ r(S, r(X)_{A,C_{S}}) one has

⟨φ⟩_{C_{S}} = ⟨φ ⊗ 1_{a}⟩_{C_{S}}. Here ⟨φ⟩_{C_{S}} is a section of ⟨X⟩_{C_{S}} (we forget about the point a), and ⟨φ ⊗ 1_{a}⟩_{C_{S}} is a section of ⟨X ⊗ ⊮_{a}⟩_{C_{S}}; the two D_{λ_{c}}-modules are identified via 4.5.5 (ii)′.

6.1.3 Now consider the two pointed curve C_{0} = (ℙ^{1}, 0, ∞, dt(0), dt^{−1}(∞)). We have coordinates t at 0 and t^{−1} at ∞. For any object X ∈ A consider the pairing

⟨ ⟩_{C_{0}} : r(∗X)_{ℂ((t))}⊗r(X)_{ℂ((t^{−1}))} = r(∗X)_{C_{0ˆ0}}⊗r(X)_{C_{0} ˆ∞} ⟶ ⟨∗X⊗X⟩_{C_{0}} _{{= \atop 4.6.2}} End X Here we write simply ℂ((t)) for (ℂ((t)), dt(0)) ∈ V_{1}. This pairing is a morphism of End X-bimodules, hence it defines a linear map

i : r(∗X)_{ℂ((t))} ⟶ Hom_{End X}(r(X)_{ℂ((t^{−1}))}, End X) =: r(X)_{ℂ((t^{−1}))}^{∗}.

Note that r(X)_{ℂ((t^{−1}))}^{∗} is a T̃_{ℂ((t^{−1}))}- module in an obvious manner. Denote by ∗r(X)_{ℂ((t^{−1}))} ⊂ r(X)_{ℂ((t^{−1}))}^{∗} the sum of generalized eigenspaces of the operator

L_{0} ∈ T̃_{ℂ((t))}. The pairing ⟨ ⟩_{C_{0}} is T (ℙ^{1} \ {0, ∞})-invariant (by definition of Δ_{ψ}, see 3.4.4), hence i commutes with the the L_{0}-action. By axiom a above we see that i(r(∗X)_{ℂ((t))} ⊂ ∗r(X)_{ℂ((t^{−1}))}. Our next axiom is

f. The map i : r(∗X)_{ℂ((t))} ⟶ ∗r(X)_{ℂ((t^{−1}))} is an isomorphism of vector spaces. It suffices to verify f for irreducible X’s only.

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 π : C_{S} → S = Spec ℂ[[q]] be a proper flat family of curves such that the generic fiber C_{η} 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 π such that the points b_{i}(0) ∈ C_{0} are pairwise different, and let ν_{i} ∈ b_{i}^{∗}ω_{C_{S}/S} be a 1-jet of coordinates at the b_{i}’s. Then C = (C_{η}, b_{i}, ν_{i}) is a ℂ((q))-point of M_{B}.

Let t_{1}, t_{2} be formal coordinates at a such that t_{1}t_{2} = q. According to 3.6.1 we get a smooth S-curve C_{S}^{∨} with points a_{1}, a_{2} ∈ C_{S}^{∨}(S) and formal coordinates t_{i} at a_{i}. Put A = B ⨆{a_{1}, a_{2}}. Then C^{∨} = (C_{η}^{∨}, b_{i}, a_{1}, a_{2}; ν_{i}; q^{−1}dt_{1}(a_{1}), dt_{2}(a_{2})) is a ℂ((q))-point of M_{A}.

The S-curves C_{S} and C_{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_{λ^{c}}.

For any object X ∈ A^{⊗B} we get the lisse D_{λ^{c}}-modules ⟨X⟩_{C} and ⟨X ⊗ R⟩_{C^{∨}} on η 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_{λ^{c}}-modules ⟨X⟩_{C} = ⟨X ⊗ R⟩_{C^{∨}}.

To formulate axiom g we need to consider a special vector in r(R). Recall that R = _{⨁I_{j}∈Irr A}I_{j} ⊗ ∗I_{j}. Choose a basis {e_{j}^{K}} in each r(I_{j})_{ℂ((t))} compatible with grading by generalized eigenspaces of L_{0}. Here, as above, we write simply ℂ((t)) for (ℂ((t)), dt(0)) ∈ V_{1}.

Below we will use the following notation: if F ∈ V is any local field, t_{F} a parameter in F, X ∈ A and e ∈ r(X)_{ℂ((t))}, then e_{(F,t_{F})} ∈ r(X)_{F,dt_{F} (0)} is a vector that corresponds to e via the isomorphism (ℂ((t)), dt(0)) \overrightarrow{∼} (F, dt_{F} (0)), t ⟼ t_{F}.

According to axiom f. above, we get the dual basis {∗e_{j}^{K}} of r(∗I_{j})_{ℂ((t))}, namely ∗e_{j}^{K} = i^{−1}e_{j}^{K∗}, where e_{j}^{K∗} ∈ ∗r(I_{j})_{(ℂ((t^{−1})),t^{−1})} is the dual basis to e_{j(ℂ((t^{−1})),t^{−1})}^{K}.

Now let φ = φ(q) be any section of r(X)_{B,ν_{B},C} = r(X)_{B,ν_{B},C^{∨}} over S. Consider the correlator a_{j}^{K} = ⟨φ ⊗ e_{j(ℂ((t_{1})),q^{−1}t_{1})}^{K} ⊗ ∗e_{j(ℂ((t_{2})),t_{2})}^{K}⟩_{C^{∨}}: this is a section of ⟨X ⊗ I_{j} ⊗ ∗I_{j}⟩_{C^{∨}}. Note that ⟨X ⊗ I_{j} ⊗ ∗I_{j}⟩_{C^{∨}} is a finite dimensional ℂ((q))-vector space. One has

6.1.5 Lemma. The series ∑_{K}a_{j}^{K} converges; its limit ⟨φ⊗c_{j}⟩_{C^{∨}} ∈ ⟨X ⊗I_{j} ⊗∗I_{j}⟩_{C^{∨}}

does not depend on a particular choice of basis {e_{j}^{K}}. □

Assuming the lemma, our final axiom is g. One has ⟨φ⟩_{C} = ⟨φ ⊗ ∑_{j}C_{j}⟩_{C^{∨}} = ∑_{j}⟨φ ⊗ C_{j}⟩_{C^{∨}} via the above canonical

isomorphism

⟨x⟩_{C} = ⟨X ⊗ R⟩_{C^{∨}} = ⊕⟨X ⊗ I_{j} ⊗ ∗I_{j}⟩_{C^{∨}}.

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 S̃ = Spec(ℂ[u, u^{−1}]) × S together with an S̃-point of M_{A} defined by the family C_{u}^{∨} = (C∨ ; S̃, b_{i}, a_{1}, a_{2}; ν_{i}, udt_{1}, dt_{2}). We get the lisse D_{λ^{c}}-module ⟨X ⊗ I_{j} ⊗ ∗I_{j}⟩_{C_{u}^{∨}} on S̃, and a collection of sections a_{j}^{K}(u, q) = ⟨φ(q) ⊗ e_{j(ℂ((t_{1})),ut_{1})}^{K} ⊗

∗e_{j(ℂ((t_{2})),t_{2})}^{K}⟩_{C_{u}^{∨}} ∈ Γ(S̃, ⟨X ⊗ I_{j} ⊗ ∗I_{j}⟩_{C^{∨}}). 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 = ∞, so we may extend it to a vector bundle V to S̃^{−} = Spec(ℂ[u^{−1}])×S invariant with respect to operator u∂_{u}. Our lemma would follow if we show that for any N ≫ 0 one has a_{j}^{K}(u, q) ∈ u^{−N}V for all but finitely many K’s. The action of the operator u∂_{u} on a_{j}^{K}(u, q) was computed in 3.4.7.1. Namely, we have u∂_{u}(a_{j}^{K}(u, q)) = ⟨φ(q)⊗L_{0}(e_{j}^{K})_{(ℂ((t_{1})),ut_{1})} ⊗∗e_{j}^{K}⟩_{C_{u}^{∨}}, hence a_{j}^{K}(u, q) is a generalized eigenvector of u∂_{u} with eigenvalue equal to an eigenvalue of L_{0} at e_{j}^{K}. Axiom a. above implies that for any \overline{μ} ∈ ℂ/ℤ and c ∈ ℝ the space ⨁_{\begin{matrix}μ=\overline{μ} mod ℤ \\ Re μ>c \end{matrix}}r(I_{j})_{ℂ(t)μ} ⊂

r(I_{j})_{ℂ((t))} is finite dimensional. On the other hand, since ⟨X ⊗ I_{j} ⊗ ∗I_{j}⟩_{C_{u}^{∨}} is a lisse module, there are only finitely many \overline{u} ∈ ℂ/ℤ such that one has a section which is a generalized eigenvector of u∂_{u} with eigenvalue mod ℤ equal to \overline{u}. This implies that for any c ∈ ℝ all but finitely many a_{j}^{K}’s are generalized eigenvectors of u∂_{u} with Re (eigenvalue) < c. This implies that all but finitely many of them lie in u^{−N}V. □

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).

6.1.7 Here is an example of how axiom g works. Let C be a fixed curve, A ⊂ C a finite set, {ν_{a}}, a ∈ A, 1-jets of coordinates at a’s, X ∈ A^{⊗A}, and φ ∈ r(X)_{a,C}. Let x ∈ C \ A be a point, t_{x} a parameter at x and λ_{1}, …, λ_{n} ∈ ℂ distinct complex numbers. Let x_{i}(q) be ℂ[[q]] points of C defined by the formula x_{i}(o) = x, t_{x}(x_{i}(q)) = λ_{i}q. Put t_{i} = t_{x/q} − λ_{i}: these are parameters at x_{i}’s for q ≠ 0. Let Y_{1}, …, Y_{n} be objects in A, ψ_{i} ∈ r(Y_{i})_{ℂ((t))}. We would like to compute ⟨φ ∈ ψ_{1}_{(ℂ((t_{1})),t_{1})} ⊗ ⋯ ⊗ ψ_{n}_{(ℂ((t_{n})),t_{n})}⟩_{C} ∈ ⟨X ⊗ Y_{1} ⊗ ⋯ ⊗ Y_{n}⟩_{(C,A,{x_{i}},ν_{A},dt_{i}(x_{i})}. To do it one should blow up the point (x, 0) ∈ C_{S} = C × 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}^{′∨} is constant: one has C_{S}^{′∨} = C_{S} ∐ ℙ_{S}^{1}; the formal parameters at a_{1} = x ∈ C_{S}, a_{2} = ∞ ∈ ℙ_{S}^{1} are t_{x}, t^{−1}, respectively. We see that C′_{S} comes from (C ∐ ℙ^{1}; x, ∞; t_{s}, t^{−1}) via the construction 3.6.4. The points A, {x_{i}} on C_{S}^{′∨} are also constant, as well as coordinates t_{i}: one has x_{i} = λ_{i} ∈ ℙ^{1}, t_{i} = t − λ_{i}. Hence

\begin{matrix}⟨X ⊗ Y_{1} ⊗ ⋯ ⊗ Y_{n}⟩_{(C;A,\{x_{i}\};ν_{A},dt_{i}(x_{i}))} = ⨁_{j}⟨Y_{1} ⊗ ⋯ ⊗ Y_{n} ⊗ I_{j}⟩_{(ℙ^{1};λ_{i},∞;dt(λ_{i}),q^{−1}dt^{−1}(∞))} \\ ⊗ ⟨∗I_{j} ⊗ X⟩_{(C;x,A;dt_{x}(x),ν_{A})} \end{matrix}

and

\begin{matrix}⟨φ ⊗ ψ_{1}_{(ℂ((t_{1})),t_{1})} ⊗ ⋯ ⊗ ψ_{n}_{(ℂ((t_{n})),t_{n})}⟩_{C} = ⟨ψ_{1}_{(ℂ((t−λ_{1})),t−λ_{1})} ⊗ ⋯ ⊗ ψ_{n}_{(ℂ((t−λ_{n})),t−λ_{n})} \\ ⊗ e_{j}^{K}(ℂ((t^{−1})), q^{−1}t^{−1})⟩_{ℙ^{1}} ⊗ ⟨∗e_{j (ℂ((tx)),tx)}^{K} ⊗ φ⟩_{C}. \end{matrix}

6.2 Global vertex operators. Assume we have an algebraic field theory as in 6.1. Let C be a smooth compact curve, A ⊂ C a finite set of points and ν_{a}, a ∈ A, a 1-jet of parameters at a’s. 6.2.1 For an object X ∈ A^{⊗A} we have a finite dimensional vector space ⟨X⟩_{C} and a linear map ⟨ ⟩_{C} : r(X)_{A_{C}} ⟶ ⟨X⟩_{C}. Also for any n-tuple of points x_{1}, ⋯, x_{n} ∈ C \ A, x_{i} ≠ x_{j} for i ≠ j, we have a linear map ⟨ ⟩_{C} : r(X)_{A,C} ⊗ r(⊮)_{x_{1},C} ⊗ … ⊗ r(⊮)_{x_{n},C} = r(X ⊗ ⊮ ⊗ ⋯ ⊗ ⊮)_{A∪{x_{1},··· ,x_{n}},C} ⟶ ⟨X ⊗ ⊮ ⊗ ⋯ ⊗ ⊮⟩_{C} = ⟨X⟩_{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(⊮) is a (T̃, V)_{c}-module (see axiom b). We may rewrite this as a linear map

V_{x_{1},··· ,x_{n}}^{A} : ⊗r(⊮)_{x_{i},C} ⟶ r(X)_{A,C}^{∗} ⊗ ⟨X⟩_{C}.

This construction may be rearranged in several ways: 6.2.2 Let the points x_{1}, ⋯, x_{n} vary. On C^{n} we have a locally free O_{C^{n}}-module r(⊮)_{C^{n}}^{⊗n} with fibers r(⊮)_{C^{n}(x_{1},··· ,x_{n})}^{⊗n} = ⊗r(⊮)_{x_{i},C}. On U = (C \ A)^{n} \ {diagonals} we have a morphism V^{A} : r⊮)_{U}^{⊗n} ⟶ Hom_{ℂ}(r(X)_{A,C}, ⟨X⟩_{C} ⊗ O_{U}) of O_{U}-modules such that the value of V^{A} at (x_{1}, ⋯, x_{n}) coincides with V_{x_{1},··· ,x_{n}}^{A}. For any open set W ⊂ U we get a map

V_{H}^{A} : Γ(W, r(⊮)_{W}^{⊗n} ⊗ Ω_{W}^{n}) ⟶ r(X)_{A,C}^{∗} ⊗ ⟨X⟩_{C} ⊗ H_{DR}^{n}(W)

which is a composition of V ⊗ id_{Ω_{W}^{n}} and the canonical projection Γ(W, Ω_{W}^{n}) → H_{DR}^{n}(W).

6.2.3 Assume that A = A_{1} ⊔ A_{2} and X = X_{1} ⊗ X_{2}, X_{i} ∈ A^{⊗A_{i}}. Then r(X)_{A,C} = r(X_{1})_{A_{1},C} ⊗ r(X_{2})_{A_{2},C}), r(X)_{A,C}^{∗} = Hom(r(X_{1})_{A_{1},C}, r(X_{2})_{A_{2},C}^{∗}). Let us fix a formal parameter t_{a} at α such that dt_{a}(a) = ν_{a}. These identify r(X_{i})_{A_{i},C} with “coordinate modules” r(X_{i})_{ℂ((t_{A_{i}}))} and r(X_{2})_{A_{i},C}^{∗} with a completion r(∗X_{2})ℂ((t_{A_{2}} ))^{∧} of r(∗X_{2})_{ℂ((t_{A_{2}} ))}. So we may rewrite the above V_{x_{1},··· ,x_{n}} as

V_{x_{1},··· ,x_{n}}^{A_{1},A_{2}} : ⊗r(⊮)_{x_{i},C} ⊗ ⟨X_{1} ⊗ X_{2}⟩_{C}^{∗} ⟶ Hom(r(X_{1})_{ℂ((t_{A_{1}} ))}, r(∗X_{2})ℂ((t_{A_{2}} ))^{∧}).

The linear operators in the image of this map are called vertex operators. 6.2.4 Now assume that X_{1} = Y, X_{2} = ∗F_{C}^{A_{1},A_{2}}(Y), where F_{C}^{A_{1},A_{2}} : A^{⊗A_{1}} → A^{⊗A_{2}} is the fusion functor from 4.6. Then ⟨X_{1} ⊗ X_{2}⟩_{C}^{∗} = Hom(F_{C}^{A_{1},A_{2}}(X_{1}), ∗X_{2}) has a canonical element id_{∗X_{2}}; hence we get

V_{x_{1},··· ,x_{n}}^{A_{1},A_{2}} : ⊗r(⊮)_{x_{i},C} ⟶ Hom(r(Y)_{ℂ((t_{A_{1}} ))}, r(F_{C}^{A_{1},A_{2}}(Y))ℂ((t_{A_{2}} ))^{∧}).

Here are the first properties of vertex operators in this setting, that follow directly from the axioms. 6.2.5 For j ∈ {1, …, n} and φ ∈ _{⨂i≠j}r(⊮)_{x_{i},C} one has V_{x_{1},... ,x̂_{j},... ,x_{n}}^{A_{1},A_{2}}(φ) = V_{x_{1},... ,x_{n}}^{A_{1},A_{2}}(φ ⊗ 1_{x_{j}}). 6.2.6 Put T (C \ A, x_{1}, …, x_{n}) = {τ ∈ T (C \ A) : τ(x_{i}) = 0} ⊂ T (C \ A). Then the linear map V_{x_{1},... ,x_{n}}^{A_{1},A_{2}} commutes with the T (C \ A, x_{1}, …, x_{n})-action. Here T (C \ A, x_{1}, …, x_{n}) acts on the left hand side via T (C \ A, x_{1}, …, x_{n}) → T_{(x_{i})o} ⊂ T̃_{(x_{i})} (= Virasoro algebra at x_{i}) and on the right hand side via the map T (C \A) → T̃_{(A)} from 2.3.4. In particular, any vertex operator F transforms via a finite dimensional representation of T (C \A, x_{1}, …, x_{n}) and F is fixed by a Lie subalgebra of T (C \A) that consists of fields vanishing to sufficiently high order at the x_{i}’s. 6.2.7 Let C′ be another curve, A′ = A_{2} ⊔ A_{3} ⊂ C′ a finite set of points, t_{a′} formal parameters at a′ ∈ A′, and {x′_{1}, ⋯, x′_{m}} ⊂ C′ \ A′. Let (C ∘ C′)_{q} be the ℂ[[q]]-curve with zero fiber obtained from C ⊔ 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} ⊔ A_{3} ⊔ {x_{1}, …, x_{n}} ⊔ {x′_{1}, …, x′_{m}} is a finite set of ℂ[[q]]-points of (C ∘ C′)_{q}, and hence we have our vertex operators map V_{x_{1},... ,x_{n},x′_{1},... ,x′_{m}}^{A_{1},A_{3}} : ⊗r(⊮)_{x_{i},C} ⊗ r(⊮)_{x′_{j},C′} ⟶ Hom(r(Y)_{ℂ((t_{A_{1}} ))},

r(F_{(C∘C′)_{q}}^{A_{1},A_{3}}(Y)ℂ((t_{A_{3}} ))^{∧}). On the other hand, it is easy to see that “topologically” (C ∘ C′)_{q} coincides with “topological” composition C_{q} ∘ C′ from 4.6.1, where

C_{q} = (C, dt_{a_{1}}(a_{1}), q^{−1}dt_{a_{2}}(a_{2})) ∈ M_{A}, a_{1} ∈ A_{1}, a_{2} ∈ A_{2}.

Hence, by 4.6.1, one has F_{(C∘C′)_{q}}^{A_{1},A_{3}} = F_{C′}^{A_{2},A_{3}} ∘ F_{C_{q}}^{A_{1},A_{2}}.

Our next property, that follows directly from axiom g, is:

for any φ ∈ ⊗r(⊮)_{x_{i},C}, φ′ ∈ ⊗r(⊮)_{x′_{j},C′} one has

V_{x_{1},··· ,x_{n},x′_{1},··· ,x′_{m}}^{A_{1},A_{3}}(φ ⊗ φ′) = V_{x′_{1},··· ,x′_{m}}^{A_{2},A_{3}}(φ′) ∘ V_{x_{1},··· ,x_{n}}^{A_{1},A_{2}},

where composition of “infinite matrixes” is understood in a way similar to 6.1.5.

6.3 Local vertex operators. Assume we have a field theory as in 6.1. 6.3.1 Let C be a smooth curve. Denote by C̃ the cotangent bundle of C with zero section removed; so a point of C̃ is a pair (x, ν_{x}), x ∈ C, ν_{x} is a 1-jet of coordinates at x. Any object X ∈ A defines a locally free OC̃-module r(X)C̃ with fibers r(X)_{(x,ν_{x})} = r(X)_{x,ν_{x},C}. A choice of a family of local parameters defines a trivialization of r(X)C̃. More precisely, let t be a function on a formal neighbourhood of the diagonal Δ : C̃ ↪ C̃ × C, Δ(x, ν_{x}) = (x, ν_{x}, x), such that t|_{Δ} = 0, d_{x_{2}}t(x, ν_{x}, x) = ν_{x} (so t_{(x,ν_{x})} = t(x, ν_{x}, ·) is a formal parameter at x); such a t defines a trivialization s^{t} : r(X)C̃) \overrightarrow{∼} r(X)_{ℂ((t))} ⊗ OC̃.

This r(X)C̃ is a DC̃-module in a canonical way; the D-module structure comes from the T_{ℂ((t))^{−1}}-action on r(X)_{ℂ((t))}. Explicitly, a vector field τ ∈ TC̃ ⊂ DC̃ acts on r(X)C̃ as follows. Choose (locally) a family t of local parameters as above. Let ∇_{0} be the flat connection that corresponds to the trivialization S^{t}. Let τ^{t} ∈ T̃_{ℂ((t))} ⊗ OC̃ be the section defined by formula τ^{t} = S_{ℂ[[t]]}(T_{(x_{1},ν_{x_{1}} )}(t)∂_{t}): here T_{(x_{1},ν_{x_{1}} )} is a vector field on C̃ × C equal to τ in the C̃-directions and to 0 in the C directions (hence T_{(x_{1},ν_{x_{1}} )}(t) is a function on the formal neighbourhood of Δ), and S_{ℂ[[t]]} : T_{ℂ[[t]]} → T̃_{ℂ((t))} was defined in 3.4.1. Now for a section φ of r(X)C̃ one has τ(φ) = ∇_{0}(τ)(φ) − τ^{t}(φ), where τ^{t}(φ) is the T̃_{ℂ((t))}-action on r(X)_{ℂ((t))}.

6.3.2 Remarks. (i) One may explain the DC̃-module structure on r(X)C̃ as follows. We have two natural actions of the Lie algebra T_{C} on r(X)C̃. The first one – “Lie derivative” – comes since r(X)C̃ is a natural sheaf, hence symmetries of C (and infinitesimal ones also) act on it. The second is an O-linear action that comes because the fibers of r(X)C̃ are Virasoro modules (using the splitting S_{O_{x}^{∧}}). Now the D-module action of vector fields is the difference of these two actions. (ii) For any étale map f : C′ → C one has a canonical isomorphism f_{r}^{∗}(X)C̃ = r(X)C̃′ of DC̃′-modules. (iii) If d_{X} = id_{X} (see 4.5), e.g., if X = ⊮, then r(X) is actually a (T̃, V)-module, hence r(X)C̃ comes from a canonical D-module r(X)_{C} on C.

6.3.3 For X_{1}, ⋯, X_{n} ∈ A consider the D-module ⊠_{i}r(X_{i})C̃ = r(X_{1})C̃ ⊠ … ⊠ r(X_{n})C̃ on C̃^{n}. If C is compact, we also have a lisse D-module ⟨X_{1} ⊗ ⋯ ⊗ X_{n}⟩C̃ on C̃ \ {diagonals} with regular singularities along the diagonals; the fiber of ⟨X_{1} ⊗ ⋯ ⊗ X_{n}⟩C̃ over (x_{1}, ν_{1}, ⋯, x_{n}, ν_{n}) is ⟨X_{1} ⊗ ⋯ ⊗ X_{n}⟩_{(C,{x_{i}},{ν_{i}})}. By 6.1.2 we have a canonical morphism of DC̃^{n}-modules ⟨ ⟩C̃ : ⊠r(X_{i})C̃ → j_{∗}⟨⊗X_{i}⟩C̃, where j : C̃^{n} \ {diagonals} ↪ C̃. 6.3.4 For a moment let us drop the compactness assumption on C; we will work locally. For X ∈ A let r(X)^{∧}_{C,C^{n}} be the completion of r(X)C̃ ⊠ O_{C^{n}} around the diagonal Δ : C̃ → C̃ × C^{n}, Δ(x, ν_{x}) = (x, ν_{x}; x, ⋯, x). A choice of a family of local parameters t = (t_{x,ν_{x}}) identifies sections of r(X)_{C̃,C^{n}}^{∧} with for-

mal power series Σm_{i_{1},··· ,i_{n}}t_{1}^{i_{1}} ⋯ t_{n}^{i_{n}}, where m_{i_{1},··· ,i_{n}} are sections of r(X)C̃ and t_{i}(x_{0}, ν_{x_{0}}, x_{1}, ⋯, x_{n}) = t_{(x_{0},ν_{x_{0}} )}(x_{i}). Then r(X)_{C̃,C^{n}}^{∧} is a (non quasicoherent)

D_{C̃×C^{n}}-module in an obvious manner. Let O_{C̃×C^{n}}^{#} ⊃ O_{C̃×C^{n}} denote the sheaf of functions having (meromorphic) singularities at diagonals x_{i} = x_{j}, i, j ≥ 0. Put r(X)_{C̃,C^{n}}^{#} := O_{C̃×C^{n}}^{#} ⊗_{O_{_{C}×C^{n}}} r(X)_{C̃,C^{n}}^{∧}: this is also a D_{C̃×C^{n}}-module. A section

of r(X)_{C̃×C^{n}}^{#} is a formal series

∏(t_{i} − t_{j})^{−a_{ij}}(Σm_{i_{1}···i_{n}}t_{1}^{i_{1}} ⋯ t_{n}^{i_{n}}), a_{ij} ≥ 0.

Now let us define the “local” vertex operators:

6.3.5 Lemma. There is a canonical morphism of D_{C̃×C^{n}}-modules

μ : r(⊮)_{C} ⊠ ⋯ ⊠ r(⊮)_{C} ⊠ r(X)C̃ ⟶ r(X)_{C̃,C^{n}}^{#}

such that (assuming C is compact) for any (x, ν_{x}; y_{1}, ν_{y_{1}}; ⋯; y_{m}, ν_{y_{m}}) ∈ C̃ × C̃^{m}, x ≠ y_{i}, y_{i} ≠ y_{j} for i ≠ j, objects Y_{i} ∈ A, an element ψ_{x} ∈ r(X)_{x,ν_{x}}, ψ_{y_{i}} ∈ r(Y_{i})_{y_{i},ν_{y_{i}}} and a section φ_{1}, ⋯, φ_{n} of r(⊮)_{C} in a neighbourhood of x one has

⟨φ_{1} ⊗ ⋯ ⊗ φ_{n} ⊗ ψ_{x} ⊗ ⋯ ⊗ ψ_{y_{m}}⟩C̃ = ⟨μ(φ_{1} ⊗ ⋯ ⊗ φ_{n} ⊗ ψ_{x}) ⊗ ψ_{y_{1}} ⊗ ⋯ ⊗ ψ_{y_{n}}⟩C̃

(as meromorphic functions on a formal neighbourhood of (x, …, x) ∈ C^{n} with values in ⟨X ⊗ Y_{1} ⊗ ⋯ ⊗ Y_{m}⟩_{(C,{x,y_{i}},{ν_{x},ν_{y_{i}} })} identified with ⟨⊮ ⊗ ⋯ ⊗ ⊮ ⊗ X ⊗ Y_{1} ⊗ ⋯ ⊗ Y_{m}⟩ via 4.5.5 (ii)′).

Proof - construction. We will write an explicit formula for μ. To do this consider first ℙ^{1} with the standard parameter t. So t defines a family of local parameters t_{x} = t − x on ℙ^{1} \ {∞}, and hence we have a trivialization s^{t} : r(⊮_{ℙ^{1}\{∞}} = r(⊮)_{ℂ((t))} ⊗ O_{ℙ^{1}\{∞}}. For φ ∈ r(⊮)_{ℂ((t))} we denote by φ^{t} the corresponding “constant” section of r(⊮)_{ℙ^{1}\{∞}}.

Now for φ_{1}, ⋯, φ_{n} ∈ r(⊮)_{ℂ((t))} and x_{1}, …, x_{n} ∈ ℙ^{1} \ {∞}, x_{i} ≠ x_{j} for i ≠ j,

consider the vertex operator V_{x_{1},··· ,x_{n}}^{0,∞}(φ_{1}^{t} ⊗ ⋯ ⊗ φ_{n}^{t}) : r(X)_{ℂ((t))} ⟶ r(X)ℂ((t))^{∧}k

from 6.2.4 (here we identified the module r(X)_{ℂ((t^{−1}))} at ∞ with r(X)_{ℂ((t))} via t^{−1} ⟼ t). In fact, this operator lies in End r(X).

[Proof. For any a ∈ ℂ^{∗} one has t_{ax} = a(t − x); hence the automorphism x ⟼ ax of ℙ^{1} acts on r(⊮)_{ℙ^{1}} (according to 6.3.2) by the formula φ^{t} ⟼ (a^{L_{0}}φ)^{t}. This implies immediately that if L_{0}φ_{i} = n_{i}φ_{i}, then V_{x_{1},··· ,x_{n}}^{0,∞}(⊗φ_{i}^{t})(L_{0}e) = (L_{0} + n_{1} + ⋯ + n_{n})V_{x_{1},··· ,x_{n}}^{0,∞}(e). Hence V_{x_{1},··· ,x_{n}}^{0,∞}(⊗φ_{i}^{t}) maps L_{0}-generalized eigenspaces in r(X)_{ℂ((t))} to ones in r(X)_{ℂ((t))}^{∧}; since the sum of these equals r(X)_{ℂ((t))}, we see that V_{x_{1},··· ,x_{n}}^{0,∞}(⊗φ_{i}^{t}) maps r(X)_{ℂ((t))} to r(X)_{ℂ((t))}.]

Clearly, V_{x_{1},··· ,x_{n}}^{0,∞}(φ_{1}^{t} ⊗ ⋯ ⊗ φ_{n}^{t}) is a meromorphic function on (ℙ^{1} \ {0, ∞})^{n} \ {diagonals} with values in End r(X)_{ℂ((t))}. Put μ(φ_{1}^{t}⊗⋯⊗φ_{n}^{t}⊗ψ_{0}) = V_{x_{1},··· ,x_{n}}^{0,∞}(φ_{1}^{t}⊗ ⋯ ⊗ φ_{n}^{t})(ψ_{0}) for ψ_{0} ∈ r(X)_{ℂ((t))}: we will consider μ( ) as a formal power series in variables t_{1}, ⋯, t_{n}, t_{i} = t(x_{i}), with poles along diagonals t_{i} = t_{j}, with values in r(X)_{ℂ((t))}.

Now consider our curve C̃. Choose a family of parameters t. It defines a

trivialization r(⊮)_{C} ⊠ ⋯ ⊠ r(⊮)_{C} ⊠ r(X)C̃ ⟶^{∼} r(⊮)_{ℂ((t))}^{⊗n} ⊗ r(X)_{ℂ((t))} ⊗ O_{C̃×C^{n}}

in a formal neighbourhood of the diagonal. We put μ(φ_{1}^{t} ⊗ ⋯ ⊗ φ_{n}^{t} ⊗ ψ_{x,t}) = μ(φ_{1}^{t} ⊗ ⋯ ⊗ φ_{n}^{t} ⊗ ψ_{ℂ((t))t})_{x,t} in obvious notations (so we write down the above μ on our curve in the coordinates t_{x} for each x ∈ C). It is easy to see that μ, 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 ℂ[[q]] and apply axiom g. □

We will often write μ(φ_{1} ⊗ ⋯ ⊗ φ_{n} ⊗ ψ) = φ_{1}(x_{1}) ⋯ φ_{n}(x_{n})ψ(x) ∈ ∏_{i,j}(x_{i} −

x_{j})^{−N}ℂ[[x_{1} − x, ⋯, x_{n} − x]] ⊗ r(X)_{x}. The composition property 6.2.7 for global vertex operators implies this associativity property of μ: 6.3.6 One has

φ_{1}(x_{1}) ⋯ φ_{n}(x_{n})ψ(x) =

φ_{1}(x_{1})(φ_{2}(x_{2})(⋯ (φ_{n}(x_{n})ψ(x)) ⋯) ∈ ℂ((x_{1} − x((⋯ ((x_{n} − x)) …))) ⊗ r(X)_{x}. Also if one of the φ_{i}’s is equal to 1, we may delete it.

6.4 Chiral algebra. Consider the three step complex L_{C^{•}} = (L_{2} → L_{1} → L_{0}) of sheaves for the Zariski or étale topology of C. Here L_{2} = r(⊮)_{C}, L_{1} = ω⊗_{O_{C}} r(⊮)_{C}, the differential d : L_{2} → L_{1} is the de Rham differential, and L_{0} = L_{1}/dL_{2} = H_{DR}^{1}(r(⊮)_{C}) is the sheaf of de Rham cohomology with coefficients in the D_{C}- module r(⊮)_{C}, and d : L_{1} → L_{0} is the projection.

6.4.1 For sections γ_{1}, γ_{2} of L_{1} we define a section γ_{1} ∗ γ_{2} of L_{1} by the formula γ_{1} ∗ γ_{2} = Res_{1}μ(γ_{1} ⊗ γ_{2}), and a section {γ_{1}, γ_{2}} ∈ L_{2} by the formula {γ_{1}, γ_{2}} = R̃esμ(γ_{1} ⊗γ_{2}). Here γ_{1} ⊗γ_{2} is a section of L_{1} ⊠L_{1} = Ω_{C×C}^{2} ⊗_{O_{C×C}} (r(⊮)_{C} ⊠r(⊮)_{C}), μ(γ_{1} ⊗γ_{2}) is a section of ω_{C} ⊠L_{1} = Ω_{C×C}^{2} ⊗p_{2}^{∗}r(⊮)_{C} with poles along the diagonal, Res_{1} is residue around the diagonal along the first variable, and R̃es was defined in 2.2.4. Now the lemma 6.3.5 implies immediately that d({γ_{1}, γ_{2}}) = γ_{1} ∗ γ_{2} + γ_{2} ∗ γ_{1} and for φ ∈ L_{2} one has (dφ) ∗ γ = 0. Define the bracket [ , ] : L_{•} ⊗ L_{•} → L_{•} by the formula [dγ_{1}, dγ_{2}]_{0,0} = d(γ_{1} ∗ γ_{2}), [dγ_{1}, γ_{2}]_{0,1} = −[γ_{2}, dγ_{1}]_{1,0} = γ_{1} ∗ γ_{2}, [γ_{1}, γ_{2}]_{1,1} = {γ_{1}, γ_{2}} for γ_{i} ∈ L_{1}. The associativity property 6.3.6 implies

6.4.2 Lemma. This bracket provides L with the structure of DG Lie algebra. □

This DG Lie algebra (or rather its zero component L_{0}) is called the chiral Lie algebra of our field theory.

6.4.3 Consider a canonical embedding i : O_{C} → r(⊮)_{C} of D_{C}-modules, i(f) = f · 1.

Denote by C_{•} the three step complex C_{2} = O_{C} ⟶^{d} C_{1} = ω_{C} → C_{0} = H; here H = H_{DR}^{1} and the differential C_{1} → C_{0} is the canonical projection. We get a canonical morphism i : C_{•} → L_{•} of complexes, i(f) = f · 1. One may see that i is actually an embedding (for i_{0} this will follow from 6.4.6), and obviously i(C_{•}) lies in the center of the chiral algebra.

6.4.4 For any x ∈ A consider the DC̃-module r(X)C̃. The formula γ(m) = Res_{1}μ(γ ⊗ m) for γ ∈ L_{0}, m ∈ r(X)C̃ defines a canonical action of L_{0} on r(X)C̃ that commutes with the DC̃-action.

6.4.5 For any local field F we may consider the “local” version L_{F^{•}} of the above L_{C^{•}}. 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 ∈ C, then L_{F_{x}^{2}} = F_{x} ⊗_{O_{C}} L_{C^{2}}, L_{F_{x}^{1}} = F_{x} ⊗_{O_{C}} L_{C^{1}}, L_{F_{x}^{0}} = H_{DR}^{1}(F_{x}, r(⊮)_{C}) = L_{F_{x}^{1}}/dL_{F_{x}^{2}}. For any X ∈ A we have a canonical map L_{F^{0}} ⊗ r(X)_{F} → r(X)_{F}, γ ⊗ m ⟼ γ(m) = Res_{0}μ(γ ⊗ m). Here μ(γ ⊗ m) ∈ H_{DR}^{1}(F) ⊗ r(X)_{F} and one has (cf. 6.4.4):

6.4.6 Lemma. This map defines a representation of the Lie algebra L_{F^{0}} on r(X)_{F}.

The central subalgebra ℂ↪^{i} 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. □

In particular, i(ℂ) ≠ 0; this implies, by degeneration arguments, that i : C_{0} → L_{0} is an embedding in the “global” situation.

Now assume that C is compact, x_{1}, ⋯, x_{n} ∈ C, x_{i} ≠ x_{j}, ν_{i} are 1-jets of parameters at x_{i}’s, and X_{1}, ⋯, X_{n} ∈ A. Put U = C \ {x_{1}, ⋯, x_{n}}. Consider the pairing ⟨ ⟩_{C} : r(X_{1})_{x,ν_{1},C} ⊗ ⋯ ⊗ r(X_{n})_{x_{n},ν_{n},C} ⟶ ⟨X_{1} ⊗ ⋯ ⊗ X_{n}⟩_{C,x_{i},ν_{i}}. We have an obvious “localization” morphism L_{0}(U) → L_{0}(F_{x_{i}}), hence a natural action of L_{0}(U) on ⊗r(X_{i})_{x_{i},ν_{i},C}.

6.4.7 Lemma. The morphism ⟨ ⟩_{C} is L_{0}(U)-invariant.

Proof. Stokes formula: we rewrite for ℓ ∈ L_{0}(U) = Ω^{1} ⊗ r(⊮)_{U} the sum Σ⟨φ_{1} ⋯ ℓ(φ_{i}) ⋯ φ_{n}⟩ as Σ Res_{x=x_{i}}⟨ℓ(x)φ(x_{1}) ⋯ φ(x_{n})⟩. □

6.5 Stress-energy tensor. TO BE REWRITTEN! POSSIBLE MISTAKES!

For any local field F consider the linear map T_{F −2}/T_{F −1} → r(⊮)_{F} /ℂ · 1, τ ⟼ τ(1) (see 3.4.1; recall that 1 is fixed by T_{F −1} by axiom c). The one-dimensional space T_{F −2}/T_{F −1} canonically coincides with the fiber at 0 of T^{⊗2}. Tensoring this map with the dual line, we get for any curve C a canonical section T of ω_{C}^{⊗2} ⊗ O_{C}(r(⊮)_{C}/O_{C}). This section is called the stress-energy tensor. Multiplication by T defines a canonical map T_{C} → ω_{C} ⊗ O_{C}(r(⊮)_{C}/O_{C}) = L_{1}/C_{1} ⟶^{d}L_{0}/C_{0} (see 6.4.3).

6.5.1 Lemma. (i) The composition T → L_{0}/C_{0} is a morphism of Lie algebras. (ii) The corresponding “local” projective action (see 6.4.5, 6.4.6) of T_{F} ⊂ L_{0F} /ℂ on r(X)_{F} coincides with the canonical Virasoro action.

Remark. One should have a canonical isomorphism between the induced extension of T by C_{0} = H and the Virasoro extension from §2, but we do not know how to establish it at a moment.

Proof. Let us sketch a proof of (ii); one proves (i) in a similar way. We may assume that F = ℂ((t)). Let us compute the action of the operator L_{K} := t^{K+1}∂_{t} · T ⊂ L_{ℂ((t))^{o}}/ℂ on r(X)_{ℂ((t))}. Take e ∈ r(X)_{ℂ((t))}, e^{∗} ∈ r(∗X)_{ℂ((t^{−1}))}. Consider the function ν(z) = ⟨\frac{1}{t−z} ∂_{t−z}(1_{z}) · e · e^{∗}⟩_{ℙ^{1}}; here z ∈ ℙ^{1} \ \{0, ∞\}, ⟨ ⟩_{ℙ^{1}} is the correlator for fields \frac{1}{t−z} ∂_{t−z}(1_{z}) ∈ r(⊮)_{ℂ((t−z)),t−z}, e, e^{∗} at points z, 0, ∞. By definition, the matrix coefficient ⟨L_{K}(e), e^{∗}⟩ is equal to Res_{z=0}z^{K+1}ν(z)dz. We have the invariance property ⟨\frac{1}{t−z} ∂_{t−z}(1_{z})·e·e^{∗}⟩+⟨(1_{z})· \frac{1}{t−z} ∂_{t}e·e^{∗}⟩+⟨(1_{z})·e· \frac{1}{t−z} ∂_{t}e^{∗}⟩ = 0. Deleting 1_{z} by ax · e, we get ⟨L_{K}(e), e^{∗}⟩ = −Res_{z=0}(⟨\frac{1}{t−z} ∂_{t}e · e^{∗}⟩ + ⟨e · \frac{1}{t−z} ∂_{t}e^{∗}⟩) · Z^{K+1}dz. To compute \frac{1}{t−z} ∂_{t}e one should expand \frac{1}{t−z} around t = 0, and to compute \frac{1}{t−z} ∂_{t}e^{∗} one should expand \frac{1}{t−z} at t = ∞.

Hence

⟨L_{K}e, e^{∗}⟩ = −Res_{z=0}z^{K+1}(−⟨∑_{n≥0} z^{−n−1}t^{n}∂_{t}e, e^{∗}⟩+⟨e, ∑_{n≥0} z^{n}t^{−n−1}∂_{t}e^{∗}⟩)dz = ⟨t^{K+1}∂_{t}e, e^{∗}⟩,

since ⟨t^{a}∂_{t}e, e^{∗}⟩ + ⟨e, t^{a}∂_{t}e^{∗}⟩ = 0. We see that L_{K} = t^{K+1}∂_{t}, q.e.d. □

6.6 Theta functions. Consider the vector spaces ⟨⊮⟩_{C}, where C is a smooth connected compact curve (with empty set of distinguished points). They are fibers of a lisse λ^{c}-twisted D-module ⟨⊮⟩ on the moduli space of smooth curves. For a point x ∈ C we have ⟨⊮⟩_{C} = ⟨⊮_{x}⟩_{C,x}, hence one has a canonical map γ_{x} : r(⊮)_{x,C} → ⟨⊮⟩_{C}. The image γ_{C} = γ_{x}(\frac{1}{x}) is independent of the choice of x (since ∂_{x}(γ_{x}(1_{x})) = 0). As C varies, the γ_{C} form a holomorphic section of ⟨⊮⟩.

Here is an explicit formula for γ 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(2πiz) is the standard parameter at infinity. The family of elliptic curves degenerates when q → 0 in the standard way described in 3.6.6. Hence on H we get a canonical trivialization ⟨⊮⟩_{H} = ⊕ℂ_{I_{j}}, horizontal with respect to the trivialization of λ^{c} described in 3.6.6. In this trivialization we have γ(q) = ∑ γ_{I_{j}}^{∨}(q), where γ_{I_{j}}^{∨}(q) = tr_{I_{jℂ((t))}}q^{−L_{0}} by axiom g. The “global” trivialization of λ^{c} given by η(q)^{c} differs from the above trivialization by q^{c/24} (see 3.6.6). In this global η-trivialization the components of γ are γ_{I_{j}}(q) = q^{c/24}tr_{I_{jℂ((t))}}q^{−L_{0}}. We

see that these are holomorphic functions on H and for any ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) the

function γ_{I_{j}} (\frac{az+b}{cz+d}) is a linear combination with constant coefficients of other γ_{I_{i}} ’s.

§7. Lisse representations

7.1 Singular support, lisse modules. Let 𝔤 be a Lie algebra, and U = U(𝔤) its universal enveloping algebra. Then U is a filtered algebra (U_{0} = ℂ, U_{1} = ℂ + 𝔤, U_{i} = U_{1}^{i} for i > 0), grU = ⊕_{i}U_{i}/U_{i−1} = S^{•}(𝔤). For φ ∈ U_{i} its symbol σ_{i}(φ) is φ mod U_{i−1} ∈ S^{i}𝔤; if φ ∈ U_{i} \ U_{i−1} we will write σ(φ) = σ_{i}(φ).

7.1.1 Let M be a finitely generated 𝔤-module. Recall that a good filtration M_{•} on M is a U_{•}-filtration such that M = ∪M_{i}, ∩M_{i} = 0 and grM_{•} is a finitely generated S^{•}(𝔤)-module. For example, if M_{0} ⊂ M is a finite dimensional vector subspace that generates M, then M_{i} = U_{i}M_{0} is a good filtration. Any two good filtrations M_{•}, M′_{•} on M are comparable, i.e., for some a one has M_{•−a} ⊂ M′_{•} ⊂ M_{•+a}.

Define the singular support SSM of M to be the support of the S^{•}(𝔤)-module grM_{•}, where M_{•} is a good filtration on M. This is a Zariski closed canonical subset of SpecS^{•}(𝔤) = 𝔤^{∗}; it does not depend on the choice of a good filtration M_{•}. If η is a generic point of SSM, then the length of the S^{•}(𝔤)-module (grM_{•})_{η} only depends on M; denote it ℓ_{η}(M). We will say that M is finite at η if ℓ_{η}(M) < ∞: this means that (grM_{•})_{η} is killed by an ideal of finite codimension in S^{•}(𝔤)_{η}.

7.1.2 Remarks. (i) If M is generated by a single vector, M ≃ U/I, then SS(M) is the zero set of symbols of elements of I. (ii) A more precise way to speak about this subject needs the microlocalization language, see e.g. [La], Appendix.

The algebra grU = S^{•}(𝔤) carries a Poisson bracket defined by the formula {f_{i}, g_{j}} = f̃_{i}g̃_{j} − g̃_{j}f̃_{i} mod U_{i+j−2}; here f_{i} ∈ S^{i}(𝔤), f̃_{i} ∈ U_{i}, f_{i} = f̃_{i} mod U_{i−1}, and the same for g_{j}, {f_{i}, g_{j}} ∈ S^{i+j−1}(𝔤). One has the following integrability theorem, due to O. Gabber [Ga]:

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 ∈ S^{•}(𝔤) vanish on SSM, then so does {f, g}. □

7.1.4 Definition. A finitely generated module M is lisse if SSM = {0}. More generally, we will say that M is lisse along a vector subspace ℓ ⊂ 𝔤 if SSM ∩ ℓ^{⊥} = {0}. □

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 𝔤 < ∞.

Explicitly, a module M is lisse if amd only if for a finite dimensional subspace V ⊂ M that generates M and any g ∈ 𝔤 there exists N ≫ 0 such that g^{N}V ⊂ U_{N−1}V.

7.2 Finiteness property. Let k ⊂ 𝔤 be a Lie subalgebra. We will say that a 𝔤module M is a (𝔤, k)-module if k acts on M in a locally finite way (i.e., for any x ∈ M one has dimU(k)x < ∞). 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} ⊂ M that generates M and put M_{i} = U_{i}M_{0}). Hence SSM ⊂ k^{⊥} = (𝔤/k)^{∗} ⊂ 𝔤^{∗}.

7.2.1 Lemma. Let M be a finitely generated (𝔤, k)-module and n ⊂ 𝔤 be a vector subspace such that dim𝔤/n+k < ∞ and M is lisse along n. Then dimM/nM < ∞.

Proof. Let M_{•} be a K-invariant good filtration on M, so grM_{•} is a finitely generated S^{•}(𝔤/k)-module. Consider the induced filtration on M/nM. It suffices to see that dimgr(M/nM) < ∞. But gr(M/nM) is a quotient of grM/ngrM (since gr_{i}M/nM = M_{i}/M_{i−1} + (M_{i} ∩ NM), (grM/ngrM)_{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^{•}(𝔤/k + n), hence it is finitely generated. □

We will use 7.3.1 as follows. Assume we are in a situation 3.3, so we have a Harish-Chandra pair (𝔤, K), an S-localization data ψ = (S^{#}, N, φ, φ_{0}) for (𝔤, K) and the corresponding S-localization functor Δ_{ψ} : (𝔤̃, K)-mod → D_{ψ}-mod. Certainly, any (𝔤̃, K)-module M is a (𝔤̃, k)-module and SSM is an Ad K-invariant closed subset of k^{⊥}. Now 7.2.1 (together with 3.3.4) implies:

7.2.2 Corollary. Assume that the following finiteness condition holds: (*) The sheaf 𝔤_{S}^{#}/k_{S}^{#} + φ(N_{(0)}) is O_{S}-coherent.

Then for a lisse (𝔤̃, K)-module M the D_{ψ}-module Δ_{ψ}(M) is lisse (see 3.2.7). More generally, if a (𝔤̃, K)-module M is lisse along any subspace φ_{0}(N_{(0)s}) ⊂ 𝔤̃, s ∈ S^{#}, then Δ_{ψ}(M) is a lisse D_{ψ}-module. □

The following corollaries of 7.1.3 will be useful.

7.2.3 Lemma. Let M be a (𝔤, k)-module such that SSM has finite codimension in k^{⊥}. Then SSM is involutive. □

7.2.4 Corollary. Assume that a Harish-Chandra pair (𝔤, K) has the property that any Zariski closed Ad K-invariant subset of k^{⊥} is either {0} or has finite codimension. Then for any (𝔤, K)-module M the SS(M) is involutive. □

7.3 Lisse modules over Virasoro algebra. Consider the Virasoro algebra T̃_{c}: this is the central ℂ-extension of Lie algebra T = ℂ((t)) that corresponds to the 2 cocycle ⟨f∂_{t}, g∂_{t}⟩_{c} = cRes(f′′′g \frac{dt}{t}). It carries the filtration T̃_{cn} : for n ≥ 1, T̃_{cn} = t^{n+1}ℂ[[t]]∂_{t}, for n ≤ 0, T̃_{cn} = ℂ + t^{n+1}ℂ[[t]]∂_{t}. Put L_{i} := t^{i+1}∂_{t} ∈ T̃_{c}. One also has the following Lie subalgebras of T̃_{c}:

n_{+} = T̃_{c1} ⊂ b_{+} = ℂ[[t]]t∂_{t} ⊂ P_{+} = ℂ[[t]]∂_{t}, n_{−} = ℂ[t^{−1}]∂_{t} ⊂ b_{−} = ℂ[t^{−1}]t∂_{t},

so b_{+} = LieK, n_{+} = Lie K_{1} (see 3.4.1). One has b_{+}⊕n_{−}⊕ℂ = T̃_{c}, b_{+}∩b_{−}−f = ℂL_{0}. 7.3.1 A higher weight T -module of central charge c is a (T̃_{c}, b_{+})-module M such that 1 ∈ ℂ ⊂ T̃_{c} acts as id_{M} and any m ∈ M is killed by some T̃_{cn} for n ≫ 0. Denote by T_{c+}-mod the category of such modules. Note that any M ∈ T_{c+}-mod is a (T̃_{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 φ on M that are finite with respect to the action of ^{t}L_{0}. The operators L_{i} := ^{t}L_{−i} define the T̃_{c}-action on ∗M. Clearly ∗M is a higher weight module called the (contravariant) dual to M. One has an obvious morphism M → ∗ ∗ 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.

7.3.2 Remark. For M ∈ T_{c+}-mod consider the monodromy operator T = exp(2πiL_{0}). Clearly T commutes with the Virasoro action, i.e., T ∈ AutM. Hence one has a canonical direct sum decomposition M = _{⨁\overline{a}∈ℂ/ℤ}M\overline{a}, where M\overline{a} is the generalized exp(2πia)-eigenspace of M. Denote by T_{c+\overline{a}}-mod the subcategory of those M’s that M = M\overline{a}. Clearly T_{c+}-mod = _{∏a∈ℂ/ℤ}T_{c+\overline{a}}-mod.

7.3.3 Lemma. For any finitely generated M ∈ T_{c+}-mod there are exactly three possibilities for SSM: it is either equal to {0}, or to T̃_{c0}^{⊥} = (ℂ + b_{+})^{⊥}, or to T̃_{c−1}^{⊥} = (ℂ + P_{+})^{⊥}.

Proof. Clearly SSM ⊂ T̃_{c0}^{⊥}. It is Ad K-invariant (the Ad K_{1}-invariance is obvious; for any t ∈ ℂ the operator exp(tL_{0}) acts on M, hence SSM is also Ad exp(tL_{0})- invariant). It is easy to see that any Ad K-invariant Zariski closed subset of T̃_{c0}^{⊥} is either {0} or coincides with one of the vector spaces T̃_{c−n}^{⊥}, n ≥ 0. According to 7.2.4 this T̃_{c−n} is the Lie subalgebra of T̃_{c}; this implies 7.3.3. □

For a higher weight module M consider the subspace M^{𝔫_{+}} of singular vectors. Clearly M^{𝔫_{+}} ≠ 0 and it is L_{0}-invariant, so we have a decomposition M^{𝔫_{+}} = ⨁_{h∈ℂ}M_{(h)}^{𝔫_{+}} by generalized eigenspaces of L_{0}. We will say that a singular vector v

has generalized weight h if v ∈ M_{(h)}^{𝔫_{+}} (i.e., if (L_{0} − h)^{n}v = 0 for n ≫ 0), and that

v has weight h if L_{0}v = hv. As usual, the Verma module M_{ch} = M_{h} ∈ T̃_{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(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].

7.3.4 Proposition. Let M = M_{h} be a Verma module, N ⊂ M is a non-zero submodule. Then

(i) N is generated by ≤ 2 singular vectors, i.e., N is either a Verma submodule or a sum of two Verma submodules.

(ii) N is an intersection of ≤ 2 Verma submodules. (iii) M/N has finite length. (iv) The spaces M_{(h′)}^{𝔫_{+}} have dimension ≤ 1, therefore, by (i), the irreducible con-

stituents of M have multiplicity 1. □

7.3.5 Lemma. Let P ∈ T̃_{c+}-mod be a finitely generated module. Then

(i) P admits a filtration of finite length ℓ with successive quotients isomorphic to a quotient of a Verma module.

(ii) The maximal semisimple quotient of P has length ≤ ℓ. (iii) Any submodule of P is finitely generated.

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). □

7.3.6 Lemma. Let M = M_{h} be a Verma module, N ⊂ M be a non-zero submodule, L = M/N. One has

(i) SSM = T̃_{c0}^{⊥} = 𝔫_{−}^{∗} (ii) SSL is either {0} or equals to T̃_{c−1}^{⊥} (iii) If SSL = 0, then L is irreducible and N is generated by two singular vectors. (iv) If N is a proper Verma submodule, then the coinvariants L_{[𝔫_{−},𝔫_{−}]} are infinite

dimensional.

Proof. (i) is obvious. To prove (ii) take a non-zero φ ∈ U(𝔫_{−}) such that φv_{h} ∈ N. The symbol σ(φ) vanishes on SSL, hence SSL ≠ 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′} ⊂ M_{h} one has SSM_{h}/M_{h′} ≠ 0. By 7.2.1 this follows from (iv).

(iv) The commutant [𝔫_{−}, 𝔫_{−}] is Lie subalgebra of 𝔫_{−} with basis L_{−3}, L_{−4}, L_{−5}, . . .. The quotient 𝔫_{−}/[𝔫_{−}, 𝔫_{−}] is abelian Lie algebra with basis L_{−1}, L_{−2}. To prove (iv) note that M_{h[𝔫_{−},𝔫_{−}]} is a free module over U(𝔫_{−}/[𝔫_{−}, 𝔫_{−}]) = ℂ[L_{−1}, L_{−2}] with generator \overline{v}_{h}, and (M_{h}/M_{h′})_{[𝔫_{−},𝔫_{−}]} is a quotient of M_{h[𝔫_{−},𝔫_{−}]} modulo the ℂ[L_{−1}, L_{−2}] submodule generated by the image \overline{v}_{h′} of v_{h′} (since M_{h′} = U(𝔫_{−})v_{h′}). Since \overline{v}_{h′} = P \overline{v}_{h}, where P is a polynomial of weight h′ − h ≠ 0, we see that our coinvariants (M_{h}/M_{h′})_{[𝔫_{−},𝔫_{−}]} = ℂ[L_{−1}, L_{−2}]/P ℂ[L_{−1}, L_{−2}] are infinite dimensional. □

7.3.7 We will say that an irreducible module L_{h} ∈ T̃_{c+}-mod is minimal, or a Belavin-Polyakov-Zamolodchikov module, if the conditions (i), (ii) below hold:

(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) (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}).

7.3.8 Proposition. ([FF] ) An irreducible module L_{h} is minimal iff both the following conditions hold:

(i) L_{h} is dominant which means that L_{h} is not isomorphic to a subquotient of any M_{h′}, h′ ≠ h.

(ii) The kernel N_{h} of the projection M_{h} → L_{h} is generated by exactly 2 singular vectors (see 7.3.4 (i)). □

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}^{∧} = lim_{←}M_{h}^{(n)} is a projective covering of L_{h} in

the category T̃_{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)^{n}v = 0.

7.3.10 Proposition. For an irreducible module L = L_{h} = M_{h}/N_{h} the following conditions are equivalent:

(i) L is lisse (ii) L is minimal (iii) The coinvariants L_{[𝔫_{−},𝔫_{−}]} are finite-dimensional (iv) The invariants L^{[𝔫_{−},𝔫_{−}]} are finite dimensional

(v) For some non-zero φ ∈ U([𝔫_{−}, 𝔫_{−}]) one has φv_{h} ∈ N_{h}

Proof. One has (i) ⟹ (iii) by 7.2.1, (iii) ⟺ (iv) by contravariant duality, (ii) ⟺ (iii) by [FF], (v) ⟹ (i) by 7.3.5 (ii) (since σ(φ) vanishes on SSL, one has SSL ≠ T̃_{c−1}^{⊥}. It remains to show that (ii) ⟹ (v). So let L_{h} be minimal. Put T = U(𝔫_{−}, 𝔫_{−}])v_{h} ⊂ M_{h}. We wish to see that the projection T → 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(2πtL_{0})) is asymptotically equivalent as t → 0 to πα/12t for some constant α < 1. On the other hand, one has log tr_{T} (exp(−2πtL_{0})) = log tr_{M_{h}}(exp(2πtL_{0}))+log(1−exp(−2πt))+log(1−exp(−4πt)) (since as L_{0}-module M_{h} is isomorphic to v_{h} ⊗ S(L_{−1}, L_{−2}, ⋯), where the generators L_{−i} of the symmetric algebra have weights i, and T is isomorphic to v_{h} ⊗ S(L_{−3}, L_{−n}, ⋯)). This function is asymptotically equivalent to π/12t. Since the spectrum of L_{0} is real, this implies that T → L_{h} is not injective. □

7.3.11 Remark. For c = c_{p,q}, h = h_{11} = 0 one may prove that (ii) ⟹ (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 φ ∈ U(𝔫_{−})_{i} one has φv_{0} ∈ N_{0} \ N′. Then the symbol of φ 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. □

7.3.12 Proposition. The following conditions on a higher weight module M are equivalent

(i) M is a finitely generated lisse module (ii) M is isomorphic to a finite direct sum of minimal irreducible modules. (iii) One has dim M^{[𝔫_{−},𝔫_{−}]} < ∞

Proof. By 7.3.10 we know that (i) ⟸ (ii) ⟹ (iii). We will use the following facts: (*) Let L_{h} be a minimal irreducible module. Then any quotient of length 2 of M_{h}^{(n)}

(see 7.3.9 (ii)) is actually a quotient of M_{h} = M_{h}^{(1)} (i.e., is L_{0}-diagonalizable). (**) If L_{h_{1}}, L_{h_{2}} are minimal and h_{1} ≠ h_{2}, then M_{h_{1}} and M_{h_{2}} have no common

irreducible component.

Here (*) follows from the fact that N_{h} ⊂ 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

(***) Ext^{1}(L_{h_{1}}, L_{h_{2}}) = 0 for any minimal L_{h_{1}}, L_{h_{2}}.

Now we may prove that (i) ⟹ (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 → Q → M/T → P → 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) ≠ 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.

Let us prove that (iii) ⟹ (ii). Let M be a module such that dim M^{[𝔫_{+},𝔫_{+}]} = r < ∞. Let M′ ⊂ 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 ≤ r, so it suffices to show thatM′ = M. Note that any non-zero submodule N ⊂ M intersects M′ non-trivially (if N ∩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^{𝔫_{+}} = ∞; since N^{𝔫_{+}} ⊂ M^{[𝔫_{+},𝔫_{+}]} this is not true). Assume that M/M′ ≠ 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 → ∗(M/M′) → ∗M → ∗M′ → 0. One has ∗M′ = ⊕L_{h_{i}}, hence, by 7.3.8, 7.3.9 (ii) the projection ⊕M_{h_{i}} → ⊕L_{h_{i}} = ∗M′ lifts to the map ⊕M_{h_{i}}^{(2)} → ∗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) ⟹ (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)_{[𝔫_{−},𝔫_{−}]} are of infinite dimension. Since (∗M)_{[𝔫_{−},𝔫_{−}]} = (M^{[𝔫_{−},𝔫_{−}]})^{∗}, we are done. □

7.3.13 Now for n ≥ 1 consider the product of Virasoro algebras T̃_{c}^{n}: this is a central ℂ-extension of T^{n} with cocycle ⟨(f_{i}∂_{t}), (g_{i}∂_{t})⟩_{c} = ∑_{i}⟨f_{i}∂_{t}, g_{i}∂_{t}⟩_{c} (see 3.4.1). The

above theory extends to T̃_{c}^{n} in an easy manner. Namely, we have a standard subalgebra 𝔫_{+} = ∏ 𝔫_{+i} ⊂ 𝔟_{+} = ∏ 𝔟_{+i} ⊂ 𝔭_{+} = ∏ 𝔭_{+i}, 𝔫_{−i} ⊂ 𝔟_{−} = ∏ 𝔟_{−i}, 𝔣 = 𝔟_{+}∩𝔟_{−} = ℂ^{n} etc. of T̃_{c}^{n}. One defines the corresponding category T_{c+}^{n}-mod of higher weight modules in an obvious manner. We have an obvious functor ⊗ : ∏ T_{c+}-mod → T_{c+}^{n}-mod, (M_{1}, …, M_{n}) ⟼ M_{1} ⊗ … ⊗ M_{n}. Clearly SSM_{1} ⊗ … ⊗ M_{n} = SSM_{1} × SSM_{2} × ⋯ × SSM_{n}.

For ℏ = (h_{i}) ∈ ℂ^{n} we have the corresponding Verma module M_{ℏ} = ⊗M_{h_{i}} and its unique irreducible quotient L_{ℏ} = ⊗L_{h_{i}}; any irreducible higher weight module is isomorphic to a unique L_{ℏ}. It follows from 7.3.4 (iv) that any submodule N ⊂ M_{ℏ} is tensor product ⊗N_{i} of submodules N_{i} ⊂ 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 T_{c+}^{n}-mod. The version of 7.3.6 for T̃_{c}^{n} case (with obvious modifications) follows immediately from the case n = 1. A module L_{ℏ} = ⊗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.

§8. Minimal models

These were defined by Belavin, Polyakov and Zamolodchikov [BPZ]. Let us start with a general representation-theoretic construction. 8.1 Fusion functors for Virasoro algebra. Let C be a compact smooth curve, A, B ⊂ C be two finite sets of points such that A ∩ B = ∅, A ≠ ∅. For a central charge c ∈ ℂ we have Virasoro algebra T̃_{c}^{A} which is central ℂ-extension of T^{A} = ∏_{a∈A}T_{a} (where T_{a} = vector fields on punctured formal disc at a) and similar

algebras T̃_{c}^{B}, T̃_{c}^{A⊔B}. One has a canonical surjective map T̃_{c}^{A} ×T̃_{c}^{B} → T̃_{c}^{A⊔B} (which is factorization by {(a, −a)} ⊂ ℂ × ℂ); the morphisms T̃_{c}^{A} → T̃_{c}^{A⊔B} ⟵ T̃_{c}^{B} are injective. One also has the canonical embedding i_{A⊔B} : T (U) ⟶ T̃_{c}^{A⊔B}, where U = C \ (A ⊔ B), and the ones i_{A} : T (C \ A) → T̃_{c}^{A}, i_{B} : T (C \ B) → T̃_{c}^{B}. There is also a canonical morphism j_{B} : T (C \ A) → T̃_{c}^{B} which is composition of the obvious embedding T (C \ A) → T_{−1}^{B} and the section s_{O_{B}} : T_{−1}^{B} → T̃_{c}^{B}. The restriction i_{A⊔B}|_{T (C\A)} : T (C \ A) ⟶ T̃_{c}^{A⊔B} coincides with i_{A} + j_{B}.

8.1.1 Assume we have a positive divisor d = ∑ n_{b}b ≥ 0 supported on B. Let T (C \ A, d) ⊂ T (C \ A, d) be the Lie subalgebra of vector fields vanishing of order ≥ n_{b} + 1 at any b ∈ B. Clearly one has T (C \ A, d_{1}) ⊂ T (C \ A, d_{2}) for d_{1} ≥ d_{2}, and T (C \ A, 0)/T (C \ A, d) = T_{0}^{B}/T_{d}^{B}, where T_{d}^{B} = ∏ T_{n_{b},b}. Let ϵ_{d} : T̃_{c}^{B} ⟶ T̃_{c}^{A}/i_{A}(T̃(C \ A, d)) be the composition

T̃_{c}^{B} ⟶ T̃_{c}^{B}/s_{O_{B}}(T_{B,d}) ⟶ T̃_{c}^{A⊔B}/i_{A⊔B}(T (U)) + s_{O_{B}}(T_{B,d})∼^{←}T̃_{c}^{A}/i_{A}(T (C \ A, d)).

The maps t_{d} are compatible, so we have ϵ = lim_{\overleftarrow{d}}ϵ_{d} : T̃_{c}^{B} ⟶ lim_{\overleftarrow{d}}T^{A}/i_{A}(T (C \A, d)).

8.1.2 Now we are able to define the (contravariant) fusion functor F_{C} : T̃_{c}^{A}−mod → T̃_{c}^{B} − mod.

Let M be any T̃_{c}^{A}-module (so 1 ∈ ℂ ⊂ T̃_{c}^{A} acts as id_{M}). Put F_{C}(M) := ⋃_{d} M^{∗}i_{A}(T (C \ A, d)) ⊂ M^{∗}; therefore an element of F_{C}(M) is a linear functional

on M invariant with respect to some i_{A}(T (C \ A, d)). For τ ∈ T̃_{c}^{B}, ℓ ∈ F_{C}(M) put τ(ℓ) = ^{t}ϵ(τ)(ℓ). It is easy to see that this formula is correct, τ(ℓ) lies in F_{C}(M) ⊂ M^{∗} and (τ, ℓ) ⟼ τ(ℓ) is T̃_{c}^{B}-action on F_{C}(M). This way F_{C}(M) becomes T̃_{c}^{B}-module. One has an easy 8.1.3 Lemma.

(i) One has F_{C}(M) = ⋃_{α}F_{C}(M)^{T_{B,d}}, and F_{C}(M)^{T_{B,d}} = (M_{T (C\A,d)})^{∗}.

(ii) Let N be any T̃_{c}^{B}-module s.t. N = ⋃_{α}N^{T_{B,d}}. Then Hom(N, F_{C}M) =

[(M ⊗ N)_{T (U)}]^{∗} (here we consider M ⊗ N as T̃_{c}^{A⊔B}-module via the surjection T̃_{c}^{A} × T̃_{c}^{B} ⟶ T̃_{c}^{A⊔B}). □

¿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 T̃_{c}^{A}-module.

8.1.4 Corollary. (i) F_{C}(M) is finitely generated lisse higher weight T̃_{c}^{B}-module.

(ii) For any finitely generated higher weight T̃_{c}^{B}-module N one has (M⊗N)_{T (U)} = (M ⊗ \overline{N})_{T (U)}, where \overline{N} is the maximal lisse quotient of N.

Proof. (i) Use 8.1.3 (i), 7.2.1, 7.3.12 (inversion 7.3.13). (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 ⊗ N)_{T (U)}^{∗} = Hom(N, F_{C}(M)) = Hom(\overline{N}, F_{C}(M)) = (M ⊗ \overline{N})_{T (U)}^{∗}, q.e.d. □

For h = (h_{b}) ∈ ℂ^{B} let L_{h}^{B} = ⨂_{b∈B}L_{c,h_{b}} be the irreducible T̃_{c}^{B}-module of higher

weight h.

8.1.5 Corollary. One has a canonical isomorphism M_{T (C\A)} = (M ⊗ L_{0}^{B})_{T (U)}.

Proof. Clearly M_{T (C\A)} = (Ind_{T (C\A)}^{T (U)}M)_{T (U)}. But Ind_{T (C\A)}^{T (U)}M coincides, as

T (U)-module, with T̃_{c}^{A⊔B}-module M ⊗ P_{o}^{B}, where P_{oB} = ⨂_{b∈B}P_{c,o,b}, P_{c,o} is a quo-

tient of Verma module M_{c,0} modulo relation L_{−1}v_{0} = 0. Clearly L_{o}^{B} is maximal lisse quotient of P_{o}^{B} (see 7.3.8), and 8.1.5 follows from 8.1.4 (ii). □

8.1.6 Corollary. Let d_{1} be the divisor ∑_{b∈B}b. Consider the action of Lie algebra

T (C \ A, 0)/T (C \ A, d_{1}) = T_{0}^{B}/T_{d_{1}}^{B} = ℂ^{B} on coinvariants M_{T (U,d_{1})}. This action is semisimple. For h = (h_{b}) ∈ ℂ^{B} the (h_{b})-component M^{(h_{b})} is equal to the coinvariants (M ⊗ L_{h}^{B})_{T (U)}. This space vanishes unless all h_{b} lie in the list 7.3.7 (ii).

Proof. Similar to 8.1.5; the semi-simplicity of ℂ^{B}-action follows from 7.3.12 (ii). □

8.1.7 Corollary. Assume that B consists of two points b_{1}, b_{2}. Let T (C \ A, B)′ ⊂ T (C \ A, 0) be the Lie subalgebra of vector fields that project to {(a, −a)} ⊂ ℂ^{2} via the projection to T (C \ A, 0)/T (C \ A, d_{1}) = ℂ^{2}. Then M_{T (C\A,B)′} = ⊕(M ⊗ L_{c,hb_{1}} ⊗ L_{c,hb_{2}})_{T (U)}, where L_{ch} runs the list 7.3.7 (ii) of irreducible lisse modules.

Proof. Similar to 8.1.6. □

8.2 Localization of lisse modules. Let π : C_{S} → S be a family of smooth projective curves, A ⊂ C_{S}(S) be a finite non-empty disjoint set of sections, ν_{a} are 1-jets of parameters at a ∈ A. By 3.4.3-3.4.7 these define the S-localization data for (T̃_{c}^{A}, v_{1}). Consider the corresponding S-localization functor Δ_{ψ_{c}} : (T̃_{c}^{A}, v_{1})_{c}-mod → D_{λ^{c}}-modules on S. Assume as above that M is a lisse (T̃_{c}^{A}, v_{1})_{c}-module.

8.2.1 Lemma. The D_{λ^{c}}-module Δ_{ψ_{0}}(M) is lisse with regular singularities at infinity.

Proof. Lissing follows from 7.2.2; the statement on regular singularities follows from 8.2.5 below. □

8.2.2 Assume now that S = Specℂ[[q]], π : C_{S} → S be a projective family of curves such that the generic fiber C_{η} is smooth and the closed fiber C_{0} has the only singular point b which is quadratic, A ⊂ C_{S}(S) be a finite non-empty disjoint set of sections, and {ν_{a}} be a 1-jet of coordinates at a ∈ A.

This collection defines an S-localization data “with logarithmic singularities at q = 0” for (T̃_{c}^{A}, v_{1}). (The definition of “S-loc. data ψ 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 σ : N → T_{S} has image equal to T_{S}^{0} = qT_{S} = ℂ[[q]]q∂_{q}. As in 3.3 such data defines an O_{S}-extension A_{ψ_{c}}^{0} of T_{S}^{0} and the corresponding associative algebra D_{ψ_{c}}^{0} which is isomorphic to the subalgebra of D_{ℂ[[q]]} generated by ℂ[[q]] = O_{S} and q∂_{q}. We have the corresponding S-localization functor Δ_{C_{S}} : (T̃_{c}^{A}, v_{1})-mod → D_{ψ_{c}}^{0}-mod. The definition of this ψ repeats word-for-word 3.4.3-3.4.7: we get the loc. data with logarithmic singularities just because T_{S}^{0} consists precisely of those vector fields that could be lifted to C_{S} \ A(S). Note that the “vertical” part N_{(0)} = ker σ ⊂ N is a free O_{S}-module and N_{(0)}/qN_{(0)} coincides with the Lie algebra T (C_{0}^{∨} \ A, B)′, where C_{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_{ψ_{C}}^{0} coincides with the algebra D_{λ_{C_{S}}^{c}}^{0} of differential operators on the determinant bundle λ_{C_{S}}^{c} generated by “q∂_{q}” and O_{S}.

Now let t_{1}, t_{2} be formal coordinates at b such that q = t_{1}t_{2}. Let C_{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} ∈ C_{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, ν_{A}, define ℂ((q))-localization data for (T̃_{c}^{A⊔B}, v_{1}). The corresponding algebra coincides with D_{λ_{C^{∨}_{η}}^{c}}, so we have the local-

ization functor Δ_{C_{η}^{∨}} : (T̃_{c}^{A⊔B}, v_{1})-mod → D_{λ_{C^{∨}_{η}}^{c}} -mod.

8.2.3 Let H be a lisse D_{λ_{C^{∨}_{η}}^{c}} -module, i.e. a finite dimensional ℂ((t))-vector space

with D-action. An h-lattice H_{h} ⊂ H, where h ∈ ℂ, is a ℂ[[t]]-lattice in H invariant with respect to the action of D_{λ_{C^{∨}_{S}}^{c}}^{0} and such that the operator q∂_{q} ∈ D_{λ_{C^{∨}_{S}}^{c}}^{0} /q acts

on H_{h}/qH_{h} as multiplication by h. Certainly, such H_{h} exists iff H has regular singularities at 0 with monodromy equal to h mod ℤ; if H_{h} exists, it is unique, so we’ll call it “the” h-lattice.

From now on let M be a lisse T̃_{c}^{A}-module.

8.2.4 Lemma. For any h ∈ ℂ, Δ_{C_{η}^{∨}}(M ⊗L_{hb_{1}} ⊗L_{hb_{2}}) is a lisse module that admits the h-lattice Δ_{C_{η}^{∨}}(M ⊗ L_{h} ⊗ L_{h})_{h}.

Proof. “lisse” follows from 8.1.4 (ii), 7.2.1. The existence of h-lattice follows easily from 3.4.7.1. □

According to 3.6.3 we have a canonical isomorphism D_{λ_{C_{S}}^{c}} = D_{λ_{C^{∨}_{S}}^{c}}. Denote this

algebra D_{λ^{c}}. So, by 8.2.4, we have for any h ∈ ℂ a D_{λ^{c}}^{0}-module D_{λ_{C^{∨}_{η}}^{c}} (M ⊗ L_{n} ⊗

L_{h})_{h}, which is zero if L_{h} is not lisse (i.e. if h ≠ h_{nm} from 7.3.7 (ii)) by 8.1.4 (ii). On the other hand, we have the D_{λ^{c}}^{0}-module Δ_{C_{S}}(M).

8.2.5 Proposition. There is a canonical isomorphism of D_{λ^{c}}^{0}-modules

Δ_{C_{S}}(M) = ⨁_{h} Δ_{C_{η}^{∨}}(M ⊗ L_{h} ⊗ L_{h})_{h}.

Proof. First, note that Δ_{C_{S}}(M) is a coherent O_{S}-module by a version of 7.2.2 “with logarithmic singularities”. Namely, Δ_{C_{S}}(M) is a coherent D_{λ^{c}}^{0}-module, and its singular support ⊂ Spec(grD_{λ^{c}}^{0}) is 0 section since M is lisse; hence Δ_{C_{S}}(M) is O_{S}-coherent.

Let e_{i} be a basis of L_{hℂ((t))} that consists of L_{0}-eigenvectors, so L_{0}e_{i} = (h − n_{i})e_{i} for n_{i} ∈ ℤ ≥ 0; let e_{i}^{∗} be the dual basis in L_{hℂ((t))} = ∗L_{hℂ((t))}. It is easy to see that Δ_{C_{η}^{∨}}(M ⊗ L_{h} ⊗ L_{h})_{h} ⊂ Δ_{C_{η}^{∨}}(M ⊗ L_{h} ⊗ L_{h}) is O_{S}-submodule generated by images of elements q_{m}^{−n_{i}} ⊗ e_{i} ⊗ e_{j}^{∗}, where m ∈ M_{A,C_{S}}, e_{i} ∈ L_{h(ℂ((t_{1})),q^{−1}t)}, e_{j}^{∗} ∈ L_{h(ℂ((t_{2})),t_{2})} (see 6.1.4 for notations).

To prove 8.2.5 it suffices to construct a morphism of D_{λ^{c}}^{0}-modules Δ_{C_{S}}(M) → ⊕Δ_{C_{η}^{∨}}( )_{h} which induces isomorphism mod q (since both are coherent 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 ∈ M_{A,C_{S}} = M_{A,C_{S}^{∨}}

in Δ_{C_{S}}(M) to the image of ∑_{i}m ⊗ e_{i} ⊗ e_{i}^{∗} in Δ_{C_{η}^{∨}}(M ⊗ L_{h} ⊗ L_{h}). It is easy

to see that this formula defines a correctly defined morphism of D_{λ^{c}}^{0}-modules (cf. 6.1.5). It induces isomorphism modulo q by 8.1.7 (since Δ_{C_{S}}(M)/q = M_{N_{(0)}/qN_{(0)}} = M_{T (C_{0}^{∨}\A,B)′}, see 8.2.2). □

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 A = A_{p,q} is category of finitely generated lisse higher weight modules for Virasoro algebra T̃_{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 ∗ : A^{0} → A is contravariant duality (see 7.3.1).

The vacuum module ⊮ is L_{c,o}; the isomorphism ∗⊮ = ⊮ is canonical one (that identifies the vacuum vectors).

The Dehn automorphism d is equal to the monodromy automorphism T = exp2πiL_{o} from 7.3.2.

We will define a canonical fusion structure on A simultaneously with the structures 6.1 of algebraic field theory. Namely, our realization functor r : A → (T̃_{c}, v_{1})- mod is “identity” embedding. The vacuum vector 1 ∈ r(⊮) = L_{0} is v_{0}.

Let π : C_{S} → S, A ⊂ C_{S}(S), ν_{A}, be as in 6.1.2. Assume that A ≠ ∅. For any X ∈ A^{⊗A} the D_{λ^{c}}-module Δ_{ψ_{c}}(X) is lisse holonomic with regular singularities at ∞. We put ⟨X⟩_{C_{S}} = Δ_{ψ_{c}}(X) and γ from 6.1.2 (iv) is identity map.

Assume now that A = ∅. We should define a canonical lisse D_{λ^{c}}-module ⟨⊮⟩_{C_{S}}. Let us make the base change and consider π_{C} : C_{C} = C_{S} ×_{S} C_{S} → C_{S}: this is a family of curves with a canonical (diagonal) section a. Consider the D_{λ^{c}}-module ⟨⊮⟩_{C_{C}}; this is a lisse D_{λ^{c}}-module on C_{S} generated by the holomorphic section ⟨1⟩_{C_{C}}. Note that ⟨1⟩_{C_{C}} is horizontal along the fibers of π : C_{S} → S. Hence there exists a (unique) D_{λ^{c}}-module ⟨⊮⟩_{C_{S}} on S together with a holomorphic section ⟨1⟩_{C_{S}} such that ⟨⊮_{a}⟩_{C_{C}} = π^{∗}⟨⊮⟩_{C_{S}}, ⟨1⟩_{C_{C}} = π^{∗}⟨1⟩_{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 = ⊕L_{h} ⊗ L_{h}).

By the way, the covariant fusion functor F_{C}^{A,B} from 4.6 is ∗F_{C} for contravariant F_{C} from 8.1 (by 8.1.3 (iii)).