LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT

LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT DENNIS GAITSGORY 1. Statement of the problem Throughout the talk, by a chiral module we shall understand a chiral D-module, unless explicitly stated otherwise. 1.1. Local fusion) definition of tensor product. Let X be a smooth but not necessarily complete) curve. 1.1.1. Let A be a chiral algebra on X. Let N 1, N 2 and N be chiral A-modules on X. We shall assume that N 1, N 2 and N all belong to the heart of the natural t-structure. We wish to define the notion of chiral pairing 1.1) N 1, N 2 ) N. The most natural definition, and one which generalizes to the case of n-fold operations, as well to the derived case, is the following: Definition 1.1.2. The datum of chiral pairing 1.1) is that of a chiral A-module N on X X, equipped with identifications j N ) j N 1 N 2 ), as chiral A-modules on X X X and as chiral A-modules on X. X )! N ) N[ 1], Note that we do not and cannot) require that N belong to the heart of the t-structure. In fact, N can live in cohomological degrees 0 and 1. We shall denote the set of chiral pairings as above by Maps loc A {N 1, N 2 }, N). 1.1.3. The above definition can be tautologically rewritten in more elementary terms, which we will use in this talk. Namely, we claim that a datum of chiral pairing as above is equivalent to a map of D-modules on X X: 1.2) j j N 1 N 2 ) X ) dr, N), which satisfies the following condition with respect to the action of A: the three maps j j A N 1 N 2 ) X ) dr, N), sum up to zero, where in the latter formula X is the main diagonal, and j is the complement of the diagonal divisor in X 3. Date: November 30, 2011. 1

2 DENNIS GAITSGORY Given N as in Definition 1.1.2, one constructs the map 1.2) from the exact triangle X ) dr,! XN ) N j j N ). Vice versa, a map 1.2) can be viewed as a map of chiral A-modules on X X, and one recovers N as its cone shifted by [ 1]. 1.1.4. Suppose that for given N 1 and N 2, the functor is co-representable. A-modD-modX)) Sets N Maps loc A {N 1, N 2 }, N) In this case we shall denote the resulting object of A-modD-modX)) by N 1 loc N 2 ) mon. Here the subscript mon refers to the fact that the above object does not capture the full fusion tensor product of N 1 and N 2, but rather the coinvariants of the monodromy in the latter. Remark 1.1.5. There seems to be no reasonable way to intrinsically define the full fusion tensor product as an object of A-modD-modX)). The problem is the same as that of making sense of the functor of total nearby cycles for D-modules. We shall take up this issue in the subsequent talks. 1.2. Taking coinvariants. In this talk, by coinvariants we shall always mean the non-derived version. 1.2.1. Assume now that X is complete, and let X be a point different from 0. Let denote the corresponding maps. ι 0 : pt X, ι : pt X, j : X {0 } := X X In what follows we shall consider the following particular case of the set-up of Sect. 1.1.1: We shall denote N 2 =: L, and assume that both L and N are supported at a fixed point 0 X. Note that in either approach to fusion, the result only depends on the restriction N 1 X =: M, so we can talk about the set Maps loc A {M, L}, N) of chiral pairings M, L) N. 1.2.2. Let P be a chiral A-module supported at. For M, L, N and P as above we can consider as chiral A-modules on and respectively. P M L and P N { } X {0} X X X { } {0} X X,

LOCAL VS GLOBAL DEFINITION OF TENSOR PRODUCT 3 In particular, we can consider the corresponding D-modules of conformal blocks C A P, M, L) D-mod X 0) and C A P, N) D-mod{ } {0}). Lemma 1.2.3. For a chiral pairing M, L) N there exists a unique map of D-modules on { } X {0} X j C A P, M, L)) ι 0 ) dr, C A P, N)) which makes the following diagram of D-modules commute: j P M L) ι 0 ) dr, P N) j C A P, M, L)) ι 0 ) dr, C A P, N)). 1.2.4. Thus, to M, L, N, and P from some fixed subcategory P A-mod ) we can attach the following two sets: A) The set Maps loc A {M, L}, N). B) The set Maps glob,p A {M, L}, N) of natural transformations between the following two functors P D-modX): P j C A P, M, L)) and P ι 0 ) dr, C A P, N)). By Lemma 1.2.3, we have a canonically defined map 1.3) Maps loc A {M, L}, N) Maps glob,p {M, L}, N). The goal of this talk is to gives sufficient conditions on 0, ) X, A, M, L, N and the category P, which will guarantee that the map 1.3) is an isomorphism. 2. Assumptions on the algebra and modules 2.1. Chiral algebras with a Harish-Chandra datum. The first set of assumptions has to do with A itself. 2.1.1. Let X be a smooth curve. Let A be a chiral algebra on X. We shall assume that A is equipped with a Sugawara data, i.e., with a homomorphism of Lie-* algebras where Θ is some fixes central extension A φ ch : Θ X A, 0 ω X Θ X Θ X 0. By [Ga], formula 6.5), we can define an action of vector fields considered as a sheaf in Zariski topology on X) on A also considered as a sheaf in Zariski topopology) by the formula 2.1) ξ a = h id){φ ch ξ), a}) a ξ. 2.1.2. Assume now that X is acted on by a group G by automorphisms. In practice, we shall take X = P 1 with fixed points 0, ) X and G = G m. In particular, from formula 2.1) we obtain an action of g := LieG) on local sections of A via the map g ΓX, T X ) given by the action of G on X.

4 DENNIS GAITSGORY 2.1.3. We shall now impose the following structure on A. We shall assume that A, viewed as a D-module on X carries a weak G-equivariant structure, such that It is compatible with the chiral bracket; The resulting action of g on local) sections of A is given by 2.1). Note that if G is connected, and A lies in the heart of the t-structure, the second condition determines the weak G-equivariant structure uniquely. I.e., it is the question of integrability of the Lie algebra action to the group.) Note that when we consider A as a weakly G-equivariant D-module on X, the obstruction to strong equivariance see [Ga, Sect. 2.7.4]), which is an action of g by endomorphisms of the D-module structure on A is by construction given by the action 2.2) ξ g, a A) h id){φ ch ξ), a}). 2.1.4. The structure of weak G-equivariance on A endows the categories A-modQCohX)) and A-modD-modX)) with natural weak G-actions, so that the forgetful functor is weakly) G-equivariant. A-modD-modX)) A-modQCohX)) Now, it follows from [Ga, Proposition 2.7.8], that the datum of φ ch satisfying the conditions of Sect. 2.1.3 upgrades the above weak G-action on A-modD-modX)) to a strong one. 2.1.5. The structure of strong G-action on A-modD-modX)) is uniquely characterized by the following property. Let M be an object of A-modD-modX)) G,weak. In particular, the D-module underlying M has a natural structure of object in D-modX) G,weak. Then the obstruction to strong G-equivariance of M as a D-module, which is a map g End D-modX) M) is the sum of the obstruction to strong equivariance of M as an object of A-modD-modX)), which is a map g End A-modD-modX)) M) and the commuting) g-action given by 2.3) ξ g, m M) h id){φ ch ξ), m}). 2.2. The category of modules. From now on we specialize to the case X = P 1, and G = G m with the standard action. 2.2.1. Recall that of C is a category acted on strongly by G m, and λ k, there exists a full subcategory C Gm,λ -mon C. Namely, consider C Gm,weak. Obstruction to strong G m -equivariance see [Ga], Sect. 2.7.4) defines on C Gm,weak a structure of module category with respect to QCohA 1 ), and we set C Gm,λ -mon := C Gm,weak QCohA 1 ) {λ}, QCohA 1 ) where QCohA 1 ) {λ} QCohA 1 ) is the full subcategory of objects set-theoretically supported at {λ} A 1.

LOCAL VS GLOBAL DEFINITION OF TENSOR PRODUCT 5 Lemma 2.2.2. The composed functor is fully faithful. C Gm,λ -mon C Gm,weak C For λ = 0 we shall denote C Gm,0 -mon simply by C Gm,mon. It is easy to see that the essential image of C Gm,mon in C is generated by the essential image of the forgetful functor C Gm,strong C. 2.2.3. It is easy to see that for λ 1 λ 2 Z, the subcategories C Gm,λ1 -mon Gm,λ2 -mon and C of C coincide. Thus, we have a well-defined subcategory for λ k/z. C Gm,λ -mon C It is also easy to see that if λ 1 λ 2 / Z, the corresponding subcategories are mutually orthogonal. We let denote the full subcategory equal to C Gm,gen-mon C C Gm,λ -mon. λ k/z 2.2.4. We let P max,0 be the following full subcategory of the heart of A-mod{0}) Gm gen-mon. Note this category will not be closed under colimits.) Note that for every object N A-mod{0}) Gm gen-mon the action of the generator L 0 := t t of LieG m ) on the underlying vector space is locally finite. We shall say that N P max,0 if the generalized eigenspaces of L 0 are finite-dimensional, and if the intersection of its spectrum with every coset in k mod Z is bounded from above. 2.2.5. Similarly, we define P max, as the full abelian subcategory of the heart of Gm gen-mon A-mod{ }) by requiring that the generalized eigenspaces of L 0 on the underlying vector space be finitedimensional, and if the intersection of its spectrum with every coset in k mod Z is bounded from below. As in [Ts, Sect. 5], there exists a canonical anti-equivalence characterized by the following property: For N P max,0 and P P max,, we have: D : P max,0 P max,, Hom Pmax, P, DN)) = C A P, N). At the level of underlying vector spaces, for λ k, we have: λdn)) λ N), where λ ) denotes the generalized eigenspace corresponding to λ.

6 DENNIS GAITSGORY 2.2.6. Let now P 0 be an arbitrary abelian subcategory of P max,0. Let P P max, be the essential image of P 0 under D. We will prove: Theorem 2.2.7. The map 1.3) is an isomorphism for N P 0 and P = P. 3.1. Construction of the inverse map. 3. Proof of the Theorem 2.2.7 3.1.1. We start with a datum of natural transformation as in B) of Sect. 1.2.4. I.e., for given M, L and N as in the theorem, we have maps of D-modules functorial in P P. j C A P, M, L)) ι 0 ) dr, C A P, N)) Our first goal is to construct a map of D-modules 3.1) j M L) ι 0 ) dr, N). Fix λ k. We will construct the corresponding composed map 3.2) j M L) ι 0 ) dr, N) ι 0 ) dr, λ N) 3.1.2. Consider the functor F λ : P Vect that sends P to λ P. It is easy to see that the functor is pro-representable. Let i Q i λ denote the corresponding filtered) projective system of objects of P. In particular, for every P P, we have λp colim i Hom P Q i λ, P). It is easy to see that we can assume that all Q i λ are λ-monodromic. 3.1.3. By construction, the vector space underlying each Q i λ comes equipped with a canonical vector qλ i, and there vectors go to one another under the transition maps for i j. Q j λ Qi λ We can replace each Q i λ by its submodule generated by qi λ, and the new projective system will pro-represent the same functor F λ. This allows to assume that the transition maps Q j λ Qi λ are surjective. In particular, for each i, the corresponding map Hom P Q i λ, P) λ P is injective. The latter property, in turn, implies that we can assume that for every P P, the vector space Hom P Q i λ, P) is finite-dimensional since F λp) is), and that the system stabilizes to λ P. i Hom P Q i λ, P)

LOCAL VS GLOBAL DEFINITION OF TENSOR PRODUCT 7 3.1.4. For N A-mod{0}), we obtain a canonical map of vector spaces and a thus a map N qi λ id Q i λ N C A Q i λ, N), 3.3) N lim i C A Q i λ, N). The following assertion will play a crucial role: Proposition 3.1.5. The projective system i C A Q i λ, N) stabilizes, its eventual value is canonically isomorphic to λ N, so that the map 3.3) identifies with the projection onto the λ eigenspace. Proof. We rewrite C A Q i λ, N) Hom P Q i λ, DN)), and as was mentioned above, the assignment i Hom P Q i λ, DN)), stabilizes to λ DN)); in particular, it is finite-dimensional. Hence, the assignment stabilizes to as required. i C A Q i λ, N) λ DN))) λ N, 3.1.6. We are now able to construct the map 3.2). For each index i consider the composed map of D-modules j M L) q i λ id Q i λ j M L) j CA Q i λ, M, L)) ι 0 ) dr, CA Q i λ, N)). Taking i large so that N C A Q i λ, N) identifies with the projection N λ N, we obtain the desired map, so that the diagram j M L) ι 0 ) dr, λ N) q i λ id q i λ id Q i λ j M L) Q i λ ι 0) dr, N) j CA Q i λ, M, L)) ι 0 ) dr, CA Q i λ, N))

8 DENNIS GAITSGORY commutes. 3.1.7. Let us now show that the diagram of D-modules P j M L) P ι 0 ) dr, N) j C A P, M, L)) ι 0 ) dr, C A P, N)) commutes for any P P. By definition, it is enough to do so after precomposing it with the map j M L) P j M L) corresponding to a spanning set of elements in the vector space underlying P. For each λ k and a vector p λ P. Consider the corresponding diagram j M L) ι 0 ) dr, N) p id p id P j M L) P ι 0 ) dr, N) j C A P, M, L)) ι 0 ) dr, C A P, N)), in which the upper square is tautologically commutative. So, we need to establish the commutativity of the outer square. Note also that the right composed arrow factors through the projection of ι 0 ) dr, N) onto ι 0 ) dr, λ N). Choose an index i so that p is the image of qλ i under a map f : Q i λ P. The map j M L) ι 0 ) dr, λ N) p id P ι 0 ) dr, N) equals ι 0 ) dr, C A P, N)) j M L) ι 0 ) dr, λ N) q i λ id Q i λ ι 0) dr, N) ι 0 ) dr, CA Q i λ, N)) f id ι 0 ) dr, C A P, N)).

LOCAL VS GLOBAL DEFINITION OF TENSOR PRODUCT 9 The map j M L) p id P j M L) equals j C A P, M, L)) ι 0 ) dr, C A P, N)) j M L) q i λ id Q i λ j M L) j CA Q i λ, M, L)) f id id j C A P, M, L)) ι 0 ) dr, C A P, N)), which by the functoriality of the datum of Maps glob,p A {M, L}, N) equals j M L) q i λ id Q i λ j M L) j CA Q i λ, M, L)) ι 0 ) dr, CA Q i λ, N)) f id ι 0 ) dr, C A P, N)), while the latter, by the construction of the map 3.1), equals j M L) ι 0 ) dr, λ N) q i λ id Q i λ ι 0) dr, N) ι 0 ) dr, CA Q i λ, N)) f id as required. ι 0 ) dr, C A P, N)),

10 DENNIS GAITSGORY 3.2. Interlude. Before we prove that the map of D-modules 3.1), constructed above, verifies the axioms of chiral pairing, as a warm-up, let us give a description of a data of chiral action of a chiral algebra on a module in terms of coinvariants. 3.2.1. Let X be an arbitrary curve with two points 0, ) X, and let A be an arbitrary chiral algebra on X. Let N be a chiral A-module supported at X. In particular, we have a map of D-modules j A) N ι 0 ) dr, N). Let W be a vector space endowed with a pairing In particular, we obtain a map of D-modules ψ : N W k. 3.4) j A left ) N W ι 0 ) dr, k). We wish to describe the map 3.4) in global terms. 3.2.2. Let us recall the following general construction. Let M be a left D-module on X. Then the data of a map jm) ι 0 ) dr, k) is equivalent to that of a map of left D-modules over X M K 0, where K 0 is the completed local field at 0 X. 3.2.3. Let P be a chiral A-module supported at X, and suppose that we are given a functional η : CN, P) k. In particular, we have a pairing of vector spaces We shall describe the map ψ η : N P Ψ : CN, P) k. 3.5) A left N P K 0 corresponding to 3.4) for W = P. Remark 3.2.4. Note that for 0, ) X and A as in Sect. 2.1, and N P max,0, there is a canonical choice for P, namely, DN). In this case, the vector space underlying P is the restricted dual of that of N with respect to the grading by L 0. 3.2.5. Consider the canonical map of D-modules on X: 3.6) A left N P C A A left, N, P) O X C A N, P) η O X. Lemma 3.2.6. The Laurent expansion of the map 3.6) at 0 X equals the map 3.5). A left N P O X

LOCAL VS GLOBAL DEFINITION OF TENSOR PRODUCT 11 Proof. Follows from the commutative diagram j A left ) N P ι 0 ) dr, N P) j CA A left, N, P) ) ι 0 ) dr, C A N P)) id j O X ) C A N P)) ι 0 ) dr, C A N P)) ψ η ι0 ) dr, k) id η ι 0 ) dr, k) id η ι 0 ) dr, k) where the left bottom horizontal arrow is the canonical map, so that the map O X C A N P) K 0 corresponding to it is given by ψ η and the tautological map O X K 0. 3.3. Verifying the relations. We shall finally show that the map of D-modules 3.1) verifies the axioms of chiral pairing. In this subsection it will be essential that we work within an abelian category. 3.3.1. We will show that any map of D-modules 3.7) j M L) ι 0 ) dr, N) such that for every P P there exists a bottom arrow completing to a commutative diagram the map 3.7) is a chiral pairing. j P M L) ι 0 ) dr, P N) j C A P, M, L)) ι 0 ) dr, C A P, N)). j P M L) ι 0 ) dr, P N) j C A P, M, L)) ι 0 ) dr, C A P, N)), As we shall see, it is enough to take P from the collection Q i λ from the previous subsection. 3.3.2. Let B be the following Lie algebra in the category of D-modules on X X: where B := p 1,2 ) dr, j j p! 3A), X X X p 1,2 X X, and where j is the embedding of the open locus p 3 X x 1, x 2, x 3 ), x 3 x 1, x 2, 0,.

12 DENNIS GAITSGORY By construction, the Lie algebra B acts on both j M L) and ι 0 ) dr, N). It is easy to see that if a map j M L) ι 0 ) dr, N) is a chiral pairing, then it also respects the action of B. Proposition 3.3.3. For M, L, N lying in the heart of the t-structure, if a map j M L) ι 0 ) dr, N) respects the action of B, then it is a chiral pairing. Remark 3.3.4. Note that the property of a map j M L) ι 0 ) dr, N) is local around 0 X, so it does not see the point. However, the assertion of Proposition 3.3.3 would have been false, had we not removed the divisor x 3 = from X X X. The latter is done in order to make the map p 1,2 affine. Proof. The assertion follows from the next general observation. Let π : Y 1 Y 2 be a map with affine fibers, and let i : Z Y 1 be a closed subset such that π i : Z Y 2 is finite. Let f : M i dr, N) be a map of D-modules. Assume that the resulting map is zero. p dr, M) p i) dr, N) Then, if M and N both lie in the heart of the t-structure, then f = 0. 3.3.5. Thus, we need to prove the commutativity of the square 3.8) B j M L) B ι 0 ) dr, N) j M L) ι 0 ) dr, N). For λ k and an index i, consider the corresponding map N qi λ Q i λ N C A Q i λ, N). By Proposition 3.1.5, it suffices to show that the two circuits of the diagram 3.8) give the same result after composing with the map ι 0 ) dr, N) ι 0 ) dr, CA Q i λ, N) ). In the sequel, instead of Q i λ we shall use an arbitrary object P P and a vector p P.

LOCAL VS GLOBAL DEFINITION OF TENSOR PRODUCT 13 3.3.6. We will show that the two arrows are equal by completing the two composed morphisms to a commutative cube: B j M L) B ι 0 ) dr, N) id p id B j P M L) id p id B ι 0 ) dr, P N) 3.9) j M L) ι 0 ) dr, N) j C A P, M, L)) ι 0 ) dr, C A P, N)) Let us specify the arrows in this diagram. The back wall equals the diagram 3.8), whose commutativity we need to establish. The top lid in 3.9) is tautologically commutative. The bottom lid is obtained as the outer square in the diagram which is commutative by assumption. j M L) ι 0 ) dr, N) p id p id j P M L) ι 0 ) dr, P N) j C A P, M, L)) ι 0 ) dr, C A P, N)), The map B j P M L) j C A P, M, L)) is the composition B j P M L) B j P M L) j C A P, M, L)), where the second arrow is the natural projection, and the first arrow is the action of B on the M and L factors. This implies that the left square in 3.9) commutes. The map B ι 0 ) dr, P N) ι 0 ) dr, C A P, N)) is defined similarly. In particular, the right square in 3.9) commutes.

14 DENNIS GAITSGORY 3.3.7. Hence, it remains to show that the front square in 3.9) commutes. Note, however, that by the definition of coinvariants, the front left vertical arrow equals also the composition B j P M L) B j P M L) j C A P, M, L)), where the second arrow is the natural projection, and the first arrow is the negative of the B-action on the P-factor. A similar remark applies to the front right vertical arrow. This shows that the front square commutes, again by the assumption on the map j M L) ι 0 ) dr, N). References [Ga] D. Gaitsgory Universal constructions of crystals, Seminar notes. [Ts] A. Tsymbalyuk, Category O, Seminar notes.