Connecting Coinvariants In his talk, Sasha taught us how to define the spaces of coinvariants: V 1,..., V n = V 1... V n g S out (0.1) for any V 1,..., V n KL κg and any finite set S P 1. In her talk, Giorgia showed us how to use this vector space to define the fusion tensor product on category O κ. In this paper, we will show how to paste the spaces (0.1) together as S varies, and introduce a connection on the corresponding space. Throughout this paper, we will work with smooth schemes X. We will always let : X X X denote the diagonal map, and j : X X \ X X denote the inclusion of the complement. In section 1.2, we will generalize the notion of diagonal and then use the notations j + (resp. j ) and + (resp. ) will mean something more general. Unless we explicitly state the contrary, a D module will refer to a right D module on X. For such a D module M, we let h denote the right exact functor: {D modules} h {sheaves}, M h M/M T X. The de Rham functor is defined as DR(X, M) = RΓ(X, Lh(M)). In certain sections we will set X = P 1, though we will explicitly remind the reader when we will do this. Whenever this will be the case, we will encounter the objects: O = C[[t]], K = C((t)) and if x is a point on X, then we let O x and K x be the ring of functions on the formal (resp. punctured formal) disk around x. They are non-canonically isomorphic to O and K.
2 Connecting Coinvariants 1. Chiral Algebras 1.1. All the constructions of this section will be on an arbitrary smooth scheme X. Definition 1.1. On a smooth scheme X, a chiral algebra A is a D module together with a map of D modules: j j (A A) {, }! A, (1.2) called the chiral bracket, which satisfies antisymmetry and the Jacobi identity 1. We call our chiral algebra unital if it comes with a map of D modules: Ω 1 A (1.3) that is compatible with the chiral bracket 2. The vector space DR(X, A) inherits a structure of Lie algebra. Definition 1.4. A chiral D module 3 M is a D module with a map of D modules: j j (A M)! M, (1.5) which must be compatible with (1.2) and with (1.3). For any open subset U X, the vector space DR(U, M) becomes a module of the Lie algebra DR(U, A). Definition 1.6. A chiral O module M is a O module with a map of O modules: j j (A M) O! M, (1.7) that is compatible with (1.2) and with (1.3) in the same sense as before. The objects in the above relation are D X O X modules, and the push-forward in the RHS is taken in this category. Explicitly, it is given by: O! M = M where is the usual push-forward for O modules. O X O X D X O X, 1 These are spelled out explicitly in [2], section 3.1 2 The compatibility relation is states in [2], section 3.1 3 We usually call this a chiral module, for short
Connecting Coinvariants 3 1.2. In [3], Nick shows us how to generalize the construction of chiral modules. We will now give a hint of that construction, which is enough for our purposes. As before, let A be a chiral algebra over a smooth scheme X and let n 1 be a natural number. Definition 1.8. A chiral module on X n is a D module M on X n with a map of D modules: j + j +(A M) {, } +!! +p! 2(M), Here the maps are as in the diagram: Z X X n p 1 X where is the diagonal consisting of points {x i, x 1,..., x i,..., x n }, i {1,..., n} and j denotes the complement. Naturally, we ask the chiral bracket (1.10) to be antisymmetric and to satisfy the Jacobi identity just like before. p 2 X n 1.3. We will not need chiral modules on X n, but a closely related concept. Let X n denote the complement to all diagonals, and we will repeat the above with the diagonals of X n removed. Definition 1.9. A chiral module on X n is a D module M on X n with a map of D modules: j j (A M) {, }! M (1.10) Here the maps are as in the diagram: X n... X n X X n, X where denotes the only remaining diagonal and j denotes the complement to it. We ask the chiral bracket (1.10) to be antisymmetric and to satisfy the Jacobi identity just like before. p 1 p 2 X n A particular example of chiral module on X n is M 1... M n X n, where M 1,..., M n are chiral modules on X. In this case it is clear how to construct the chiral bracket (1.10) out of the chiral brackets of each M i, because the locus splits up nicely into n connected components corresponding to each i {1,..., n}. If we had not restricted to X n, then it would have been impossible to define the chiral bracket.
4 Connecting Coinvariants 2. Lie Algebras Definition 2.1. A Lie algebra L is a D module with a map: L L {, }! L, (2.2) called the Lie bracket, which satisfies antisymmetry and the Jacobi identity just like in the case of chiral algebras. Note that any chiral algebra is automatically a Lie algebra. Definition 2.3. A Lie (respectively chiral) module for L is a D module M with a map of D modules: L M! M (respectively j j (L M)! M) that is compatible with the Lie bracket (2.2). 2.1. In the study of finite dimensional Lie algebras, a major role is played by universal enveloping algebras, which are defined by a universal property. The exact same construction works in the chiral setting, where to any Lie algebra L we associate a unital chiral algebra U(L) defined by the same universal property: Hom Lie (L, A) = Hom chiral (U(L), A), (2.4) functorially in the chiral algebra A. Our goal now is to present an explicit construction of U(L), but first we need to see what it should be. One expects its! fiber over x X to be 4 the universal enveloping algebra of L! x. One can write this as a cokernel of Lie algebras: DR(X, L) DR(X\x, L) L! x[1] 0 and therefore we expect that the! fiber of U(L) at x should equal: U(DR(X\x, L)) C (2.5) U(DR(X,L)) 2.2. Let p 1 : X X X be the first projection, and then the D modules: L small = p 1 (Ω L) p 1 j j (Ω L) = L big. have fibers over x equal to DR(X, L) and DR(X\x, L), respectively. They are both Lie algebras in the category of D modules, so we can construct their classical universal enveloping algebras in the category of D modules. Then we define: U(L) = U(L big ) Ω X, U(L small ) 4 in some vague, motivational sense
Connecting Coinvariants 5 as an algebra in the category of D modules. It has precisely the fibers we want from (2.5), and this is the explicit description of the universal enveloping chiral algebra. It has a PBW filtration coming from U(L big ) such that: U(L) 0 = Ω X, U(L) 1 = L Ω X. 3. Examples 3.1. Specialize now X to be a curve. Given a finite-dimensional Lie algebra g and a non-degenerate symmetric bilinear form κ : g g C, we construct the Kac-Moody Lie algebra: where the Lie bracket kills Ω X and is otherwise given by: L g,κ = g D X ΩX, (3.1) {x 1, x 1} = [x, x ] 1 + κ(x, x ) 1, where 1 D X! (D X ) and 1 is the section of! (Ω X ) given by: 1 = dt dt (t t ) 2 (Ω X Ω X )(2 ) (Ω X Ω X )( ) =! (Ω X ) Ω X Ω X Ω X Ω X Note that this section does not depend on the choice of the coordinate t on X; we will encounter it again when we will discuss the Sugawara construction. For any point x X, we have: DR 0 (Spec(O x ), L g,κ ) = g O x, DR 0 (Spec(K x ), L g,κ ) = g K x C, (3.2) where the latter has the Lie algebra structure of our good old Kac-Moody Lie algebra ĝ k. Moreover, for any finite subset S X we have: DR 0 (X\S, L g,κ ) = ĝ S κ, (3.3) as introduced by Sasha and Giorgia. Finally, our main player will be the chiral Kac-Moody algebra 5 : A g,κ = U(L g,κ )/(1 1 ), where we are setting equal two units 1, 1 : Ω X A g,κ. The first comes from the definition of a chiral algebra, and the second one comes from the central factor Ω X of L g,κ. 5 The two units 1, 1 : Ω A g,κ we are setting equal to each other are (1.3) and the one coming from (3.1)
6 Connecting Coinvariants 3.2. Still taking X to be a curve, consider T = T X OX D X and look at the map: T T = (T X OX D X ) (T X OX D X ) T X OX X D X X =! (T), where the middle arrow is minus the usual Lie bracket on vector fields. This endows T with a structure of a Lie algebra, such that: DR 0 (Spec(K x ), T) = Vir, DR 0 (Spec(O x ), T) = Vir +, (3.4) where Vir + and Vir are the Lie algebras of infinitesimal automorphisms of the formal, respectively punctured formal, disk. 4. Conformal blocks 4.1. Let us now go back to a chiral module M for a chiral algebra A on a smooth projective scheme X. If we take (1.2) and push it forward under the second projection p 2, we get a map of D modules on X: p 2 j j (A M) ρ M. (4.1) The cokernel of the above map is denoted by H (X, M) and is naturally a D module on X called conformal blocks 6. The effect of p 2 above is to kill the tangent vectors in the direction of the first factor of X X, so the fiber of (4.1) at some point x X gives us an exact sequence: DR 0 (X\x, A) M x ρ x Mx H (X, ρ) x 0. (4.2) The reason why we have X\x is j. The Jacobi identity for the chiral bracket implies that ρ x is a Lie algebra action on the vector space M x, and so we can write conformal blocks as a space of coinvariants: H (X, ρ) x = M x /(DR 0 (X\x, A) M x ). A particularly important case is the vacuum module M = A, in which case H (X, A) is the trivial D module on X. In other words, it is not only globally free as a O module, but the connection is also trivial. This is actually true for any algebra in the category of D modules, as we proved in Proposition 3 of my talk in the Fall of 2009. 6 If M were only a chiral O module instead of a chiral D module, then the above discussion would go through with the only exception that H (X, M) would only be an O module on X
Connecting Coinvariants 7 4.2. The setting in which we will be using conformal blocks is for a D module M on X n as in section 1.2. Then we start from the chiral bracket (1.10), push it forward under the projection p 2, and we define conformal blocks as the D module on X n : H (X n, M) = Coker(p 2 j + j +(A M) p 2 +!! +p! 2M) But it we are looking at D modules on X n as in section 1.3, then note that p 2 is simply a trivial n fold cover. Then the above becomes: H (X n, M) = Coker(p 2 j j (A M) M n ). If we take the fiber of this above a point (x 1,..., x n ) X n, we see that: ( ) H (X n, M) (x1,...,x n) = Coker DR(X\{x 1,..., x n }, A) M x1,...,x n ) Ψ M n (x 1,...,x n). (4.3) Lemma 4.4. The above fiber can also be realized as the cokernel: H (X n, M) = Coker ( DR(X\{x 1,..., x n }, A) M x1,...,x n ) M (x1,...,x n)), (4.5) with the map being the sum of the n component maps in (4.3). Proof. Indeed, the addition map M n (x M 1,...,x n) (x 1,...,x n) induces an a priori surjective map between the cokernel of (4.3) and the cokernel (4.5). To prove that it is injective, one needs to show that anything in the kernel of the addition map is in the image of the arrow in (4.3). Since the kernel is spanned by permutations of the following vectors, it s enough to prove that the vectors (m, m, 0,..., 0) M n (x 1,...,x n) lie in the image of the map (4.3). However, it s easy to see that: (m, m, 0,..., 0) = Ψ(ω m), where ω is a differential form on X with the residue 1 at x 1 and 1 at x 2. If M = M 1... M n X n for chiral modules M 1,..., M n on X, then the above lemma says that: H (X n, M) (x1,...,x n) = (M 1 x1... M n xn )/ DR 0 (X\{x 1,...,x n},a) (4.6) with the action of DR 0 being the diagonal one. In the last section, we will show that this description of conformal blocks coincides with the coinvariants introduced by Sasha and Giorgia.
8 Connecting Coinvariants 4.3. In the Fall of 2009, we saw that conformal blocks have certain functorial interpretations in the category of D schemes. However, for our purposes now, we are only interested in the fact that (4.6) realizes them as coinvariants, much in the same way as Sasha and Giorgia talked about in their talks. This analogy will be made explicit later, but let us make a technical point. We are in the case A = A g,κ = U(L g,κ ) of section 3.1. Sasha and Giorgia take coinvariants with respect to the Lie algebra, whose chiral analogue is DR(X\x, L g,κ ), whereas in (4.6) we are taking coinvariants with respect to the universal enveloping DR(X\x, A g,κ ). The following proposition shows that the two constructions actually produce the same thing. Proposition 4.7. For a module M over a Lie algebra L, the natural surjection: M x /(DR 0 (X\x, L) M x ) M x /(DR 0 (X\x, A) M x ) is an isomorphism, where A = U(L). Proof. It is enough to look at the PBW filtration of A, under which A 1 = L Ω X, and to prove that the natural surjection: M x /(DR 0 (X\x, A n ) M x ) M x /(DR 0 (X\x, A n+1 ) M x ) is an isomorphism for all n 1. We need to show it is injective, so we need to show that any element α m DR 0 (X\x, A n+1 ) M x also lies in DR 0 (X\x, A n ) M x. The chiral bracket of the universal enveloping algebra respects the PBW filtration, and actually: j j L A n! A n+1. Because X\x is affine, the above is still a surjection when we pass to DR, so there is a section f(t, t )(α 1 α 2 ) which projects onto α under the chiral bracket, where α 1 DR 0 (X\x, L) and α 2 DR 0 (X\x, A n ). The Jacobi identity then implies that α m lies in DR 0 (X\x, A n ) M x. References [1] Beilinson, Alexander; Drinfeld, Vladimir, Chiral Algebras [2] Gaitsgory, Dennis, Notes on 2D Conformal Field Theory and String Theory [3] Rozenblyum, Nick, Modules over a Chiral Algebra