Remarks on deformation quantization of vertex Poisson algebras Shintarou Yanagida (Nagoya) Algebraic Lie Theory and Representation Theory 2016 June 13, 2016
1 Introduction Vertex Poisson algebra (VPA) is a classical limit of vertex algebra (VA). e.g. 1. Poisson structure on the space of connections on trivial G-bundle over punctured disc 2. The classical W algebras (Poisson structure on the space of opers, or Hamiltonian reduction of e.g. 1) One can ask the deformation problem: given a VPA, can one obtain a VA as a deformation? In this talk we give two remarks on this deformation problem. 1. The dg Lie algebra controlling the VA structures 2. Existence of the deformation.
Contents 1 Introduction 2 Vertex Poisson algebras Definition Limit construction 3 Deformation problem Ad-hoc formulation Recollection of (usual) deformation quantization 4 Operads in 15 minutes Operads Koszul dual (co)operads Convolution dg Lie algebra 5 Chiral algebras Chiral algebras Coisson algebras 6 Chiral dg Lie algebra Definition Chiral deformation quantization, revisited Existence/Uniqueness of chiral deformation quantization
2.1 Definition of VPA c.f. [Frenkel and Ben-zvi, Chap. 16] A vertex Lie algebra (VLA) is a triple (L, T, Y ) consisting of 1 a vector space L 2 T End(L) 3 Y (A, z) = n 0 A (n)z n 1 End(L) z 1 C[[z 1 ]] such that Y (TA, z) = z Y (A, z) Y (A, z)b = (e zt Y (B, z)a) [A (m), Y (B, w)] = ) n 0 (w m n Y (A (n) B, w)) ( m n For any VA (V, 0, T, Y), its polar part (V, T, Y ) is a VLA. Y (A, z) := Y(A, z) negative power part of the series Y(A, z).
A VA (V, 0, T, Y) is called commutative if for any A, B V [Y(A, z), Y(B, w)] = 0. A commutative VA (V, 0, T, Y) is equivalent to a commutative C-algebra (V, ) with unit 0 and a derivation T. The equivalence is given by A B = A ( 1) B; Y(A, z)b = e zt A B A vertex Poisson algebra is a quintuple (V, 0, T, Y +, Y ) of (V, 0, T, Y + ) is a comm. VA (V, T, Y ) is a VLA all the coefficients of Y (A, z) are derivations of the the commutative product on V induced by Y +.
2.2 Limit construction c.f. [Frenkel and Ben-zvi, Chap. 16] Assume (V ħ, Y ħ ) is a flat family of VA over C[[ħ]]. If V 0 := V ħ /ħv ħ is a comm. VA over C with Y 0 := Y ħ then V 0 is a VPA with (mod ħ), Y (A, z) := 1 ħ Yħ (A, z) (mod ħ). Examples of limit construction of VPA 1 V (g) := V K (g)/k 1 V K (g), Poisson structure on the space Conn G (D ) of connections on the trivial G-bundles on the punctured disc D, 2 W (g) := W K (g, e reg )/K 1 W K (g, e reg ), Poisson structure on the space of opers.
3.1 Ad-hoc formulation of our deformation problem Given a VPA (V 0, 0, T, Y +, Y ), classify VAs (V ħ, 0, T, Y ħ ) over C[[ħ]] such that Y = Y + + ħy + ħ 2 Y 2 +. Similar to the deformation quantization problem: Given a Poisson algebra (A,, { }) over C, classify associative algebras (A, ) over C[[ħ]] such that a b = a b + ħ{a, b} + ħ 2 α 2 (a, b) + = n 0 ħ n α n (a, b).
3.2 Recollection of deformation quantization c.f. [Kontsevich, LMP (2003)] For a given Poisson algebra (A,, { }), two deformations (A, 1 ) and (A, 2 ) are called equivalent if there exists φ = n 0 ħn φ n End(A)[[ħ]] such that φ(a 1 b) = φ(a) 2 φ(b). The equivalent class of deformations are described by the Hochschild complex C (A, A) = ( n 0 C n (A, A), d), and the Hochschild cohomology H (A, A). 1 the equivalence class of α 1 = {} is an element of H 2 (A, A). 2 The associativity of is rewritten as dα m + 1 2 [α i, α j ] = 0, i+j=m where [ ] is the Gerstenhaber bracket.
The Hochschild complex C n (A, A) := Hom(A n, A), df(a 0,..., a n ) :=a 0 f(a 1,..., a n ) n + ( 1) i f(a 0,..., (a i 1 a i ),..., a n ) i=1 The Gerstenhaber bracket + ( 1) n+1 f(a 0,..., a n 1 ) a n. 1 2 [α i, α j ] (a, b, c) := α i (α j (a, b), c) α i (a, α j (b, c))
The equations dα m + 1 2 i+j=m [α i, α j ] = 0 can also be rewritten as the Maurer-Cartan equation dα + [α, α] = 0, α = α n g 1 of the dg Lie algebra g = (C (A; A), [ ], d). The grading is g = n 1 g n, g n := C n+1 (A, A) = Hom(A (n+1), A). The Lie bracket [ ] is [α, β] := α β ( 1) α β β α α β(a 0,..., a α + β ) := α ( 1) r β r=0 α(a 0,..., a r 1, β(a r,..., a r+ β ), a r+ β +1,..., a α + β ).
Thus a deformation quantization of a Poisson algebra is the problem to find/classify solutions α = n 0 α n of Maurer-Cartan equation of the dg Lie algebra C (A, A) with α 0 and α 1 equal to the given and { }. Back to our situation: What is the dg Lie algebra controlling the deformation of VPA? Hints in the literature: 1 According to the theory of operad, one can construct a dg Lie algebra controlling the deformations of P-algebras for any Koszul operad P. 2 According to the theory of chiral algebra by Beilinson-Drinfeld, VAs and VPAs can be formulated in terms of operads.
4.1 Operads c.f. [Loday and Vallette, Algebraic Operads (2012)] S-module An S-module M is a series {M(n)} n 1 of right S n -modules. I = C 0 0 the identity S-module. M N := n M(n) Sn N n for two S-modules M and N, An operad is a triple (P, γ, η) of 1 An S-module P = n 1 P(n) (spaces of n-ary operations), 2 An S-module morphism γ : P P P (composition map). 3 An S-module morphism η : I P (unit). satisfying some compatibility conditions. A cooperad (C,, ε) with the decomposition map : C C C is defined similarly.
Examples: 1 Com: the operad of commutative algebras Com(n) = Cµ n, the trivial S n -rep. for any n 1. The generator µ n corresponds to the multiplication operation µ n (x 1,..., x n ) = x 1 x n C[x 1,..., x n ]. The composition map is γ : Com(n) Com n Com, γ(µ n ; µ in,..., µ in ) = µ i1 + +i n. 2 Assoc: the operad of (non-comm.) associative algebras Assoc(n) = regular rep. The composition map is γ(σ; σ 1,..., σ n ) := σ (σ 1,..., σ n ) where σ is the block permutation associated to σ.
1 Lie: the operad of Lie algebras µ Lie Lie(2): the Lie bracket. Lie(n) Ind Z/nZ ζ n, ζ n is the 1-dim. rep. by the root of unity. 2 End V : the operad of endomorphisms on a vector space V. End V (n) := Hom C (V n, V). γ is the usual composition of linear maps. 3 coend V : the cooperad of endomorphisms on a vector space V. coend V (n) := Hom C (V, V n ). A P-algebra structure on a vector space V is an operad morphism P End V.
4.2 Koszul operads The free operad F(E) of an S-module E. Its underlying S-module is given by F(E) := n F n (E), F 0 (E) := I, F 1 (E) := I E, F n 1 (E) F n (E) := I E F n 1 (E). It has a weight grading F(E) = F(E) (d) with wt(µ) = 1 for µ E. A quadratic operad A quadratic data (E, R) is a pair of an S-module E and a sub-s-module R F(E) (2) P(E, R) := F(E)/(R): the quadratic operad for (E, R). C(E, R): the quadratic cooperad.
s denotes the shift of a complex. (sm) p = M p 1. The Koszul dual cooperad of a quadratic operad P = P(E, R): P c! := C(sE, sr). The Koszul dual operad of P is P! := (coend sc P c! ). P c! and P! have weight gradings like F(E). Common examples: Com! Lie, Lie! Com. Assoc! Assoc.
4.3 Convolution dg Lie algebra c.f. [Loday and Vallette, Chapter 6] There is a good class of quadratic operads called Koszul operads. (The natural projection ΩP c! P is a quasi-isomorphism.) Com, Assoc and Lie are Koszul operads. For a Koszul operad P and a vector space V, the S-module g g P,V := Hom(P c!, End V ) = n 1 Hom(P c! (n), End V (n)) has a natural structure of dg Lie algebra (g, [ ], d), called the convolution dg Lie algebra. It has a weight grading g = d 0 g (d) induced by that on P c!. For any dg Lie algebra g, set MC(g) := {degree n = 1 sol. of the Maurer-Cartan equation of g}.
Fact (Loday-Vallette) { P-alg. structures on V} 1:1 MC 1 (g P,V ) := {weight d = 1 elements in MC(g P,V )} Once a weight 1 solution µ of MC eqn. is given, we have a twisted dg Lie algebra g µ P,V := (Hom(Pc!, End V ), [ ], d µ := d + [µ, ]). and it encodes the deformation of P-algbera structure µ: MC 1 (g µ P,V ) 1:1 {P-alg structure deforming µ}
examples P = Assoc: g µ Assoc,V (C (A, A), [ ], d) with A = (V, µ). P = Lie: g µ Lie,V the Chevalley complex with the Nijenhuis-Richardson bracket.
5.1 Chiral algebras c.f. [Beilinson-Drinfeld, Chiral algebras (2006)] D-modules and functors M(X): the category of right D X -modules (quasi-coherent as O X -modules) on a smooth variety X. For a morphism f : X Y of smooth varieties, denote the standard functors as f : DM(X) DM(Y), f! : DM(Y) DM(X). For f : X Y a LCI over Y of pure dim. d, f := f! [ d] : DM(Y) DM(X), in fact M(Y) M(X).
chiral operad (chiral pseudo-tensor structure in [BD]) X: smooth curve For n Z 1, (n) : X X n : diagonal embedding j (n) : U (n) := {(x i ) X n x i x j ( i j)} X n. For M M(X), one can define the chiral operad End ch M, whose underlying S-module is given by Remark: End ch M (n) := Hom M(X n )(j (n) j(n) M n, (n) M) j (2) j(2) M 2 M 2 ( diag ),
A chiral algebra (CA) structure (without unit) on M M(X) is an operad morphism φ : Lie End ch M. The operation φ(µ Lie ) End ch M (2) is called the chiral bracket.
VA and CA For a VA V, denote by V the VA bundle on a smooth curve X. V is a left D-module if V is quasi-conformal. Denote by Y 2 : j (2) j(2) V 2 (2) (V) the morphism of left D-modules induced by Y. Locally, Y 2 x (f(z, w)a B) = f(z, w)y(a, z w)b (mod V[[z, w]]). Fact (Beilinson-Drinfeld, c,.f. [Frenkel-Ben-zvi,Chap. 19]) For a quasi-conformal VA V, the right D-module V r := V ω X has a structure of CA. The chiral bracket µ End ch V r (2) is given by µ = (Y 2 ) r.
5.2 Coisson algebras End c M : the compound operad for M M(X). Its underlying S-module is End c M (n) = S Q(n) End c M (n) S, End c M (n) S := Hom M(XS )(M S, (S) M) ( t SLie( π 1 (t) )) Q(n) is the set of equivalence classes of decompositions of the set {1,..., n}. π S : {1,..., n} S A coisson algebra structure on M M(X) is an operad morphism Lie End c M. S
Chiral algebras and coisson algebras End ch M has a filtration W such that End ch M (n) = W0 W 1 W n W n+1 = 0, and we have an inclusion of operads gr End ch M Endc M. The filtration comes from the Cousin complex of ω X. Corollary: given a family A t of chiral algebras flat over C[[t]], A 0 := A t /ta t has a structure of coisson algebra. The limit construction of VPA is a special case. Namely, VPA yields a coisson algebra.
6.1 Definition of chiral dg Lie algebra Definition For M M(X), the S-module g ch M := Hom(Liec!, End ch M ) has a structure of dg Lie algebra (g ch M, [ ], d). We call it the chiral dg Lie algebra. Cororally (of 4 Operads) {CA structures on M} MC 1 (g ch M ).
Similarly, we have the coisson dg Lie algebra g c M := Hom(Liec!, End c M ) A coisson algebra structure corresponds to a weight 1 element of MC(g c M ) bijectively.
6.2 Deformation problem, revisited By 5 (chiral algebras), we have morphisms of operads End ch M gr Endch M Endc M, Proposition The above induces a morphism of dg Lie algebras g ch M gc M and we have ψ : MC 1 (g ch M ) MC1 (g c M ). Definition We call α MC 1 (g c M ) a chiral deformation quantization of α cois MC 1 (g c M ) if ψ(α) = αcois.
How does the complex g c M look like? As an S n -module, g c M (n) n := Hom(Lie c!, End c M (n) n) Hom(Lie c!, Hom M(X n )(M n, M)) Hom M(Xn )((sm) n ( diag ), s M)) s 1 n Hom M(X n )( n M( diag ), M). Given α cois MC(g c V 0,r ) corresponding to VPA (V 0, Y +, Y ), d = d + [α cois, ] on f g c V 0,r (n) n Hom( n V 0, V) is n 1 df(a 0,..., a n ) = ( 1) r Y + (a r )f(a 0,..., â r,..., a n ) + 0 r<s n 1 r=0 ( 1) n+r+s f(a 0,..., â r,..., â s,..., a n, Y (a r )a s )
Remark 1 Lie conformal algebra cohomology by De Sole and Kac is a special case of our construction. 2 Tamarkin constructed a dg Lie algebra in Deformation quantization of chiral algebras (Proc. ICM, 2002) It seems that his algebra coincides with ours.
6.3 Existence/Uniqueness of chiral deformation When the map ψ : MC 1 (g ch M ) MC1 (g c M ) is surjective, then a chiral deformation quantization exists. Theorem ψ is an isomorphism if if M is a projective D-module, or if M arises from the (g, K)-setting [BD, 2.9.7]. Remark The second case includes the poisson structures over connections on G-bundles on a curve. (In particular, the classical limit of V k (g).) We conjecture that the bijectivity holds for Hamiltonian reductions. (In particular, the classical W algebras.)