W-algebras, moduli of sheaves on surfaces, and AGT

W-algebras, moduli of sheaves on surfaces, and AGT MIT 26.7.2017

The AGT correspondence Alday-Gaiotto-Tachikawa found a connection between: [ ] [ ] 4D N = 2 gauge theory for Ur) A r 1 Toda field theory

The AGT correspondence Alday-Gaiotto-Tachikawa found a connection between: [ ] [ ] 4D N = 2 gauge theory for Ur) A r 1 Toda field theory Specifically, the Nekrasov partition function for the LHS should match conformal blocks for the W algebra of type gl r.

The AGT correspondence Alday-Gaiotto-Tachikawa found a connection between: [ ] [ ] 4D N = 2 gauge theory for Ur) A r 1 Toda field theory Specifically, the Nekrasov partition function for the LHS should match conformal blocks for the W algebra of type gl r. In the pure gauge theory, the mathematical connection was established by Maulik-Okounkov and Schiffmann-Vasserot,

The AGT correspondence Alday-Gaiotto-Tachikawa found a connection between: [ ] [ ] 4D N = 2 gauge theory for Ur) A r 1 Toda field theory Specifically, the Nekrasov partition function for the LHS should match conformal blocks for the W algebra of type gl r. In the pure gauge theory, the mathematical connection was established by Maulik-Okounkov and Schiffmann-Vasserot, who defined an action of the W algebra on the cohomology group H of the moduli space M of sheaves/instantons on A 2.

The AGT correspondence Alday-Gaiotto-Tachikawa found a connection between: [ ] [ ] 4D N = 2 gauge theory for Ur) A r 1 Toda field theory Specifically, the Nekrasov partition function for the LHS should match conformal blocks for the W algebra of type gl r. In the pure gauge theory, the mathematical connection was established by Maulik-Okounkov and Schiffmann-Vasserot, who defined an action of the W algebra on the cohomology group H of the moduli space M of sheaves/instantons on A 2. Then the Nekrasov partition function is 1, 1, where 1 is the unit cohomology class, which one can uniquely determine based on how the W algebra acts on it the Gaiotto state).

The Ext operator Things are even more interesting in the presence of matter. Mathematically, matter in the bifundamental representation: C r 1 C r 2 Ur 1 ) Ur 2 )

The Ext operator Things are even more interesting in the presence of matter. Mathematically, matter in the bifundamental representation: C r 1 C r 2 Ur 1 ) Ur 2 ) is encoded in the Ext operator of Carlsson-Okounkov: ) ] A m : H H, A m = p 1 [c Ext F, F), m p2

The Ext operator Things are even more interesting in the presence of matter. Mathematically, matter in the bifundamental representation: C r 1 C r 2 Ur 1 ) Ur 2 ) is encoded in the Ext operator of Carlsson-Okounkov: ) ] A m : H H, A m = p 1 [c Ext F, F), m p2 Therefore, AGT for linear quiver gauge theories follows if one presents A m as an intertwiner of the W algebra action on H

The Ext operator Things are even more interesting in the presence of matter. Mathematically, matter in the bifundamental representation: C r 1 C r 2 Ur 1 ) Ur 2 ) is encoded in the Ext operator of Carlsson-Okounkov: ) ] A m : H H, A m = p 1 [c Ext F, F), m p2 Therefore, AGT for linear quiver gauge theories follows if one presents A m as an intertwiner of the W algebra action on H We will prove this by q deforming everything, redefining the W algebra action, and proving that A m commutes with it.

The Ext operator Things are even more interesting in the presence of matter. Mathematically, matter in the bifundamental representation: C r 1 C r 2 Ur 1 ) Ur 2 ) is encoded in the Ext operator of Carlsson-Okounkov: ) ] A m : H H, A m = p 1 [c Ext F, F), m p2 Therefore, AGT for linear quiver gauge theories follows if one presents A m as an intertwiner of the W algebra action on H We will prove this by q deforming everything, redefining the W algebra action, and proving that A m commutes with it. Since our construction is purely geometric, it makes sense for sheaves on surfaces S more general than the affine plane A 2

Deformation Mathematically, deformation refers to two related processes:

Deformation Mathematically, deformation refers to two related processes: Replacing the cohomology of M by its algebraic K theory: H K = K equiv M) = K equiv M n ) n=0 which entails replacing 4D N = 2 by 5D N = 1 gauge theory.

Deformation Mathematically, deformation refers to two related processes: Replacing the cohomology of M by its algebraic K theory: H K = K equiv M) = K equiv M n ) n=0 which entails replacing 4D N = 2 by 5D N = 1 gauge theory. Replacing Lie algebras by quantum groups, e.g. g U q g) or: Heisenberg q Heisenberg = Z[q 1 ±1, q±1 2 ] a n n Z\0 [a m, a n ] = δ n m n1 q n 1 )1 qn 2 )1 qrn ) 1 q n

Deformation Mathematically, deformation refers to two related processes: Replacing the cohomology of M by its algebraic K theory: H K = K equiv M) = K equiv M n ) n=0 which entails replacing 4D N = 2 by 5D N = 1 gauge theory. Replacing Lie algebras by quantum groups, e.g. g U q g) or: Z[q 1 ±1 Heisenberg q Heisenberg =, q±1 2 ] a n n Z\0 [a m, a n ] = δm n n1 q n 1 )1 qn 2 )1 qrn ) 1 q n More generally, we will consider the deformed W algebra of Feigin-Frenkel and Awata-Kubo-Odake-Shiraishi. In type gl 1, this algebra is q Heisenberg, while for sl 2, it deforms Virasoro.

The deformed W algebra The original definition of the q W algebra is via free fields: qw r H r = Z[q 1 ±1, q±1 2 ] bi) n 1 i r n Z\0 [b i) n, bj) m ] = δmn1 n q1 n)1 qn 2 )1 δ i jq nδ i<j )

The deformed W algebra The original definition of the q W algebra is via free fields: qw r H r = Z[q 1 ±1, q±1 2 ] bi) n 1 i r n Z\0 [b i) n, bj) m ] = δmn1 n q1 n)1 qn 2 )1 δ i jq nδ i<j ) q W r is defined as the subalgebra generated by W d,k given by: [ k b s ] [ i ) ] n b s i ) n W k x) = : u si exp nx n exp nx n : 1 s 1... s k r i=1 n=1 n=1

The deformed W algebra The original definition of the q W algebra is via free fields: qw r H r = Z[q 1 ±1, q±1 2 ] bi) n 1 i r n Z\0 [b i) n, bj) m ] = δmn1 n q1 n)1 qn 2 )1 δ i jq nδ i<j ) q W r is defined as the subalgebra generated by W d,k given by: [ k b s ] [ i ) ] n b s i ) n W k x) = : u si exp nx n exp nx n : 1 s 1... s k r i=1 n=1 n=1 More intrinsically, q W r can be described as the algebra generated by symbols W d,k modulo relations such as: ) ) x y W k x)w 1 y)ζ yq k W 1 y)w k x)ζ = xq = q [ ) ) ] 1 1)q 2 1) x y δ q 1 yq k W k+1 x) δ W k+1 y) xq

The Verma module and the vertex operator Let u 1,..., u r C. Define the Verma module of q W r as: M = q W r / right ideal W d,k, W 0,k e k u 1,..., u r )) d>0 1 k r and similarly define M with the parameters u i replaced by u i.

The Verma module and the vertex operator Let u 1,..., u r C. Define the Verma module of q W r as: M = q W r / right ideal W d,k, W 0,k e k u 1,..., u r )) d>0 1 k r and similarly define M with the parameters u i replaced by u i. For m C, the vertex operator will almost be an intertwiner: Φ m : M M

The Verma module and the vertex operator Let u 1,..., u r C. Define the Verma module of q W r as: M = q W r / right ideal W d,k, W 0,k e k u 1,..., u r )) d>0 1 k r and similarly define M with the parameters u i replaced by u i. For m C, the vertex operator will almost be an intertwiner: Φ m : M M as it isn t required to commute with q W r on the nose. Instead: ) [Φ m, W k x)] m k 1 mr u 1...u r q r k x u 1 = 0 1)...u r for all k 1, where [A, B] s = AB sba.

The Verma module and the vertex operator Let u 1,..., u r C. Define the Verma module of q W r as: M = q W r / right ideal W d,k, W 0,k e k u 1,..., u r )) d>0 1 k r and similarly define M with the parameters u i replaced by u i. For m C, the vertex operator will almost be an intertwiner: Φ m : M M as it isn t required to commute with q W r on the nose. Instead: ) [Φ m, W k x)] m k 1 mr u 1...u r q r k x u 1 = 0 1)...u r for all k 1, where [A, B] s = AB sba. Lemma: the endomorphism Φ m is uniquely determined, up to constant multiple, by property 1).

The main Theorem Theorem N) There exists an action q W r K, for which the latter is isomorphic to the Verma module M. Moreover, the Ext operator A m : K K is equal to the vertex operator Φ m : M M, up to a simple exponential.

The main Theorem Theorem N) There exists an action q W r K, for which the latter is isomorphic to the Verma module M. Moreover, the Ext operator A m : K K is equal to the vertex operator Φ m : M M, up to a simple exponential. The second part is an intersection-theoretic computation, once we give a geometric definition of the action in first part.

The main Theorem Theorem N) There exists an action q W r K, for which the latter is isomorphic to the Verma module M. Moreover, the Ext operator A m : K K is equal to the vertex operator Φ m : M M, up to a simple exponential. The second part is an intersection-theoretic computation, once we give a geometric definition of the action in first part. We define the action of W x, y) = k>0 d Z W d,k x d y) k W x, yd x ) = Lx, yd x ) Ey) Ux, yd x ) where D x is the difference operator f x) f xq) on K as:

The main Theorem Theorem N) There exists an action q W r K, for which the latter is isomorphic to the Verma module M. Moreover, the Ext operator A m : K K is equal to the vertex operator Φ m : M M, up to a simple exponential. The second part is an intersection-theoretic computation, once we give a geometric definition of the action in first part. We define the action of W x, y) = k>0 d Z W d,k x d y) k W x, yd x ) = Lx, yd x ) Ey) Ux, yd x ) where D x is the difference operator f x) f xq) on K as: and L, E, U are geometric endomorphisms of K that are lower triangular, diagonal, and upper triangular, respectively.

The moduli space of sheaves Explicitly, our M is the moduli space of framed sheaves: ) F torsion-free sheaf on P 2 φ, F = O r

The moduli space of sheaves Explicitly, our M is the moduli space of framed sheaves: ) F torsion-free sheaf on P 2 φ, F = O r There is an action of T = C C C ) r on M, where the first two factors act on P 2 and the third factor acts on φ.

The moduli space of sheaves Explicitly, our M is the moduli space of framed sheaves: ) F torsion-free sheaf on P 2 φ, F = O r There is an action of T = C C C ) r on M, where the first two factors act on P 2 and the third factor acts on φ. Let K denote the equivariant K theory of M, whose elements are formal differences of T equivariant vector bundles on M.

The moduli space of sheaves Explicitly, our M is the moduli space of framed sheaves: ) F torsion-free sheaf on P 2 φ, F = O r There is an action of T = C C C ) r on M, where the first two factors act on P 2 and the third factor acts on φ. Let K denote the equivariant K theory of M, whose elements are formal differences of T equivariant vector bundles on M. Since vector bundles can be tensored with T representations, K is a module over the ring Z[q 1 ±1, q±1 2, u±1 1,..., u±1 r ].

K theory and partitions An r partition will be a collection of r usual partitions: λ = λ 1),..., λ r) ) where λ k) = λ k) 1 λ k) 2...) and we will write λ for the sum of the sizes of the λ k).

K theory and partitions An r partition will be a collection of r usual partitions: λ = λ 1),..., λ r) ) where λ k) = λ k) 1 λ k) 2...) and we will write λ for the sum of the sizes of the λ k). T fixed points of M are indexed by r partitions, specifically: F λ = I λ 1)... I λ r) where I λ C[x, y] is the monomial ideal of shape λ.

K theory and partitions An r partition will be a collection of r usual partitions: λ = λ 1),..., λ r) ) where λ k) = λ k) 1 λ k) 2...) and we will write λ for the sum of the sizes of the λ k). T fixed points of M are indexed by r partitions, specifically: F λ = I λ 1)... I λ r) where I λ C[x, y] is the monomial ideal of shape λ. A convenient basis of a localization of) K is given by: ) λ = skyscraper skeaf of F λ K

K theory and partitions An r partition will be a collection of r usual partitions: λ = λ 1),..., λ r) ) where λ k) = λ k) 1 λ k) 2...) and we will write λ for the sum of the sizes of the λ k). T fixed points of M are indexed by r partitions, specifically: F λ = I λ 1)... I λ r) where I λ C[x, y] is the monomial ideal of shape λ. A convenient basis of a localization of) K is given by: ) λ = skyscraper skeaf of F λ K Given a box at coordinates i, j) in the Young diagram of the constituent partition λ k) of an r partition λ, we call: z = u k q i 1q j 2 the weight of.

The correspondences The operator Ey) is multiplication by the exterior algebra of the universal sheaf, and therefore its matrix coefficients are: λ Ey) µ = δ µ λ r i=1 1 u ) i ζ y λ z y )

The correspondences The operator Ey) is multiplication by the exterior algebra of the universal sheaf, and therefore its matrix coefficients are: λ Ey) µ = δ µ λ r i=1 1 u ) i ζ y λ z For any d 1, let us consider the following locus: Z d = {F 0... F d sheaves, x A 2, s.t. F k 1 /F k = Ox k} y )

The correspondences The operator Ey) is multiplication by the exterior algebra of the universal sheaf, and therefore its matrix coefficients are: λ Ey) µ = δ µ λ r i=1 1 u ) i ζ y λ z For any d 1, let us consider the following locus: Z d = {F 0... F d sheaves, x A 2, s.t. F k 1 /F k = Ox k} y ) This space comes endowed with maps π d), πd) + : Z d M that remember only F 0 and F d, respectively,

The correspondences The operator Ey) is multiplication by the exterior algebra of the universal sheaf, and therefore its matrix coefficients are: λ Ey) µ = δ µ λ r i=1 1 u ) i ζ y λ z For any d 1, let us consider the following locus: Z d = {F 0... F d sheaves, x A 2, s.t. F k 1 /F k = Ox k} y ) This space comes endowed with maps π d), πd) + : Z d M that remember only F 0 and F d, respectively, and with line bundles L 1,..., L d on Z d that keep track of the length one quotients F 0 /F 1,..., F d 1 /F d.

The operators Lx, y) and Ux, y) are adjoint, so let s focus on the first one: ) Lx, y) := π d) x d + 1 y π d) : K K[[x, y 1 ]] L 1 d=0

The operators Lx, y) and Ux, y) are adjoint, so let s focus on the first one: ) Lx, y) := π d) x d + 1 y π d) : K K[[x, y 1 ]] L 1 d=0 Therefore, the matrix coefficients of Lx, y) are given by: λ Lx, y) µ = 1 i<j λ\µ ζ T a standard Young tableau of shape λ\µ zj z i ) λ\µ i=1 µ ζ zi z x λ\µ 1 y z 1 ) λ\µ 1 i=1 ) r j=1 1 z ) iq u j ) 1 qz i z i+1

The operators Lx, y) and Ux, y) are adjoint, so let s focus on the first one: ) Lx, y) := π d) x d + 1 y π d) : K K[[x, y 1 ]] L 1 d=0 Therefore, the matrix coefficients of Lx, y) are given by: λ Lx, y) µ = 1 i<j λ\µ ζ T a standard Young tableau of shape λ\µ zj z i ) λ\µ i=1 µ ζ zi z x λ\µ 1 y z 1 ) λ\µ 1 i=1 ) r j=1 1 z ) iq u j ) 1 qz i z i+1 where we recall that a standard Young tableau is a labeling of the boxes of λ\µ with the numbers 1, 2,... such that the numbers increase as we go up and to the right. Also z i = z i.

Replacing A 2 with a general surface S Up to some conjectures, everything can be made sense of when our sheaves live over a general smooth surface S.

Replacing A 2 with a general surface S Up to some conjectures, everything can be made sense of when our sheaves live over a general smooth surface S. The moduli space M will parametrize stable sheaves on S.

Replacing A 2 with a general surface S Up to some conjectures, everything can be made sense of when our sheaves live over a general smooth surface S. The moduli space M will parametrize stable sheaves on S. The q W algebra that acts on K M still has generators W d,k, but the structure constants Z[q ±1 ] now depend on: 1, q±1 2 {q 1, q 2 } = Chern roots of the cotangent bundle of S

Replacing A 2 with a general surface S Up to some conjectures, everything can be made sense of when our sheaves live over a general smooth surface S. The moduli space M will parametrize stable sheaves on S. The q W algebra that acts on K M still has generators W d,k, but the structure constants Z[q ±1 ] now depend on: 1, q±1 2 {q 1, q 2 } = Chern roots of the cotangent bundle of S To be completely precise, the generators W d,k q W r will give rise not to endomorphisms of K M, but to maps K M K M S

Replacing A 2 with a general surface S Up to some conjectures, everything can be made sense of when our sheaves live over a general smooth surface S. The moduli space M will parametrize stable sheaves on S. The q W algebra that acts on K M still has generators W d,k, but the structure constants Z[q ±1 ] now depend on: 1, q±1 2 {q 1, q 2 } = Chern roots of the cotangent bundle of S To be completely precise, the generators W d,k q W r will give rise not to endomorphisms of K M, but to maps K M K M S Finally, the parameter m that defines the Ext bundle and the operator A m, will now be a K theory class on S.