Heriot-Watt University. AGT relations for abelian quiver gauge theories on ALE spaces Pedrini, Mattia; Sala, Francesco; Szabo, Richard Joseph

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1 Heriot-Watt University Heriot-Watt University Research Gateway AGT relations for abelian quiver gauge theories on ALE spaces Pedrini, Mattia; Sala, Francesco; Szabo, Richard Joseph Published in: Journal of Geometry and Physics DOI: /j.geomphys Publication date: 016 Document Version Peer reviewed version Link to publication in Heriot-Watt University Research Portal Citation for published version APA: Pedrini, M., Sala, F., & Szabo, R. J AGT relations for abelian quiver gauge theories on ALE spaces. Journal of Geometry and Physics, 103, DOI: /j.geomphys General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

2 Accepted Manuscript AGT relations for abelian quiver gauge theories on ALE spaces Mattia Pedrini, Francesco Sala, Richard J. Szabo PII: S X DOI: Reference: GEOPHY 678 To appear in: Journal of Geometry and Physics Received date: 3 June 014 Revised date: 3 November 015 Accepted date: 14 January 016 Please cite this article as: M. Pedrini, F. Sala, R.J. Szabo, AGT relations for abelian quiver gauge theories on ALE spaces, Journal of Geometry and Physics 016, This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

3 Manuscript EMPG AGT RELATIONS FOR ABELIAN QUIVER GAUGE THEORIES ON ALE SPACES MATTIA PEDRINI, FRANCESCO SALA and RICHARD J. SZABO Scuola Internazionale Superiore di Studi Avanzati SISSA, Via Bonomea 65, Trieste, Italia; Istituto Nazionale di Fisica Nucleare, Sezione di Trieste Department of Mathematics, The University of Western Ontario, Middlesex College, London N6A 5B7, Ontario, Canada; Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, United Kingdom; Maxwell Institute for Mathematical Sciences, Edinburgh, United Kingdom; The Tait Institute, Edinburgh, United Kingdom ABSTRACT. We construct level one dominant representations of the affine Kac-Moody algebra ĝl k on the equivariant cohomology groups of moduli spaces of rank one framed sheaves on the orbifold compactification of the minimal resolution X k of the A k 1 toric singularity C /Z k. We show that the direct sum of the fundamental classes of these moduli spaces is a Whittaker vector for ĝl k, which proves the AGT correspondence for pure N = U1 gauge theory on X k. We consider Carlsson-Okounkov type Ext-bundles over products of the moduli spaces and use their Euler classes to define vertex operators. Under the decomposition ĝl k h ŝl k, these vertex operators decompose as products of bosonic exponentials associated to the Heisenberg algebra h and primary fields of ŝl k. We use these operators to prove the AGT correspondence for N = superconformal abelian quiver gauge theories on X k. Date: November Mathematics Subject Classification: 14D0, 14D1, 14J80, 81T13, 81T60 Keywords: framed sheaves, ALE spaces, Kac-Moody algebras, vertex operators, supersymmetric gauge theories, AGT relations, conformal field theories mattia.pedro@gmail.com, salafra83@gmail.com, R.J.Szabo@hw.ac.uk

4 Contents 1 Introduction and summary AGT relations and ALE spaces Summary of results Outline Acknowledgements Combinatorial preliminaries 9.1 Partitions and Young tableaux Symmetric functions Infinite-dimensional Lie algebras Heisenberg algebras Affine algebra of type Âk Frenkel-Kac construction AGT relations on R Equivariant cohomology of Hilb n C Heisenberg algebra Vertex operators Integrals of motion N = gauge theory Quiver gauge theories  r theories A r theories Moduli spaces of framed sheaves Orbifold compactification of X k Rank one framed sheaves Equivariant cohomology Representations of ĝl k Overview Equivariant cohomology of Hilb n X k Heisenberg algebras Dominant representation of ĝl k on W j Chiral vertex operators for ĝl k Ext-bundles and bifundamental hypermultiplets Vertex operators and primary fields Integrals of motion N = quiver gauge theories on X k 51

5 8.1 N = gauge theory Quiver gauge theories  r theories A r theories A Virasoro primary fields 59 B Edge contributions 63 1 Introduction and summary 1.1 AGT relations and ALE spaces In this paper we study a new occurrence of the deep relations between the moduli theory of sheaves and the representation theory of affine/vertex algebras. We are particularly interested in the kind of relations which come from gauge theory considerations. An important example of these relations is the AGT correspondence for gauge theories on R 4 : in 3] Alday, Gaiotto and Tachikawa conjectured a relation between the instanton partition functions of N = supersymmetric quiver gauge theories on R 4 and the conformal blocks of two-dimensional A r 1 Toda conformal field theories see also 6, 4]; this conjecture has been explicitly confirmed in some special cases, see e.g. 41,, 59, 1]. From a mathematical perspective, this correspondence implies: 1 the existence of a representation of the W-algebra Wgl r on the equivariant cohomology of the moduli spaces Mr, n of framed sheaves on the projective plane P of rank r and second Chern class n such that the latter is isomorphic to a Verma module of Wgl r ; the fundamental classes of Mr, n give a Whittaker vector of Wgl r pure gauge theory; 3 the Ext vertex operator is related to a certain intertwiner of Wgl r under the isomorphism stated in 1 quiver gauge theory. The instances 1 and were proved by Schiffmann and Vasserot 56], and independently by Maulik and Okounkov 40]. For r = 1, the moduli space M1, n is isomorphic to the Hilbert scheme of n points on C and Wgl 1 is the W-algebra associated with an infinite-dimensional Heisenberg algebra; the AGT correspondence for pure U1 gauge theory reduces to the famous result of Nakajima 46, 47] in the equivariant case 60, 36, 44]. Presently, 3 has been proved only in the rank one case 18] and in the rank two case 17, 49]. In this paper we are interested in the AGT correspondence for N = quiver gauge theories on ALE spaces associated with the Dynkin diagram of type A k 1 for k. The corresponding instanton partition functions are defined in terms of equivariant cohomology classes over Nakajima quiver varieties of type the affine Dynkin diagram Âk 1. These quiver varieties depend on a real stability parameter ξ R, which lives in an open subset of R k having a chambers decomposition: if two real stability parameters belong to the same chamber, the corresponding quiver varieties are equivariantly isomorphic; otherwise, the corresponding quiver varieties are only C -diffeomorphic. Therefore, the pure gauge theories partition functions should be all nontrivially equivalent, while the partition functions for quiver gauge theories should satisfy wall-crossing formulas cf. 7, 31]. By looking at instanton partition functions of pure gauge theories associated with moduli spaces of Z k -equivariant framed sheaves on P which are quiver varieties depending on a so-called level zero chamber, the authors of 9, 53, 6] conjectured an extension of the AGT correspondence in the A-type ALE case as a relation between instanton partition functions of N = quiver gauge 3

6 theories and conformal blocks of Toda-like conformal field theories with Z k parafermionic symmetry. In particular, the pertinent algebra to consider in this case is the coset Ar, k := ĝl N ĝl N k acting at level r, where N is related to the equivariant parameters. For r = 1 the algebra A1, k is simply ĝl k acting at level one. In general, Ar, k is isomorphic to the direct sum of the affine Lie algebra ĝl k acting at level r and the Z k -parafermionic Wgl r -algebra. Checks of the conjecture has been done 61, 30] by using partition functions of pure gauge theories associated with moduli spaces of Z k -equivariant framed sheaves on P. In 10, 11] the authors studied in details N = quiver gauge theories on the minimal resolution X of the Kleinian singularity C /Z and provided evidences for the conjecture: in this case, the quiver variety depends on a so-called level infinity chamber and corresponds to moduli spaces of framed sheaves on a suitable stacky compactification of X. In the k = case, a comparison of these approaches using different stability chambers is done in 5]; further speculations in the arbitrary k case are in 1]. Mathematically, this correspondence should imply: 1 the existence of a representation of the coset Ar, k on the equivariant cohomology of Nakajima quiver varieties associated with the affine A-type Dynkin diagram such that the latter is isomorphic to a Verma module of Ar, k; the fundamental classes of the quiver varieties give a Whittaker vector of Ar, k pure gauge theory; 3 the Ext vertex operator is related to a certain intertwiner of Ar, k under the isomorphism stated in 1 quiver gauge theory. As pointed out in 5], different chambers should provide different realizations of the action conjectured in 1. On the other hand, the conjectural wall-crossing behavior of the instanton partition functions for quiver gauge theories 31] should be related by a similar behavior of the Ext vertex operators by varying of the stability chambers. The ALE space we consider in this paper is the minimal resolution X k of the simple Kleinian singularity C /Z k. In 14] an orbifold compactification X k of X k is constructed by adding a smooth divisor D, which lays the foundations for a new sheaf theory approach to the study of Ur instantons on X k cf. 3]. Moduli spaces of sheaves on X k framed along D are also constructed in 14]; by using these moduli spaces we have a new sheaf theory approach to the study of Nakajima quiver varieties with the stability parameter of X k and, consequently, of Ur gauge theories on ALE spaces of type A k 1 which are isomorphic to X k. In the present paper we use this new approach to study the AGT correspondence for abelian quiver gauge theories on X k : from a physics point of view we prove the relations between instanton partition functions and conformal blocks and from a mathematical point of view we prove 1, and Summary of results Let us now summarize our main results. Recall that the compactification X k is a two-dimensional projective toric orbifold with Deligne-Mumford torus T := C C ; the complement X k \ X k is a smooth Cartier divisor D endowed with the structure of a Z k -gerbe. There exist line bundles O D j on D, for j = 0, 1,..., k 1, endowed with unitary flat connections associated with the irreducible unitary representations of Z k. Hence by 3, Theorem 6.9] locally free sheaves on X k which are isomorphic along D to O D j correspond to U1 instantons on X k with holonomy at infinity given by the j-th irreducible unitary representation of Z k, for j = 0, 1,..., k 1. Fix j = 0, 1,..., k 1. A rank one D, O D j-framed sheaf on X k is a pair E, φ E, where E is a rank one torsion free sheaf on X k, locally free in a neighbourhood of D, and φ E : E D 4

7 O D j is an isomorphism. Let M u, n, j be the fine moduli space parameterizing isomorphism classes of rank one D, O D j-framed sheaves on X k, with first Chern class given by u Z k 1 and second Chern class n. As explained in Remark 5.11, the vector u is canonically associated with an element γ u + ω j Q + ω j, where Q is the root lattice of the Dynkin diagram of type A k 1 and ω j is the j-th fundamental weight of type A k 1. We denote by U j the set of vectors u associated with γ + ω j for some γ Q. The moduli space M u, n, j is a smooth quasi-projective variety of dimension n. On M u, n; j there is a natural T -action induced by the toric structure of X k. Let ε 1, ε be the generators of the T -equivariant cohomology of a point and consider the localized equivariant cohomology W u,j := n 0 H T M u, n, j Cε1,ε ] Cε 1, ε. Define also the total localized equivariant cohomology by summing over all vectors u U j : W j := u U j W u,j. The affine Lie algebra ĝl k acts on W j as follows see Proposition 6.4 and Proposition 6.8. Proposition. There exists a ĝl k-action on W j under which it is the j-th dominant representation of ĝl k at level one, i.e., the highest weight representation of ĝl k with fundamental weight ω j of type Âk 1. Moreover, the weight spaces of W j with respect to the ĝl k-action are the W u,j with weights γ u + ω j. The vector spaces W u,j also have a representation theoretic intepretation. Corollary Corollary 6.7. W u,j is a highest weight representation of the Virasoro algebra associated with ĝl k of conformal dimension u := 1 u C 1 u, where C is the Cartan matrix of type A k 1. The representation is constructed by using a vertex algebra approach via the Frenkel-Kac construction. A similar construction for the cohomology groups of moduli spaces of rank one torsion free sheaves over smooth projective surfaces is outlined in 47, Chapter 9]. In 4], Nagao analysed vertex algebra realizations of representations of ŝl k on the equivariant cohomology groups of Nakajima quiver varieties associated with the affine Dynkin diagram Âk 1, for an integer k, with dimension vector corresponding to the trivial holonomy at infinity j = 0; in this case the pertinent representation is the basic representation of ŝl k. In the following we describe our AGT relations, which connect together W j for j = 0, 1,..., k 1, the action of ĝl k on W j and abelian quiver gauge theories on X k. The first relation we obtain concerns the pure gauge theory. Let Z Xk ε1, ε ; q, ξ be the instanton partition function for the j pure N = U1 gauge theory on the ALE space X k with fixed holonomy at infinity given by the j-th irreducible representation of Z k see Section 8.1. It has the following representation theoretic characterization. Theorem AGT relation for pure N = U1 gauge theory. The Gaiotto state G j := ] M u, n, j T u U j n 0 5

8 is a Whittaker vector for ĝl k. Moreover, the weighted norm of the weighted Gaiotto state G j q, ξ := q n+ 1 u C 1 u ξ C 1 u M u, n, j ] T is exactly Z Xk ε1, ε ; q, ξ j. u U j n 0 We also consider N = superconformal quiver gauge theories with gauge group U1 r+1 for some r 0. By the ADE classification in 5, Chapter 3] the admissible quivers in this case are the linear quivers of the finite-dimensional A r -type Dynkin diagram and the cyclic quivers of the affine  r -type extended Dynkin diagram. In order to state AGT relations in these cases, we introduce Ext vertex operators 18, 17, 49]. Consider the element E µ K M u 1, n 1, j 1 M u, n, j whose fibre over a point E, φ E ], E, φ E ] is Eµ E,φ E ], E,φ E ] = Ext1 E, E O Xk µ O Xk D, where O Xk µ is the trivial line bundle on X k on which the torus T µ = C acts by scaling the fibres with HT µ pt; C = Cµ]. By using the Euler class of E µ we define a vertex operator V µ x, z End k 1 j=0 W j z ± 1, x ± 1 1,..., x ± 1 k 1 ]] see Section 7.1. Under the decomposition ĝl k = h ŝl k, we have the following characterization of V µ x, z in terms of vertex operators depending respectively on h and ŝl k. Theorem Theorem 7.6. The vertex operator V µ x, z can be expressed in the form V µ x, z = V µ, µ+ε 1 +ε z k ε1 ε k ε1 ε k 1 j 1,j =0 u 1 U j1, u U j Vµ v 1, x, z z u u1 exp log z c γ 1 exp γ1 W u1,,j1 where V α,β z denotes a generalized bosonic exponential associated with the Heisenberg algebra h see Definition 3.3, exp log z c γ 1 exp γ1 is the vertex operator on Wj1 defined in Equation 7.4, and V µ v 1, x, z is the primary field 7.5 of the Virasoro algebra associated with ŝl k with conformal dimension u u 1 = 1 v 1 C v 1, where v 1 := C 1 u u 1 for j 1, j = 0, 1,..., k 1 and u 1 U j1, u U j. For j 1, j = 0, 1,..., k 1 denote by V j 1,j µ x, z the restriction of the vertex operator V µ x, z to HomW j1, W j z ± 1, x ± 1 1,..., x ± 1 k 1 ]]. Let ZÂr X ε1 k, ε, µ; q, ξ be the instanton partition function for the N = superconformal j U1 r+1 quiver gauge theory of type Âr with holonomy at infinity associated with j := j 0, j 1,..., j r, topological couplings q υ C and ξ υ C k 1 for υ = 0, 1,..., r, and masses µ := µ 0, µ 1,..., µ r. We prove the following AGT relation. Theorem AGT relation for N = U1 r+1 quiver gauge theory of type Âr. The partition function of the Âr-theory on X k is given by ZÂr X k ε1, ε, µ; q, ξ j = Tr W j0 q L 0 ξ C 1 h r υ=0 V jυ,j υ+1 µ υ x υ, z υ δ conf υ,υ+1, 6

9 where q := q 0 q 1 q r, ξ i := ξ 0 i ξ 1 i ξ r i, z υ := z 0 q 1 q υ, and x υ i := x 0 i ξ 1 i ξ υ i for υ = 1,..., r and i = 1,..., k 1. Here L 0 is the Virasoro energy operator associated to ĝl k, h = h1,..., h k 1 are the generators of the Cartan subalgebra of sl k, and δυ,υ+1 conf is the conformal restriction operator defined in Equation 8.6. We also get a characterization of ZÂr X ε1 k, ε, µ; q, ξ j function on C and a part depending only on ŝl k. in terms of the corresponding partition Corollary. Let V µ v 1, x, z := z u u1 Vµ v 1, x, z exp log z c γ 1 exp γ1. Then we have ZÂr X ε1 k, ε, µ; q, ξ j = ZÂr ε C 1, ε, µ; q 1 k q k ηq 1 k 1 r Tr V ωj0 q Lŝl k 0 ξ C 1 h υ=0 u υ U conf jυ V µυ v υ,υ+1, x υ, z υ W uυ,jυ, where ηq is the Dedekind function, V ω j0 is the j 0 -th dominant representation of ŝl k and U conf j υ is the subset of U jυ defined in Equation 8.4. Let Z Ar X ε1 k, ε, µ; q, ξ be the instanton partition function for the N = superconformal j U1 r+1 quiver gauge theory of type A r with holonomy at infinity associated with j := j 0, j 1,..., j r. We also prove the following AGT relation. Theorem AGT relation for N = U1 r+1 quiver gauge theory of type A r. The partition function of the A r -theory on X k is given by Z Ar X ε1 k, ε, µ; q, ξ j r = 0 conf, V µ0 x 0, z 0 υ=1 Vµ jυ 1,jυ υ x υ, z υ δυ 1,υ conf V µr+1 x r+1, z r+1 0 conf k 1, j=0 W j where z υ := z 0 q 0 q 1 q υ and x υ i := x 0 i ξ 0 i ξ 1 i ξ υ 1 i for υ = 1,..., r + 1, i = 1,..., k 1, and 0 conf := r υ=0 δconf 0,υ, 0 ] with, 0 ] the vacuum vector of the fixed point basis of k 1 j=0 W j. Denote by V the direct sum of the k level one dominant representations of ŝl k. Similarly to before, we have the following characterization. Corollary. We have Z Ar X k ε1, ε, µ; q, ξ j = ZAr C ε 1, ε, µ; q 1 k 0 conf, k 1 j 0,j 0 =0 u 0 U j0, u 0 U j 0 r υ=1 V µ0 v 0,0, x 0, z 0 W u0,j 0 V µυ v υ 1,υ, x υ, z υ W uυ,jυ u υu conf jυ 7

10 k 1 j r+1,j r+1 =0 u 1 U j1, u 1 U j 1 V µr+1 v 1,1, x r+1, z r+1 W u1,j 1 0 conf V. Another important aspect of the AGT correspondence that we address in this paper is the relation of our construction with quantum integrable systems. In particular, for any j = 0, 1,..., k 1 we define an infinite system of commuting operators which are diagonalized in the fixed point basis of W j ; geometrically these operators correspond to multiplication by equivariant cohomology classes see Section 7.3. The eigenvalues of these operators with respect to this basis can be decomposed into a part associated with k non-interacting Calogero-Sutherland models and a part which can be interpreted as particular matrix elements of the vertex operators V µ x, z in highest weight vectors of ĝl k. The significance of this property is that this special orthogonal basis manifests itself in the special integrable structure of the two-dimensional conformal field theory and yields completely factorized matrix elements of composite vertex operators explicitly in terms of simple rational functions of the basic parameters, which from the gauge theory perspective represent the contributions of bifundamental matter fields. The study of the AGT relation for pure N = U1 gauge theories and the problem of constructing commuting operators associated with ĝl k is also addressed in 8] from another point of view: there they consider the conformal limit of the Ding-Iohara algebra, depending on parameters q, t, for q, t approaching a primitive k-th root of unity and relate the representation theory of this limit to the AGT correspondence. However, their point of view is completely algebraic, so unfortunately it is not clear to us how to geometrically construct the action of the conformal limit on the equivariant cohomology groups. 1.3 Outline This paper is structured as follows. In Section we briefly recall the relevant combinatorial notions that we use in this paper. In Section 3 we collect preliminary material on Heisenberg algebras and affine Lie algebras of type Âk 1, giving particular attention to the Frenkel-Kac construction of level one dominant representations of ŝl k and ĝl k. In Section 4 we review the AGT relations for N = superconformal abelian quiver gauge theories on R 4. In Section 5 we briefly recall the construction of the orbifold compactification X k and of moduli spaces of framed sheaves on X k from 14]. Section 6 addresses the construction of the action of ĝl k on W j for j = 0, 1,..., k 1: we perform a vertex algebra construction of the representation by using the Frenkel-Kac theorem. In Section 7 we define the virtual bundle E µ and the vertex operator V µ x, z, and we characterize it in terms of vertex operators of an infinite-dimensional Heisenberg algebra h and primary fields of ŝl k under the decomposition ĝl k = h ŝl k; moreover, we geometrically define an infinite system of commuting operators. In Section 8 we prove our AGT relations, and furthermore provide expressions for our partition functions in terms of the corresponding partition functions on C and a part depending only on ŝl k. The paper concludes with two Appendices containing some technical details of the constructions from the main text: in Appendix A we give the proof that the vertex operator V µ v 1, x, z is a primary field, while in Appendix B we recall the expressions from 14] for the edge factors which appear in the definition of V µ v 1, x, z as well as in the eigenvalues of the integrals of motion. 8

11 1.4 Acknowledgements We are grateful to M. Bershtein, A. Konechny, O. Schiffmann and E. Vasserot for helpful discussions. Also, we are indebted to the anonymous referee, whose remarks helped to improve the paper. This work was supported in part by PRIN Geometria delle varietà algebriche, by GNSAGA-INDAM, by the Grant RPG-404 from the Leverhulme Trust, and by the Consolidated Grant ST/J000310/1 from the UK Science and Technology Facilities Council. The bulk of this paper was written while the authors were staying at Heriot-Watt University in Edinburgh and at SISSA in Trieste. The last draft of the paper was written while the first and second authors were staying at IHP in Paris under the auspices of the RIP program. We thank these institutions for their hospitality and support. Combinatorial preliminaries.1 Partitions and Young tableaux A partition of a positive integer n is a nonincreasing sequence of positive numbers λ = λ 1 λ λ l > 0 such that λ := l a=1 λ a = n. We call l = lλ the length of the partition λ. Another description of a partition λ of n uses the notation λ = 1 m 1 m, where m i = #{a N λ a = i} with i i m i = n and i m i = lλ. On the set of all partitions there is a natural partial ordering called dominance ordering: For two partitions µ and λ, we write µ λ if and only if µ = λ and µ µ a λ λ a for all a 1. We write µ < λ if and only if µ λ and µ λ. One can associate with a partition λ its Young tableau, which is the set Y λ = {a, b N 1 a lλ, 1 b λ a }. Then λ a is the length of the a-th column of Y λ ; we write Y λ = λ for the weight of the Young tableau Y λ. We shall identify a partition λ with its Young tableau Y λ. For a partition λ, the transpose partition λ is the partition whose Young tableau is Y λ := {b, a N a, b Y λ }. The elements of a Young tableau Y are called the nodes of Y. For a node s = a, b Y, the arm length of s is the quantity As := A Y s = λ a b and the leg length of s the quantity Ls := L Y s = λ b a. The arm colength and leg colength are respectively given by A s := A Y s = b 1 and L s := L Y s = a 1.. Symmetric functions Here we recall some preliminaries about the theory of symmetric functions in infinitely many variables which we shall use later on. Our main reference is 37]. Let F be a field of characteristic zero. The algebra of symmetric polynomials in N variables is the subspace Λ F,N of Fx 1,..., x N ] which is invariant under the action of the group of permutations σ N on N letters. Then Λ F,N is a graded ring: Λ F,N = n 0 Λn F,N, where Λn F,N is the ring of homogeneous symmetric polynomials in N variables of degree n together with the zero polynomial. For any M > N there are morphisms ρ MN : Λ F,M Λ F,N that map the variables x N+1,..., x M to zero. They preserve the grading, and hence we can define ρ n MN : Λn F,M Λn F,N ; this allows us to define the inverse limits Λ n F := lim Λ n F,N, N 9

12 and the algebra of symmetric functions in infinitely many variables as Λ F := n 0 Λn F. In the following when no confusion is possible we will denote Λ F resp. Λ n F simply by Λ resp. Λn. Now we introduce a basis for Λ. For this, we start by defining a basis in Λ N. Let µ = µ 1,..., µ t be a partition with t N, and define the polynomial m µ x 1,..., x N = x µτ1 1 x µ τn N, τ σ N where we set µ j = 0 for j = t + 1,..., N. The polynomial m µ is symmetric, and the set of m µ for all partitions µ with µ N is a basis of Λ N. Then the set of m µ, for all partitions µ with µ N and i µ i = n, is a basis of Λ n N. Since for M > N t we have ρn MN m µx 1,..., x M = m µ x 1,..., x N, by using the definition of inverse limit we can define the monomial symmetric functions m µ. By varying over the partitions µ of n, these functions form a basis for Λ n. Next we define the n-th power sum symmetric function p n as p n := m n = i x n i. The set consisting of symmetric functions p µ := p µ1... p µt, for all partitions µ = µ 1,..., µ t, is another basis of Λ. We now set F = C throughout and we fix a parameter β C though everything works for any field extension C F and β F. Define an inner product, β on the vector space Λ Qβ with respect to which the basis of power sum symmetric functions p λ x are orthogonal with the normalization p λ, p µ β = δ λ,µ z λ β lλ,.1 where δ λ,µ := a δ λ a,µ a and This is called the Jack inner product. z λ := j 1 j m j m j!. Definition.. The monic forms of the Jack functions J λ x; β 1 Λ Qβ for x = x 1, x,... are uniquely defined by the following two conditions 37]: i Triangular expansion in the basis m µ x of monomial symmetric functions: J λ x; β 1 = m λ x + µ<λ ψ λ,µ β m µ x with ψ λ,µ β Qβ. ii Orthogonality: J λ, J µ β = δ λ,µ β Ls + As + 1 β Ls As. s Y λ Lemma.3. For any integer n 1 we have p 1 n = n! λ =n 1 β Ls + As + 1 J λ. s Y λ 10

13 Proof. The assertion follows straightforwardly from 58, Proposition.3 and Theorem 5.8] after normalizing our Jack functions: the Jack functions considered in 58] are given by J λ = β λ ] β Ls As Jλ, s Y λ where the normalization factor is computed by using 58, Theorem 5.6]. 3 Infinite-dimensional Lie algebras 3.1 Heisenberg algebras In this section we recall the representation theory of Heisenberg algebras and the affine Lie algebras ŝl k. Since the Lie algebra gl k coincides with F id sl k, as a by-product we get the representation theory of ĝl k. Let C F be a field extension of C. Let L be a lattice, i.e., a free abelian group of finite rank d equipped with a symmetric nondegenerate bilinear form, L : L L Z. Fix a basis γ 1,..., γ d of L. Definition 3.1. The lattice Heisenberg algebra h F,L associated with L is the infinite-dimensional Lie algebra over F generated by q i m, for m Z \ {0} and i {1,..., d}, and the central element c satisfying the relations { q i m, c ] = 0 for m Z \ {0}, i {1,..., d}, q i m, q j ] 3. n = m δm, n γ i, γ j L c for m, n Z \ {0}, i, j {1,..., d}. For any element v L we define the element q v m h F,L by linearity, with q i m := qγ i m. Set h + F,L := m>0 d Fq i m and h F,L := m<0 d Fq i m. Let us denote by Uh F,L resp. Uh ± F,L the universal enveloping algebra of h F,L resp. h ± F,L, i.e., the unital associative algebra over F generated by h F,L resp. h ± F,L. We introduce some terminology similar to that used in 5, Section 5..5]. Definition 3.3. For v L, define free bosonic fields as the elements ϕ v z := m=1 z m m qv m and ϕ v + z := in h F,L z]] and h+ F,L z 1 ]], respectively. For α, β F, define the generalized bosonic exponential m=1 z m m V v α,β z := exp α ϕ v z exp β ϕ v + z =: : exp α ϕ z + β ϕ + z : in h F,L z, z 1 ]], where the symbol : : denotes normal ordering with respect to the decomposition h F,L = h F,L h+ F,L, i.e., all negative generators qv m are moved to the left of all positive generators q v m for m > 0. When β = α, we call Vv α, α z a normal-ordered bosonic exponential. 11 qv m

14 Remark 3.4. The bosonic exponentials are vertex operators, i.e., they are uniquely characterized by their commutation relations in the Heisenberg algebra h F,L : For v, v L one has { α v, v L z m V v α,β z for m > 0, q v m, V v α,β z] = β v, v L z m V v α,β z for m < 0. The compositions of vertex operators V v 1 α 1,β 1 z 1 V vn α n,β n z n in h F,L z 1 ± 1,..., z n ± 1 ]] can be easily calculated as n V v i α i,β i z i = 1 z i αi β j v i,v j L n : V v i z α i,β i z i :, 3.5 j 1 j<i n where the factors 1 z i z j α i β j v i,v j L are understood as formal power series in z i z j. Remark 3.6. When v = γ i for i = 1,..., d, we simply denote ϕ i ± z := ϕγ i ± z; if d = 1, we further simply write ϕ ± z. We use analogous notation for the generalized free boson exponentials. Example 3.7. Consider the lattice L := Z k with the symmetric nondegenerate bilinear form v, w L = k v i w i. In this case h F,L is called the Heisenberg algebra of rank k over F, and we denote it by h k F. It is generated by elements pi m, m Z \{0}, i = 1,..., k, and the central element c satisfying the relations 3. with γ i, γ j L = δ ij. When k = 1, h F,L is simply the infinite-dimensional Heisenberg algebra h F over the field F. Example 3.8. Fix an integer k and let Q be the root lattice of type A k 1 endowed with the standard bilinear form, Q see Remark 3.16 below. Let h F,Q be the lattice Heisenberg algebra over F associated to Q; we call h F,Q the Heisenberg algebra of type A k 1 over F. It can be realized as the Lie algebra over F generated by q i m for m Z \ {0}, i = 1,..., k 1, and the central element c satisfying the relations { q i m, c ] = 0 for m Z \ {0}, i = 1,..., k 1, q i m, q j ] n = m δm, n C ij c for m Z \ {0}, i, j = 1,..., k 1, where C = C ij is the Cartan matrix of type A k Virasoro generators We construct the Viraroso algebra associated with the Heisenberg algebra h F. Define elements L h 0 = m=1 q m q m and L h n = 1 m Z q m q m+n for n Z \ {0} in the completion of the enveloping algebra Uh F, where we set q 0 := 0. They satisfy the relations L h n, L h m] = n m L h n+m + n n 1 δ m+n,0 c, 1 hence c and L h n with n Z generate a Virasoro algebra Vir F over F. Remark 3.9. It is well-known see Appendix A that the generalized bosonic exponential V α,β z is a primary field of the Virasoro algebra Vir F generated by L h n with conformal dimension α, β = 1 α β, i.e., it satisfies the commutation relations L h n, V α,β z ] = z n z z + α, β n + 1 V α,β z. 1

15 3.1. Fock space We are interested in a special type of representation of a given lattice Heisenberg algebra h F,L over F. Definition Let W be the trivial representation of h + F,L i.e., the one-dimensional F-vector space with trivial h + F,L -action. The Fock space representation of the Heisenberg algebra h F,L is the induced representation F F,L := h F,L h + W. F,L The Fock space is an irreducible highest weight representation whereby any element w 0 W is a highest weight vector, i.e., h + F,L annihilates w 0 and the elements in W of the form q v m 1 q v m l w 0 generate F F,L for v L, l 1 and m i 1 for i = 1,..., l. Example For the Heisenberg algebra h F, the Fock space F F is isomorphic to the polynomial algebra Λ F = F p 1, p,...] in the power sum symmetric functions introduced in Section.. In this realization, the actions of the generators are given for m > 0 by p m f := p m f, p m f := m f p m and c f := f 3.1 for any f Λ F. Example The Fock space FF k of the rank k Heisenberg algebra hk F can be realized as the tensor product of k copies of the polynomial algebra Λ F : F k F Λ k F. In this realization, the action of the generators p i m is obvious: each copy of the Heisenberg algebra generated by p i m for m Z \ {0} acts on the i-th factor Λ F as in Equation Whittaker vectors We give the definition of Whittaker vector for Heisenberg algebras following 19, Section 3]; in conformal field theory it has the meaning of a coherent state. Definition Let χ: Uh + F,L F be an algebra homomorphism such that χ h + 0, and let V F,L be a Uh F,L -module. A nonzero vector w V is called a Whittaker vector of type χ if η w = χη w for all η Uh + F,L. Remark By 19, Proposition 10], if w, w are Whittaker vectors of the same type χ, then w = λ w for some nonzero λ F. 3. Affine algebra of type Âk 1 Let k be an integer and let sl k := slk, F denote the finite-dimensional Lie algebra of rank k 1 over F generated in the Chevalley basis by E i, F i, H i for i = 1,..., k 1 satisfying the relations E i, F j ] = δ ij H j, H i, H j ] = 0, H i, E j ] = C ij E j, H i, F j ] = C ij F j, where C = C ij is the Cartan matrix type A k 1 see Remark 3.16 below. 13

16 An explicit realization of the generators of sl k in the algebra Mk, F of k k matrices over F is given in the following way. Let E i,j denote the k k matrix unit with 1 in the i, j entry and 0 everywhere else for i, j = 1,..., k. Define E i := E i,i+1, F i := E i+1,i and H i := E i,i E i+1,i+1 for i = 1,..., k 1. One sees immediately that E i, F i, H i satisfy the defining relations for sl k. Let us denote by t the Lie subalgebra of sl k generated by H i for i = 1,..., k 1 and by n + resp. n the Lie subalgebra of sl k generated by E i resp. F i for i = 1,..., k 1. Then there is a triangular decomposition sl k = n t n + as a direct sum of vector spaces. Remark For i = 1,..., k, define e i t by e i diaga1,..., a k = a i. The elements γ i := e i e i+1 for i = 1,..., k 1 form a basis of t. The root lattice Q is the lattice Q := k 1 Zγ i. The elements of Q are called roots, and in particular γ i are called the simple roots. The lattice of positive roots is Q + := k 1 Nγ i. Since e i corresponds to the i-th coordinate vector in Z k, there is a description of Q and Q + in Z k given by Q = { } e i e j i, j = 1,..., k and Q + = { } e i e j 1 i < j k. By setting γ i, γ j Q := γ i H j = C ij, we define a nondegenerate symmetric bilinear form, Q on Q. The fundamental weights ω i of type A k 1 are the elements of t defined by ω i H j = δ ij for i, j = 1,..., k 1. In the standard basis of Z k, they are given explicitly by ω i := i e l i k l=1 for i = 1,..., k 1. Let P := k 1 Zω i be the weight lattice. Then Q P, as γ i = k 1 j=1 C ij ω j. The set of dominant weights is P + := k 1 Nω i. There is a coset decomposition of P given by P = k 1 j=0 k l=1 e l Q + ω j, 3.17 where we set ω 0 := 0. The coroot lattice is the lattice Q := k 1 ZH i. We now introduce the Kac-Moody algebra ŝl k of type Âk 1, first via its canonical generators and then as a central extension of the loop algebra of sl k. Definition The Kac-Moody algebra ŝl k of type Âk 1 over F is the Lie algebra over F generated by e i, f i, h i for i = 0, 1,..., k 1 satisfying the relations e i, f j ] = δ ij h j, h i, h j ] = 0, h i, e j ] = Ĉij e j, h i, f j ] = Ĉij f j, where Ĉ = Ĉ ij is the Cartan matrix of the extended Dynkin diagram of type  k 1. 14

17 The matrix Ĉ is given for k 3 by Ĉ = Ĉ ij = and for k = by Ĉ = Ĉ ij =. Let us denote by t the Lie subalgebra of ŝl k generated by h i for i = 0, 1,..., k 1 and by n + resp. n the Lie subalgebra of ŝl k generated by e i resp. f i for i = 0, 1,..., k 1. Then there is a triangular decomposition ŝl k = n t n + as a direct sum of vector spaces. Now we describe the relation between sl k and ŝl k. Define in sl k the elements E 0 := E k,1, F 0 := E 1,k and H 0 := E k,k E 1,1. Consider next the loop algebra sl k := sl k Ft, t 1 ]. Set ẽ 0 := E 0 t, ẽ i := E i 1, f 0 := F 0 t 1, fi := F i 1, h 0 := H 0 1, hi := H i 1, for i = 1,..., k 1. Let us denote by c the central element of ŝl k given by c = k 1 i=0 h i. Then we can realize ŝl k as a one-dimensional central extension 0 Fc ŝl k π sl k 0, where the homomorphism π is defined by π : e i ẽ i, f i f i, h i h i, for i = 0, 1,..., k 1, and the Lie algebra structure of ŝl k is obtained through M t m, N t n ] = M, N] t m+n + m δ m, n trm N c 3.19 for every M, N sl k and m, n Z. Thus the canonical generators of ŝl k are e 0 := E 0 t, e i := E i 1, f 0 := F 0 t 1, f i := F i 1, h 0 := H c, h i := H i 1, and we can realize t as the one-dimensional extension 0 Fc t π t 0. 15

18 Remark 3.0. Let γ 0 := k 1 γ i. For i = 1,..., k 1, let e i be as in Remark 3.16; then γ 0 = e k e 1. We extend e i from t to t by setting e i c = 0. Then γ i c = 0 for i = 0, 1,..., k 1. Thus the root lattice Q of ŝl k is the lattice Q = k 1 i=0 Zγ i = Zγ 0 Q. In a similar way, one can define the lattice of positive roots and a nondegenerate symmetric bilinear form on Q. Let ω 0 be the element in t defined by ω 0 t = 0 and ω 0 c = 1. Define ω i := ω i + ω 0 for i = 1,..., k 1. We call ω 0, ω 1,..., ω k 1 the fundamental weights of type Âk 1. Set P := k 1 i=0 Z ω i. Any weight λ = k 1 i=0 λ i ω i P can be written as λ = λ + k λ ω 0, where λ P and k λ = λc = k 1 i=0 λ i is the level of λ Highest weight representations By declaring the degrees of generators deg e i = deg f i = 1 and deg h i = 0 for i = 0, 1,..., k 1, we endow ŝl k with the principal grading ŝl k = n Z ŝlk n. The principal grading of ŝl k induces a Z-grading of its universal enveloping algebra U ŝl k over F, which is written as U ŝl k = U n. Set b := t n +. Let λ be a linear form on t. We define a one-dimensional b-module Fv λ by n + v λ = 0 and h i v λ = λh i v λ for i = 0, 1,..., k 1. Consider the induced ŝl k-module Ṽ λ := U ŝl k U b Fv λ. Setting Ṽn := U n v λ, we define the principal grading Ṽ λ = n Z Ṽn. The ŝl k-module Ṽ λ contains a unique maximal proper graded ŝl k-submodule I λ. Definition 3.1. The quotient module V λ := Ṽ λ / I λ is called the highest weight representation of ŝl k at level k λ. The nonzero multiples of the image of v λ in V λ are called the highest weight vectors of V λ. The principal grading on Ṽ λ induces an N-grading n Z V λ = n 0 V n called the principal grading of V λ. 16

19 Definition 3.. The i-th dominant representation of ŝl k at level one is the highest weight representation V ω i of ŝl k for i = 0, 1,..., k 1. The module V ω 0 is also called the basic representation of ŝl k. Remark 3.3. One can define the Lie algebra ĝl k as the one-dimensional extension 0 Fc ĝl k π gl k Ft, t 1 ] 0. Since gl k = F id sl k, the representation theory of ĝl k is obtained by combining the representation theory of the Heisenberg algebra h F with that of ŝl k. For example, all highest weight representations of ĝl k are of the form F F V λ for some weight λ P. 3.. Whittaker vectors Let us denote by q i m the element H i t m for i {1,..., k 1} and m Z. By Equation 3.19, these elements satisfy q i m, q j ] n = m δm+n,0 C ij c and qm, c ] = 0, for i, j {1,..., k 1} and m, n Z. For a root γ, we denote by q γ m the element H γ t m where H γ t is defined by H, H γ Q Z R = γh for any H t. The subalgebra of ŝl k generated by q i m, for i {1,..., k 1} and m Z\{0}, and c is isomorphic to the Heisenberg algebra h F,Q. This motivates the following definition of Whittaker vector for ŝl k cf. 19, Section 6]. Definition 3.4. Let χ: Uh + F,Q F be an algebra homomorphism such that χ h + 0, and let V F,Q be a U ŝl k -module. A nonzero vector w V is called a Whittaker vector of type χ if η w = χη w for all η Uh + F,Q. 3.3 Frenkel-Kac construction Let V be a representation of h F,Q. We say that it is a level one representation if the central element c acts by the identity map. Henceforth we let V be a level one representation of h F,Q such that for any v V there exists an integer mv for which q l 1m1 q la m a v = if m i > 0 and i m i > mv. Fix an index j {0, 1,..., k 1} and consider the coset Q + ω j. Denote by FQ + ω j ] the group algebra of Q + ω j over F. For a root γ Q we define the generating function V γ, z EndV FQ + ω j ]z, z 1 ]] of operators on V FQ + ω j ] by the bosonic vertex operator V γ, z = V γ 1, 1 z explog z c + γ = exp m=1 z m m qγ m exp m=1 z m m qγ m explog z c + γ, 17

20 where explog z c + γ is the operator defined by explog z c + γ v β + ω j ] := z 1 γ,γ Q+ γ,β+ω j Q v β + γ + ω j ] for v β + ω j ] V FQ + ω j ]. Remark 3.6. Here for the operator explog z c + γ we follow the notation in 4, Section 3..1]. In the existing literature, this operator is denoted in various different ways. Let V m γ EndV FQ + ω j ] denote the operator defined by the formal Laurent series expansion V γ, z = m Z V mγ z m. We define a map ǫ: Q Q {±1} by ǫγ i, γ j = { 1, j = i, i + 1, 1, otherwise, with the properties ǫγ + γ, β = ǫγ, β ǫγ, β and ǫγ, β + β = ǫγ, β ǫγ, β. Theorem 3.7 6, Theorem 1]. Let j {0, 1,..., k 1} and let V be a level one representation of h F,Q satisfying the condition 3.5. Then the vector space V FQ + ω j ] carries a level one ŝl k -module structure given by H i 1 v β + ω j ] = γ i, β Q + δ ij v β + ωj ], H i t m v β + ω j ] = q i m v β + ω j ], E i t m v β + ω j ] = ǫγ i, β V m+δij γ i v β + ω j ], F i t m v β + ω j ] = ǫβ, γ i V m δij γ i v β + ω j ], for i {1,..., k 1} and m Z \ {0}. If V is the Fock space of h F,Q, then V FQ + ω j ] is the j-th dominant representation of ŝl k Virasoro operators Let {η i } k 1 be an orthonormal basis of the vector space Q Z R. The Virasoro algebra associated with h F,Q ŝl k has generators c and Lŝl k n for n Z defined by 6, Section.8] k 1 Lŝl k 0 = Lŝl k n = 1 m=1 k 1 m Z q η i m qη i m + 1 k 1 q η i 0, q η i m qη i m+n for n Z \ {0}. Note that distinct orthonormal bases of Q Z R give rise to the same Virasoro algebra Vir F. 18

21 4 AGT relations on R Equivariant cohomology of Hilb n C In the following we shall give a brief survey of results concerning the equivariant cohomology of the Hilbert schemes Hilb n C and representations of Heisenberg algebras thereon 43, 9, 47, 60, 36, 55, 44]. Let us consider the action of the torus T := C on the complex affine plane C given by t 1, t x, y = t 1 x, t y, and the induced T -action on the Hilbert scheme of n points Hilb n C which is the fine moduli space parameterizing zero-dimensional subschemes of C of length n; it is a smooth quasi-projective variety of dimension n. Following ], the T -fixed points of Hilb n C are zero-dimensional subschemes of C of length n supported at the origin 0 C which correspond to partitions λ of n. We shall denote by Z λ the fixed point in Hilb n C T corresponding to the partition λ of n. For i = 1, denote by t i the T -modules corresponding to the characters χ i : t 1, t T t i C, and by ε i the equivariant first Chern class of t i. Then H T pt; C = Cε 1, ε ] is the coefficient ring for the T -equivariant cohomology. The equivariant Chern character of the tangent space to Hilb n C at a fixed point Z λ is given by ch T TZλ Hilb n C = s Y λ e Ls+1 ε 1 As ε + e Ls ε 1+As+1 ε. The equivariant Euler class is therefore given by eu T TZλ Hilb n C = 1 n eu + λ eu λ, where eu + λ = Ls+1 ε1 As ε and eu λ = Ls ε1 As+1 ε. s Y λ s Y λ Remark 4.1. By 44, Corollary 3.0], eu + λ is the equivariant Euler class of the nonpositive part T 0 Z λ of the tangent space to Hilb n C at the fixed point Z λ. Let ı λ : {Z λ } Hilb n C be the inclusion morphism and define the class λ] := ı λ 1 HT 4n Hilb n C. 4. By the projection formula we get λ] µ] = δ λ,µ eu T TZλ Hilb n C λ] = 1 n δ λ,µ eu + λ eu λ λ]. Denote ı n := ı λ : Hilb n C T Hilb n C. Z λ Hilb n C T Let ı! n : H T Hilb n C T loc H T Hilb n C be the induced Gysin map, where loc H T loc := H T Cε 1,ε ] Cε 1, ε 19

22 is the localized equivariant cohomology. By the localization theorem, ı! n is an isomorphism and its inverse is given by ı! 1 ı n : A λ A eu T TZλ Hilb n C. Z λ Hilb n C T Henceforth we denote H C,n := H T Hilbn C loc. Define the bilinear form, HC,n : H C,n H C,n Cε 1, ε, 4.3 where p n is the projection of Hilb n C T to a point. A, B 1 n p! n ı! n 1A B, Remark 4.4. Our sign convention in defining the bilinear form is different from the one used e.g. in 47, 18]. We choose this convention because, under the isomorphism to be introduced later on in 4.1, the form 4.3 becomes exactly the Jack inner product.1. This convention produces various sign changes compared to previous literature. Hence every time we state that a given result coincides with what is known in the literature, the reader should keep in mind up to the sign convention we choose. Following 36, Section.] we define the distinguished classes For λ, µ partitions of n one has α λ ] := s Y λ 1 eu + λ λ] Hn T Hilb n C loc. αλ ], α µ ] eu λ = δ HC,n λ,µ 4.5 eu + λ Ls ε 1 As + 1 ε Ls β + As + 1 = δ λ,µ = δ λ,µ, Ls + 1 ε1 Asε s Y Ls + 1 β + As λ where β = ε ε Remark 4.7. In 44, Section 3v], Nakajima gives a geometric interpretation of the class α λ ]. By the localization theorem and Equation 4.5, the classes α λ ] form a Cε 1, ε -basis for the infinite-dimensional vector space H C := n 0 H C,n. Hence the symmetric bilinear form 4.3 is nondegenerate. The forms, HC,n define a symmetric bilinear form, HC : H C H C Cε 1, ε by imposing that H C,n 1 and H C,n are orthogonal for n 1 n. Then, HC is also nondegenerate. The unique partition of n = 1 is λ = 1. Let us denote by α] := α 1 ] the corresponding class. Then α], α] H = β 1. C Let us denote by D x and D y respectively the x and y axes of C. By localization, the corresponding equivariant cohomology classes in H T C loc are given by D x ] T = 0] ε 1 = 0] eu + 1 = α] and D y] T = 0] ε = β α]. 0

23 4. Heisenberg algebra Following 43, 47], for an integer m > 0 define the Hecke correspondences D x n, m := { Z, Z Hilb n+m C Hilb n C Z Z, suppi Z /I Z = {y} D x }, where I Z, I Z are the ideal sheaves corresponding to Z, Z respectively. Let q 1, q denote the projections of Hilb n+m C Hilb n C to the two factors, respectively. Define linear operators p m D x ] T EndH C by p m D x ] T A := q 1! q A D x n, m] T for A H T Hilbn C loc. We also define p m D x ] T EndH C to be the adjoint operator of p m D x ] T with respect to the inner product, HC on H C. As the class D x] T spans H T C loc over the field Cε 1, ε, we can define operators p m η EndH C for every class η H T C loc. Theorem 4.8 see 43, 44]. The linear operators p m η, for m Z\{0} and η H T C loc, satisfy the Heisenberg commutation relations pm η 1, p n η ] = m δ m, n η 1, η HC,1 id and pm η, id ] = 0. The vector space H C becomes the Fock space of the Heisenberg algebra h HC,1 modelled on H C,1 = H T C loc with the unit 0 in H 0 T Hilb0 C loc as highest weight vector. Remark 4.9. Since D x ] T = α], we have p m α] = p m D x ] T. Henceforth we denote by h C the Heisenberg algebra h HC,1, and we define so that one has the nonzero commutation relations p m := p m D x ] T for m Z \ {0}, 4.10 p m, p m ] = m β 1 id. Since D x ] T generates H T C loc over Cε 1, ε, the operators p m generate h C. Let λ = 1 m 1 m be a partition. Define p λ := i pm i i. Then pλ 0, p µ 0 HC = δ λ,µ z λ β lλ. Let us denote by Λ β the ring of symmetric functions in infinitely many variables Λ Cε1,ε over the field Cε 1, ε, equiped with the Jack inner product.1. Theorem 4.11 see 43, 36, 18]. There exists a Cε 1, ε -linear isomorphism preserving bilinear forms such that φ : H C Λ β 4.1 φp λ 0 = p λ x and φα λ ] = J λ x; β 1. Via the isomorphism φ, the operators p m act on Λ β as multiplication by p m for m < 0 and as m β 1 p m for m > 0. 1

24 4..1 Whittaker vectors We characterize a particular class of Whittaker vectors cf. Definition 3.14 which will be useful in our studies of gauge theories. Proposition Let η Cε 1, ε. In the completed Fock space n 0 H C,n, every vector of the form Gη := exp η p 1 0 is a Whittaker vector of type χ η, where the algebra homomorphism χ η : Uh + C Cε 1, ε is defined by χ η p 1 = η β 1 and χ η p n = 0 for n > 1. Proof. The statement follows from the formal expansion Gη = n=0 η n n! p 1 n with respect to the vector 0, together with the relation p m 0 = 0 for m > 0 and the identity in Uh C for m 1. p m p 1 n = n β 1 δ m,1 p 1 n 1 + p 1 n p m 4.3 Vertex operators Let T µ = C and H T µ pt; C = Cµ]. Let us denote by O C µ the trivial line bundle on C on which T µ acts by scaling the fibers. In 18], Carlsson and Okounkov define a vertex operator VL, z for any smooth quasi-projective surface X and any line bundle L on X. Here we shall describe only VO C µ, z; see 18] for a complete description of such types of vertex operators. Let Z n Hilb n C C be the universal subscheme, whose fiber over a point Z Hilb n C is the subscheme Z C itself. Consider Z i := p i3o Zni K Hilb n 1 C Hilb n C C for i = 1,, where p ij denotes the projection to the i-th and j-th factors. Define the virtual vector bundle E n 1,n µ = p 1 Z 1 + Z Z 1 Z p 3O C µ K Hilb n 1 C Hilb n C, where p 3 is the projection to C. The fibre of E n 1,n µ over Z 1, Z Hilb n 1 C T Hilb n C T is given by E n 1,n µ Z1,Z = χ O C, O C µ χ I Z1, I Z O C µ, where χe, F := i=0 1i Ext i E, F for any pair of coherent sheaves E, F on C, while its rank is rk E n 1,n µ = n1 + n. Define the operator VO C µ, z EndH C z, z 1 ]] by its matrix elements 1 n VO C µ, za 1, A H C