AGT relations for abelian quiver gauge theories on ALE spaces
EMPG–14–10
Mattia Pedrini, Francesco Sala and Richard J. Szabo
[8pt] Scuola Internazionale Superiore di Studi Avanzati (SISSA),
Via Bonomea 265, 34136 Trieste, Italia;
[3pt] Istituto Nazionale di Fisica Nucleare, Sezione di Trieste
[3pt] Department of Mathematics, The University of Western Ontario,
Middlesex College, London N6A 5B7, Ontario, Canada;
[3pt] Department of Mathematics, HeriotWatt University,
Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, United Kingdom;
[3pt] Maxwell Institute for Mathematical Sciences, Edinburgh, United Kingdom;
[3pt] The Tait Institute, Edinburgh, United Kingdom
Abstract. We construct level one dominant representations of the affine KacMoody algebra on the equivariant cohomology groups of moduli spaces of rank one framed sheaves on the orbifold compactification of the minimal resolution of the toric singularity . We show that the direct sum of the fundamental classes of these moduli spaces is a Whittaker vector for , which proves the AGT correspondence for pure gauge theory on . We consider CarlssonOkounkov type Extbundles over products of the moduli spaces and use their Euler classes to define vertex operators. Under the decomposition , these vertex operators decompose as products of bosonic exponentials associated to the Heisenberg algebra and primary fields of . We use these operators to prove the AGT correspondence for superconformal abelian quiver gauge theories on .
Contents
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 : in [3] Alday, Gaiotto and Tachikawa conjectured a relation between the instanton partition functions of supersymmetric quiver gauge theories on and the conformal blocks of twodimensional Toda conformal field theories (see also [62, 4]); this conjecture has been explicitly confirmed in some special cases, see e.g. [41, 2, 59, 1]. From a mathematical perspective, this correspondence implies: (1) the existence of a representation of the Walgebra on the equivariant cohomology of the moduli spaces of framed sheaves on the projective plane of rank and second Chern class such that the latter is isomorphic to a Verma module of ; (2) the fundamental classes of give a Whittaker vector of (pure gauge theory); (3) the Ext vertex operator is related to a certain “intertwiner” of under the isomorphism stated in (1) (quiver gauge theory). The instances (1) and (2) were proved by Schiffmann and Vasserot [56], and independently by Maulik and Okounkov [40]. For , the moduli space is isomorphic to the Hilbert scheme of points on and is the Walgebra associated with an infinitedimensional Heisenberg algebra; the AGT correspondence for pure 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 quiver gauge theories on ALE spaces associated with the Dynkin diagram of type for . The corresponding instanton partition functions are defined in terms of equivariant cohomology classes over Nakajima quiver varieties of type the affine Dynkin diagram . These quiver varieties depend on a real stability parameter , which lives in an open subset of 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 diffeomorphic. Therefore, the pure gauge theories partition functions should be all nontrivially equivalent, while the partition functions for quiver gauge theories should satisfy “wallcrossing” formulas (cf. [7, 31]).
By looking at instanton partition functions of pure gauge theories associated with moduli spaces of equivariant framed sheaves on (which are quiver varieties depending on a socalled “level zero chamber”), the authors of [9, 53, 6] conjectured an extension of the AGT correspondence in the Atype ALE case as a relation between instanton partition functions of quiver gauge theories and conformal blocks of Todalike conformal field theories with parafermionic symmetry. In particular, the pertinent algebra to consider in this case is the coset
(1.1) 
acting at level , where is related to the equivariant parameters. For the algebra is simply acting at level one. In general, is isomorphic to the direct sum of the affine Lie algebra acting at level and the parafermionic algebra. Checks of the conjecture has been done [61, 30] by using partition functions of pure gauge theories associated with moduli spaces of equivariant framed sheaves on . In [10, 11] the authors studied in details quiver gauge theories on the minimal resolution of the Kleinian singularity and provided evidences for the conjecture: in this case, the quiver variety depends on a socalled “level infinity chamber” and corresponds to moduli spaces of framed sheaves on a suitable stacky compactification of . In the case, a comparison of these approaches using different stability chambers is done in [5]; further speculations in the arbitrary case are in [12].
Mathematically, this correspondence should imply: (1) the existence of a representation of the coset on the equivariant cohomology of Nakajima quiver varieties associated with the affine Atype Dynkin diagram such that the latter is isomorphic to a Verma module of ; (2) the fundamental classes of the quiver varieties give a Whittaker vector of (pure gauge theory); (3) the Ext vertex operator is related to a certain “intertwiner” of 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 wallcrossing 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 of the simple Kleinian singularity . In [14] an orbifold compactification of is constructed by adding a smooth divisor , which lays the foundations for a new sheaf theory approach to the study of instantons on (cf. [23]). Moduli spaces of sheaves on framed along 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 and, consequently, of gauge theories on ALE spaces of type which are isomorphic to . In the present paper we use this new approach to study the AGT correspondence for abelian quiver gauge theories on : 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), (2) and (3).
1.2. Summary of results
Let us now summarize our main results. Recall that the compactification is a twodimensional projective toric orbifold with DeligneMumford torus ; the complement is a smooth Cartier divisor endowed with the structure of a gerbe. There exist line bundles on , for , endowed with unitary flat connections associated with the irreducible unitary representations of . Hence by [23, Theorem 6.9] locally free sheaves on which are isomorphic along to correspond to instantons on with holonomy at infinity given by the th irreducible unitary representation of , for .
Fix . A rank one framed sheaf on is a pair , where is a rank one torsion free sheaf on , locally free in a neighbourhood of , and is an isomorphism. Let be the fine moduli space parameterizing isomorphism classes of rank one framed sheaves on , with first Chern class given by and second Chern class . As explained in Remark 5.20, the vector is canonically associated with an element , where is the root lattice of the Dynkin diagram of type and is the th fundamental weight of type . We denote by the set of vectors associated with for some .
The moduli space is a smooth quasiprojective variety of dimension . On there is a natural action induced by the toric structure of . Let be the generators of the equivariant cohomology of a point and consider the localized equivariant cohomology
(1.2) 
Define also the total localized equivariant cohomology by summing over all vectors :
(1.3) 
The affine Lie algebra acts on as follows (see Proposition 6.57 and Proposition 6.73).
Proposition.
There exists a action on under which it is the th dominant representation of at level one, i.e., the highest weight representation of with fundamental weight of type . Moreover, the weight spaces of with respect to the action are the with weights .
The vector spaces also have a representation theoretic intepretation.
Corollary (Corollary 6.71).
is a highest weight representation of the Virasoro algebra associated with of conformal dimension , where is the Cartan matrix of type .
The representation is constructed by using a vertex algebra approach via the FrenkelKac 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 [42], Nagao analysed vertex algebra realizations of representations of on the equivariant cohomology groups of Nakajima quiver varieties associated with the affine Dynkin diagram , for an integer , with dimension vector corresponding to the trivial holonomy at infinity ; in this case the pertinent representation is the basic representation of .
In the following we describe our AGT relations, which connect together for , the action of on and abelian quiver gauge theories on . The first relation we obtain concerns the pure gauge theory. Let be the instanton partition function for the pure gauge theory on the ALE space with fixed holonomy at infinity given by the th irreducible representation of (see Section 8.1). It has the following representation theoretic characterization.
Theorem (AGT relation for pure gauge theory).
The Gaiotto state
(1.4) 
is a Whittaker vector for . Moreover, the weighted norm of the weighted Gaiotto state
(1.5) 
is exactly .
We also consider superconformal quiver gauge theories with gauge group for some . By the ADE classification in [52, Chapter 3] the admissible quivers in this case are the linear quivers of the finitedimensional type Dynkin diagram and the cyclic quivers of the affine type extended Dynkin diagram. In order to state AGT relations in these cases, we introduce Ext vertex operators [18, 17, 49]. Consider the element whose fibre over a point , is
(1.6) 
where is the trivial line bundle on on which the torus acts by scaling the fibres with . By using the Euler class of we define a vertex operator
Theorem (Theorem 7.14).
The vertex operator can be expressed in the form
(1.7) 
where denotes a generalized bosonic exponential associated with the Heisenberg algebra (see Definition 3.6),
For denote by the restriction of the vertex operator to .
Let be the instanton partition function for the superconformal quiver gauge theory of type with holonomy at infinity associated with , topological couplings and for , and masses . We prove the following AGT relation.
Theorem (AGT relation for quiver gauge theory of type ).
The partition function of the theory on is given by
(1.8) 
where , , , and for and . Here is the Virasoro energy operator associated to , are the generators of the Cartan subalgebra of , and is the conformal restriction operator defined in Equation (8.30).
We also get a characterization of in terms of the corresponding partition function on and a part depending only on .
Corollary.
Let
(1.9) 
where is the Dedekind function, is the th dominant representation of and is the subset of defined in Equation (8.23).
Let be the instanton partition function for the superconformal quiver gauge theory of type with holonomy at infinity associated with . We also prove the following AGT relation.
Theorem (AGT relation for quiver gauge theory of type ).
The partition function of the theory on is given by
(1.10) 
where and for , , and with the vacuum vector of the fixed point basis of .
Denote by the direct sum of the level one dominant representations of . Similarly to before, we have the following characterization.
Corollary.
We have
(1.11) 
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 we define an infinite system of commuting operators which are diagonalized in the fixed point basis of ; 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 noninteracting CalogeroSutherland models and a part which can be interpreted as particular matrix elements of the vertex operators in highest weight vectors of . The significance of this property is that this special orthogonal basis manifests itself in the special integrable structure of the twodimensional 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 gauge theories and the problem of constructing commuting operators associated with is also addressed in [8] from another point of view: there they consider the “conformal” limit of the DingIohara algebra, depending on parameters , for approaching a primitive 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 2 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 , giving particular attention to the FrenkelKac construction of level one dominant representations of and . In Section 4 we review the AGT relations for superconformal abelian quiver gauge theories on . In Section 5 we briefly recall the construction of the orbifold compactification and of moduli spaces of framed sheaves on from [14]. Section 6 addresses the construction of the action of on for : we perform a vertex algebra construction of the representation by using the FrenkelKac theorem. In Section 7 we define the virtual bundle and the vertex operator , and we characterize it in terms of vertex operators of an infinitedimensional Heisenberg algebra and primary fields of under the decomposition ; 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 and a part depending only on . 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 is a primary field, while in Appendix B we recall the expressions from [14] for the edge factors which appear in the definition of as well as in the eigenvalues of the integrals of motion.
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 GNSAGAINDAM, by the Grant RPG404 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 HeriotWatt 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.
2. Combinatorial preliminaries
2.1. Partitions and Young tableaux
A partition of a positive integer is a nonincreasing sequence of positive numbers such that We call the length of the partition Another description of a partition of uses the notation , where with and . On the set of all partitions there is a natural partial ordering called dominance ordering: For two partitions and , we write if and only if and for all . We write if and only if and
One can associate with a partition its Young tableau, which is the set is the length of the th column of ; we write for the weight of the Young tableau . We shall identify a partition with its Young tableau . For a partition , the transpose partition is the partition whose Young tableau is . . Then
The elements of a Young tableau are called the nodes of . For a node , the arm length of is the quantity and the leg length of the quantity . The arm colength and leg colength are respectively given by and .
2.2. 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 be a field of characteristic zero. The algebra of symmetric polynomials in variables is the subspace of which is invariant under the action of the group of permutations on letters. Then is a graded ring: , where is the ring of homogeneous symmetric polynomials in variables of degree (together with the zero polynomial).
For any there are morphisms that map the variables to zero. They preserve the grading, and hence we can define ; this allows us to define the inverse limits
(2.1) 
and the algebra of symmetric functions in infinitely many variables as In the following when no confusion is possible we will denote (resp. ) simply by (resp. ).
Now we introduce a basis for For this, we start by defining a basis in Let be a partition with , and define the polynomial
(2.2) 
where we set for . The polynomial is symmetric, and the set of for all partitions with is a basis of Then the set of , for all partitions with and , is a basis of . Since for we have , by using the definition of inverse limit we can define the monomial symmetric functions By varying over the partitions of , these functions form a basis for .
Next we define the th power sum symmetric function as
(2.3) 
The set consisting of symmetric functions , for all partitions , is another basis of .
We now set throughout and we fix a parameter (though everything works for any field extension and ). Define an inner product on the vector space with respect to which the basis of power sum symmetric functions are orthogonal with the normalization
(2.4) 
where and
(2.5) 
This is called the Jack inner product.
Definition 2.6.
The monic forms of the Jack functions for are uniquely defined by the following two conditions [37]:

Triangular expansion in the basis of monomial symmetric functions:
(2.7) 
Orthogonality:
(2.8)
Lemma 2.9.
For any integer we have
(2.10) 
Proof.
3. Infinitedimensional Lie algebras
3.1. Heisenberg algebras
In this section we recall the representation theory of Heisenberg algebras and the affine Lie algebras . Since the Lie algebra coincides with , as a byproduct we get the representation theory of .
Let be a field extension of . Let be a lattice, i.e., a free abelian group of finite rank equipped with a symmetric nondegenerate bilinear form . Fix a basis of .
Definition 3.1.
The lattice Heisenberg algebra associated with is the infinitedimensional Lie algebra over generated by , for and , and the central element satisfying the relations
(3.2) 
For any element we define the element by linearity, with . Set