Generalized Jack polynomials and the AGT relations for the SU(3) group

Abstract

We find generalized Jack polynomials for the SU (3) group and verify that their Selberg averages for several first levels are given by Nekrasov functions. To compute the averages we derive recurrence relations for the sl3 Selberg integrals. IntroductionThe AGT relations [1] provide an extremely interesting link between the four dimensional N = 2 gauge theories of class S [2] and two dimensional conformal field theories. Moreover, these relations offer a new view on a variety of related fields, such as integrable systems The most surprising property of the AGT relations is that they give an additional and unexpected structure on the conformal field theory Hilbert space. One usually writes the states in this space as descendants of some primary field: L −Y |α , all the correlators therefore become sums over Young diagrams Y . However, the Nekrasov function is a sum over pairs of Young diagrams Y . Finding the corresponding basis | Y in CFT is an interesting problem. This basis was found explicitly in the case of c = 1 [6] and it was argued to exist in the general case [7]. Concretely, if one performs the bosonization of the Virasoro algebra, the basis vectors are expressed through the generalized Jack polynomials which are defined as the polynomial eigenfunctions of a certain differential operator 1 [8]. One can also compute the correlators in CFT using the Dotsenko-Fateev approach, in which they are given by certain multiple integrals related to the Selberg integrals [9]. In this setting the generalized Jack polynomials also play a distinguished role: their Selberg averages are factorised into a product of linear functions of momenta. More precisely, the averages are given by the Nekrasov functions.More generally, the AGT relations map the gauge theory with the SU (N ) group to the four point conformal block in the Toda field theory with two general and two degenerate fields [10]. The same factorisation of the sl N Selberg averages should happen in this case as well.In this Letter we explicitly find generalized Jack polynomials for the group SU (3) and check that their Selberg averages indeed reproduce the Nekrasov functions on the first levels. To compute the averages we derive the W -constraints for the β-deformed A 2 quiver matrix model. In section 2 we introduce the differential operator whose polynomial eigenfunctions are given by the generalized Jack polynomials, compute them explicitly and check their elementary properties. In section 3 we derive the DotsenkoFateev representation of the conformal block in Toda field theory and show that the AGT relations hold if certain Selberg averages of generalised Jack polynomials are given by the Nekrasov functions. Using the W constraints we compute the averages and check the relevant formulas for the first levels. The Nekrasov functions and AGT relations are provided in Appendix A. The W constraints are presented in Appendix B. * sa.mironov_1@physics.msu.ru † andrey.morozov@itep.ru ‡ yegor.zenkevich@gmail.com, zenkevich@ms2.inr.ac.ru 1 It is suspected that g...

Generalized Macdonald polynomials, spectral duality for conformal blocks and AGT correspondence in five dimensions

“…Similar polynomials for the four dimensional case (i.e. generalized Jack polynomials) were introduced in [21] and also studied in [22].…”

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

“…All the quantities in matrix model are analytically continued from integer values of N and β, what is made unambiguously due to Selberg nature of the integrals [113]. This quadruple decomposition is recently presented in some detail in [62] based on the number of previous developments [59][60][61][113][114][115][116][117][118][119][120][121][122], see section 2.1.1 below. The group theory symmetry behind the whole picture [64] is encoded in the 2-site U q (gl 2 ) XXZ spin chain integrable system [20,74] (reduced to XXX in 4d, when q = 1 [71]).…”

On factorization of generalized Macdonald polynomials

“…At the same time they are relatively new special functions, far from being thoroughly understood and clearly described. They are deformations of the generalized Jack polynomials introduced in [59,60]. Even the simplest questions about them are yet unanswered.…”