Towards elliptic deformation of q,t-matrix models

Andrei Mironov,Alexei Morozov

doi:10.1016/j.physletb.2021.136221

Abstract

As a necessary step in construction of elliptic matrix models, which preserve the superintegrability property <char>∼char, we suggest an elliptic deformation of the peculiar loci pkΔn, which play an important role in precise formulation of this property. The suggestion is to define the pkΔn-loci as elliptic functions with the right asymptotics at τ→i∞. If this hypothesis is correct, one can move to substituting the Schur and Macdonald functions in the role of characters by the elliptic GNS and study their behavior at the deformed topological and Δ loci.

Highlights

Eigenvalue matrix models can be defined in a number of different ways: through matrix integrals, through various equations (Virasoro and W-like constraints) imposed on partition functions, through recursion relations on averages and so on
We demonstrate that a proper choice of the elliptic deformation of loci completely preserves the factorization property of the Schur polynomials
Note that expression for μR is unrelated to ellipticity: it arises at the level of t deformation, eq(4): if (20) is already known, the indices νR and μR remain intact under the further elliptic deformation to (20)

Summary

Introduction

Eigenvalue matrix models can be defined in a number of different ways: through matrix integrals, through various equations (Virasoro and W-like constraints) imposed on partition functions, through recursion relations on averages and so on. There is more than that: these theories appear to be superintegrable, i.e. correlators can be evaluated explicitly in terms of reasonably-elementary functions [2] This is true for a cleverly adjusted basis of observables [3], which are usually not the naive multi-trace operators, but their peculiar combinations described by “characters” χR, the polynomials from the SchurMacdonald family, see [4] for various examples. An accompanying discussion of the elliptic matrix model itself can be found in [6] It is explained in [6] that the factorization at the elliptic loci already at the level of the Schur functions is required in the elliptic matrix model. We will see that this remains true after suggested elliptic deformation

Trigonometric case

On the structure of μR

Conclusion