Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function

TL;DR

This paper introduces elliptic liftings of Shiraishi functions, creating a new class of functions (ELS-functions) that serve as solutions to non-stationary double elliptic equations, with applications in gauge theories and elliptic polynomials.

Contribution

It proposes elliptic generalizations of Shiraishi functions, conjectures their role in double elliptic systems, and connects them to elliptic Macdonald polynomials and 6d gauge theory partition functions.

Findings

ELS-functions are symmetric elliptic polynomials extending Macdonald polynomials.

Explicit plethystic formula for 6d partition function in the U(1) case.

Demonstrated that ELS-functions relate to elliptic genera of affine Laumon spaces.

Abstract

As a development of arXiv:1912.12897, we note that the ordinary Shiraishi functions have an insufficient number of parameters to describe generic eigenfunctions of double elliptic system (Dell). The lacking parameter can be provided by substituting elliptic instead of the ordinary Gamma functions in the coefficients of the series. These new functions (ELS-functions) are conjectured to be functions governed by compactified DIM networks which can simultaneously play the three roles: solutions to non-stationary Dell equations, Dell conformal blocks with the degenerate field (surface operator) insertion, and the corresponding instanton sums in $6d$ SUSY gauge theories with adjoint matter. We describe the basics of the corresponding construction and make further conjectures about the various limits and dualities which need to be checked to make a precise statement about the Dell description by double-periodic network models with DIM symmetry. We also demonstrate that the ELS-functions provide symmetric polynomials, which are an elliptic generalization of Macdonald ones, and compute the generation function of the elliptic genera of the affine Laumon spaces. In the particular $U(1)$ case, we find an explicit plethystic formula for the $6d$ partition function, which is a non-trivial elliptic generalization of the $(q,t)$ Nekrasov-Okounkov formula from $5d$.