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

Abstract

As a development of [1], 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.

Factorisation of 3d $$ \mathcal{N} $$ = 4 twisted indices and the geometry of vortex moduli space

“…This data specifies a tuple of Young diagrams, where k 20 The elliptic/4d lift of this statement was recently discussed by the authors of [39].…”

On a complete solution of the quantum Dell system

“…Hamiltonians and their eigenfunctions arise from Nekrasov functions in the 2 = 0 (Nekrasov-Shatashvili) limit, but the Shiraishi functions can appear applicable even beyond it. In the forthcoming paper [63], we discuss an improved version of our claim, with an additional elliptic deformation of the Shiraishi series. Our work is supported in part by Grants-in-Aid for Scientific Research (17K05275) (H.A.…”

Characteristic determinant and Manakov triple for the double elliptic integrable system

“…Solution to the eigenvalue problem forĤ n was suggested in [5,6] by extending the Shiraishi functions [64] -solutions to a non-stationary Macdonald-Ruijsenaars quantum problem.…”

“…as A x and A p respectively. Introduce the bilinear pairing: 6) by defining it on the basis elements…”

“…A more universal classification picture arises if we slightly change the definition of f λ (w) as in transition from θ p (e w ) to ϑ(w) (A.4) together with additional normalization factor ϑ (0)/ϑ(log(λ)). Then the function f λ (w) turns into the Kronecker elliptic function [70,71] depending on the moduliτ (defined through ω = e 2πıτ ): 6) where u = log(λ). In trigonometric and rational limits (when Im(τ) → 0) it is as follows:…”