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.
Highlights
ELS-functionThe Shiraishi function Pn(xi; p|yi; s|q, t) is originally defined [39] to be a formal power series
In the language of network systems, one needs to compactify the network in both the vertical and horizontal directions [32,33,34], see figure 1
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 double elliptic system (Dell) description by double-periodic network models with DIM symmetry
Summary
The Shiraishi function Pn(xi; p|yi; s|q, t) is originally defined [39] to be a formal power series. Where we have made a change s → s−1 to facilitate the comparison with six dimensional gauge theory We see this is nothing but the elliptic lift of the Nekrasov factor that appears in six dimensional instanton partition function [41, 42]. Note that when we compare the Shiraishi function with the Nekrasov function, not t but s, which is introduced as a non-stationary parameter in [39], plays the role of the omega background parameter as it should be [12] This is the case in [13] and [14], where the instanton counting was discussed from the viewpoint of affine Lie algebras. Which should be compared with eq (4.21) of [43]
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