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

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 $5d$.

21 pages

References in corpus (10)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• On AGT relation in the case of U(3)
• Defects and Quantum Seiberg-Witten Geometry
• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• Instanton counting with a surface operator and the chain-saw quiver
• Quantum Elliptic Calogero-Moser Systems from Gauge Origami
• Factorisation of 3d $\mathcal{N}=4$ Twisted Indices and the Geometry of Vortex Moduli Space
• On complete solution of the quantum Dell system
• Instantons and the 5D U(1) gauge theory with extra adjoint
• Modular properties of 6d (DELL) systems

Cited by in corpus (10)

• Dualities in quantum integrable many-body systems and integrable probabilities -- I
• 3-Schurs from explicit representation of Yangian $Y({\widehat{\mathfrak{gl}}}_1)$. Levels 1-5
• On the status of DELL systems
• Characteristic determinant and Manakov triple for the double elliptic integrable system
• Summing up perturbation series around superintegrable point
• Quasi-Hopf twist and elliptic Nekrasov factor
• On Cherednik and Nazarov-Sklyanin large N limit construction for integrable many-body systems with elliptic dependence on momenta
• A basic triad in Macdonald theory
• 5D AGT conjecture for circular quivers
• Twisted Baker-Akhiezer function from determinants