Spherical Hecke algebra in the Nekrasov-Shatashvili limit

TL;DR

This paper explores the NS limit of the Spherical Hecke algebra in relation to Nekrasov instanton partition functions, revealing connections to integrability, Bethe ansatz, and TBA equations.

Contribution

It unifies the SHc algebra framework with the Nekrasov-Shatashvili limit, expressing partition functions via Bethe roots and defining operators that generate root variations.

Findings

Expressed instanton partition functions in terms of Bethe roots.

Defined operators obeying SHc algebra at first order in epsilon_2.

Connected SHc algebra with TBA and Mayer cluster expansion approaches.

Abstract

The Spherical Hecke central (SHc) algebra has been shown to act on the Nekrasov instanton partition functions of $\mathcal{N}=2$ gauge theories. Its presence accounts for both integrability and AGT correspondence. On the other hand, a specific limit of the Omega background, introduced by Nekrasov and Shatashvili (NS), leads to the appearance of TBA and Bethe like equations. To unify these two points of view, we study the NS limit of the SHc algebra. We provide an expression of the instanton partition function in terms of Bethe roots, and define a set of operators that generates infinitesimal variations of the roots. These operators obey the commutation relations defining the SHc algebra at first order in the equivariant parameter $\epsilon_2$. Furthermore, their action on the bifundamental contributions reproduces the Kanno-Matsuo-Zhang transformation. We also discuss the connections with the Mayer cluster expansion approach that leads to TBA-like equations.