Quantum integrability of $\mathcal{N}=2$ 4d gauge theories

TL;DR

This paper reveals the quantum integrable structure underlying the Nekrasov-Shatashvili TBA-like equations for 4d $ ext{N}=2$ SYM theories, connecting instanton partition functions with integrability concepts.

Contribution

It demonstrates that the instanton partition function is characterized by a TQ-equation and introduces a dual TQ-equation, linking the NS TBA-like equations to known integrability structures.

Findings

Derivation of a dual TQ-equation sharing the same T-polynomial.

Evaluation of the quantum Wronskian to unity.

Reproduction of the NS TBA-like equation as a non-linear integral equation.

Abstract

We provide a description of the quantum integrable structure behind the Thermodynamic Bethe Ansatz (TBA)-like equation derived by Nekrasov and Shatashvili (NS) for $\mathcal{N}=2$ 4d Super Yang-Mills (SYM) theories. In this regime of the background, -- we shall show --, the instanton partition function is characterised by the solution of a TQ-equation. Exploiting a symmetry of the contour integrals expressing the partition function, we derive a 'dual' TQ-equation, sharing the same T-polynomial with the former. This fact allows us to evaluate to $1$ the quantum Wronskian of two dual solutions (for $Q$) and, then, to reproduce the NS TBA-like equation. The latter acquires interestingly the deep meaning of a known object in integrability theory, as its two second determinations give the usual non-linear integral equations (nlies) derived from the 'dual' Bethe Ansatz equations.