Finite $\epsilon_2$-corrections to the $\mathcal{N}=2$ SYM prepotential

TL;DR

This paper computes the first $ ext{epsilon}_2$-correction to the instanton partition functions of four-dimensional $ ext{N}=2$ Super Yang-Mills theory in the Nekrasov-Shatashvili limit, introducing a new operator to handle singularities.

Contribution

It presents the derivation of the $ ext{epsilon}_2$-correction to the prepotential, combining field theory and integrability techniques with a novel operator for singularity analysis.

Findings

Derived the first $ ext{epsilon}_2$-correction to the instanton partition function.

Introduced a new operator $ abla$ to distinguish singularities.

Connected the correction to Thermodynamic Bethe Ansatz-like equations.

Abstract

We derive the first $\epsilon_2$-correction to the instanton partition functions of $\mathcal{N}=2$ Super Yang-Mills (SYM) in four dimensions in the Nekrasov-Shatashvili limit $\epsilon_2\rightarrow 0$. In the latter we recall the emergence of the famous Thermodynamic Bethe Ansatz-like equation which has been found by Mayer expansion techniques. Here we combine efficiently these to field theory arguments. In a nutshell, we find natural and resolutive the introduction of a new operator $\nabla$ that distinguishes the singularities within and outside the integration contour of the partition function.