Recursion relation for instanton counting for $SU(2)$ ${\cal N}=2$ SYM in NS limit of $\Omega$ background

TL;DR

This paper introduces a new, efficient method for calculating A-cycle periods in ${ m SU}(2)$ ${ m N}=2$ SYM in the NS limit, enabling higher-order instanton expansions and confirming results with a numerical approach.

Contribution

A novel method for instanton counting that improves efficiency and extends the order of calculations for ${ m SU}(2)$ ${ m N}=2$ SYM in the NS limit.

Findings

New method surpasses standard techniques in efficiency.

Explicit higher-order instanton calculations demonstrated.

Numerical approach confirms large $q$ asymptotics with existing conjectures.

Abstract

In this paper we investigate different ways of deriving the A-cycle period as a series in instanton counting parameter $q$ for ${\cal N}=2$ SYM with up to four antifundamental hypermultiplets in NS limit of $\Omega$ background. We propose a new method for calculating the period and demonstrate its efficiency by explicit calculations. The new way of doing instanton counting is more advantageous compared to known standard techniques and allows to reach substantially higher order terms with less effort. This approach is applied for the pure case as well as for the case with several hypermultiplets. We also investigate a numerical method for deriving the $A$-cycle period valid for arbitrary values of $q$. Analyzing large $q$ asymptotic we get convincing agreement with an analytic expression deduced from a conjecture by Alexei Zamolodchikov in a different context.