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

Abstract

Abstract In this paper we investigate different ways of deriving the A-cycle period as a series in instanton counting parameter q for $$ \mathcal{N} $$ N = 2 SYM with up to four antifundamental hypermultiplets in NS limit of Ω background. We propose a new recursive 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.In addition we suggest a numerical method for deriving the A-cycle period for arbitrary values of q. In the case when one has no hypermultiplets for the A-cycle an analytic expression for large q asymptotics is obtained using a conjecture by Alexei Zamolodchikov. We demonstrate that this expression is in convincing agreement with the numerical approach.