Recursion relation for instanton counting for SU(2) N=2 SYM in NS limit of Ω background

In this paper we investigate different ways of deriving the A-cycle period as a series in instanton counting parameter q for N = 2 SYM with up to four antifundamental hypermultiplets in NS limit of Omega 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.

Automated summary

The paper presents a new recursive method for calculating the A-cycle period in N = 2 SYM models with antifundamental hypermultiplets, demonstrating its efficiency compared to standard techniques. Additionally, a numerical method for deriving the A-cycle period for arbitrary q values is suggested, with an analytic expression obtained for large q asymptotics when no hypermultiplets are present. The paper shows convincing agreement between this expression and the numerical approach.