Zamolodchikov recurrence relation and modular properties of effective coupling in $\mathcal{N}=2$ SQCD

In this work, we present a recurrence relation for the instanton partition function of $\mathcal{N}=2$ SYM $SU(N)$ gauge theory with $2N$ fundamental multiplets. The main difficulty lies in determining the asymptotic behaviour of the partition function in the regime of large vacuum expectation values of the Higgs field. We demonstrate that, in this limit, the partition function is governed by the Quantum Seiberg-Witten curves, as it is in the Nekrasov-Shatashvili limit, up to a normalisation constant. With the found asymptotic behaviour, we are able to write the recurrence relation for the partition function and to find the effective infrared coupling constant. The resulting effective constant is an inverse to a modular function with respect to a certain triangle group, and the asymptotic itself is a product of modular functions and forms with respect to the same group.