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 the $\mathcal{N}=2$ 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. Using the saddle point method and the $qq$-characters technique, we demonstrate that, in this limit, the partition function is governed by the Quantum Seiberg-Witten curves, as in the Nekrasov-Shatashvili limit, up to a normalisation constant. With the asymptotic behaviour found, 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 of 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 triangle groups.