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

TL;DR

This work derives a Zamolodchikov-like recurrence for the instanton partition function of $ ext{SU}(N)$ gauge theories with $2N$ fundamental hypermultiplets by analyzing the large-Higgs-VEV asymptotics. The authors fuse saddle-point methods with $qq$-characters to show that Quantum Seiberg-Witten curves control the dominant contributions, from which they extract a finite effective infrared coupling $q_{ m IR}$ with triangle-group modularity and modular forms entering the vacuum energy. They formulate a renormalized recurrence that resums the partition function in powers of $q_{ m IR}$ and demonstrate consistency with known results for $N=2,3$, and with decoupling limits for higher $N$, while addressing the limited accuracy of the naive saddle-point when $N ext{≥}4$. The modular structure, contraction to the coupling matrix, and the appearance of Hauptmoduls tie the field-theoretic analysis to the algebraic and number-theoretic properties of the underlying conformal/AGT framework, offering a controlled path to compute nonperturbative data in these conformal $ ext{N}=2$ theories. Overall, the paper advances understanding of nonperturbative dynamics in $ ext{N}=2$ SQCD through a rigorous blend of saddle-point analysis, qq-characters, and modularity, with explicit recurrence relations and modular-function descriptions of the effective coupling.

Abstract

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.

Figures (3)

• Figure 1: Boundary of a large Young diagram (a), and its zoomed pieces, with slope more (b) and less (c) than one. On the latter two pictures are shown the poles (red crosses) and zeroes (blue circles) of the function in the left-hand sides of (\ref{['eqn:omegaomega']}).
• Figure 2: Cuts of $\omega_0(y)$
• Figure 3: Singular areas and their shadows