Zamolodchikov asymptotic formula and instanton expansion in N=2 SUSY N_f=2N_c QCD

TL;DR

The paper uses the AGT correspondence to translate Zamolodchikov's large-$\Delta$ asymptotics of the Virasoro four-point conformal block into the Nekrasov instanton partition function for the $N_f=4$, $N_c=2$ Seiberg-Witten theory, and extracts the instanton part of the prepotential in the $\epsilon_1,\epsilon_2\to0$ limit. It derives explicit first- and second-order instanton terms and provides an exact all-orders formula for $\mathcal{F}_{inst}$, showing that the effective coupling is nonperturbatively renormalized despite conformality, with the bare coupling $x$ related to the spectral torus via theta-constants. This establishes a concrete nonperturbative RG in a conformal SUSY gauge theory and demonstrates a powerful algebro-geometric bridge between 2d CFT data and 4d gauge dynamics, with implications for generalizations to higher rank. The results underscore the practical utility of AGT in obtaining explicit instanton corrections and deepen the understanding of the geometry underlying Seiberg-Witten theory.

Abstract

The AGT relations allow one to convert the Zamolodchikov asymptotic formula for the conformal block into the instanton expansion of the Seiberg-Witten prepotential for theory with two colors and four fundamental flavors. This provides an explicit formula for the instanton corrections in this model, resolving in this way an old problem in Seiberg-Witten theory. The answer is especially elegant for vanishing matter masses, then the bare charge of gauge theory 16q_0 = 16e^{iπτ_0} plays the role of a branch point on the spectral torus. The exact prepotential at this point is F a^2\log q with q\neq q_0, unlike the case of another conformal theory, with massless adjoint field. Instead, 16q_0 = θ_{10}^4/θ_{00}^4(q) = 16q(1+O(q)), i.e. the Zamolodchikov asymptotics gives rise, in particular, to an exact non-perturbative beta-function so that the effective coupling differs from the bare charge by infinite number of instantonic corrections.