Modular anomaly equations in N=2* theories and their large-N limit

Abstract

We propose a modular anomaly equation for the prepotential of the N=2* super Yang-Mills theory on R^4 with gauge group U(N) in the presence of an Omega-background. We then study the behaviour of the prepotential in a large-N limit, in which N goes to infinity with the gauge coupling constant kept fixed. In this regime instantons are not suppressed. We focus on two representative choices of gauge theory vacua, where the vacuum expectation values of the scalar fields are distributed either homogeneously or according to the Wigner semi-circle law. In both cases we derive an all-instanton exact formula for the prepotential. As an application, we show that the gauge theory partition function on S^4 at large N localises around a Wigner distribution for the vacuum expectation values leading to a very simple expression in which the instanton contribution becomes independent of the coupling constant.

arXiv commentLaTeX, 32 pages, no figures

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• Exact results in a N=2 superconformal gauge theory at strong coupling
• N=2* Super-Yang-Mills Theory at Strong Coupling
• A slow review of the AGT correspondence
• Strong-Coupling Phases of Planar N=2* Super-Yang-Mills Theory
• N=2 gauge theories and quantum phases
• Modular and duality properties of surface operators in N=2* gauge theories
• S-duality, triangle groups and modular anomalies in N=2 SQCD
• Non-perturbative studies of N=2 conformal quiver gauge theories
• $\mathcal{N}=2^{\star }$ from Topological Amplitudes in String Theory
• On the large $\Omega$-deformations in the Nekrasov-Shatashvili limit of $\mathcal{N}=2^{\ast }$ SYM
• Chiral observables and S-duality in N = 2* U(N) gauge theories
• Exact partition functions for the $\Omega$-deformed $\mathcal{N}=2^{\ast }$ $SU(2)$ gauge theory
• New non-linear equations and modular form expansion for double-elliptic Seiberg-Witten prepotential
• Modular properties of 6d (DELL) systems
• Chiral trace relations in $\Omega$-deformed $\mathcal{N}=2$ theories
• Partition function of N=2* SYM on a large four-sphere
• Exact partition functions for deformed $\mathcal{N}=2$ theories with $N_f=4$ flavours
• Effective Gravitational Couplings of Higher-Rank Supersymmetric Gauge Theories
• On new exact conformal blocks and Nekrasov functions
• Instanton Corrections for m and Omega
• Modular properties of surface operators in N=2 SQCD