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.
Highlights
Original N = 4 theory, but a remnant of this symmetry manifests in the fact that the expansion coefficients of the N = 2∗ prepotential in the limit of small mass are almost modular forms of the bare gauge coupling
We propose a modular anomaly equation for the prepotential of the N = 2∗ super Yang-Mills theory on R4 with gauge group U(N ) in the presence of an Ω-background
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
Summary
Using localization methods [5,6,7] In this framework, the instanton prepotential is viewed as the free energy. The localization techniques allow to push the calculation to higher instanton numbers without major problems, but the resulting explicit expressions for the F (k)’s quickly become rather cumbersome. We notice that there are no nonperturbative corrections to the logarithmic term (2.9a), as expected, and that f1inst is independent of the vacuum expectation values ai’s and that its k-th instanton coefficient is σ1(k)/k where σ1(k) is the sum of the divisors of k. We see that new structures, which were not present in the 1-loop results (2.9), start appearing in the non-perturbative sector; for example f3inst contains the triple sum. The symbol [i1, i2, . . . , in] denotes a sum over n non-coinciding positive integers (see appendix B for details)
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