Instanton Expansions for Mass Deformed N=4 Super Yang-Mills Theories

TL;DR

This paper develops a method to efficiently compute instanton expansions in mass-deformed N=4 super Yang-Mills theories using modular anomaly equations and recursion relations, enabling high-order calculations for various gauge groups.

Contribution

It introduces a new recursive approach based on modular anomaly equations to derive instanton expansions in mass-deformed N=4 theories, extending to groups with unknown Seiberg-Witten curves.

Findings

Efficient high-order instanton expansion computation for SU(3).

Instanton expansion coefficients are integers after differentiation.

Postulation of anomaly equations for SO(2n) and E_n groups.

Abstract

We derive modular anomaly equations from the Seiberg-Witten-Donagi curves for softly broken N=4 SU(n) gauge theories. From these equations we can derive recursion relations for the pre-potential in powers of m^2, where m is the mass of the adjoint hypermultiplet. Given the perturbative contribution of the pre-potential and the presence of ``gaps'' we can easily generate the m^2 expansion in terms of polynomials of Eisenstein series, at least for relatively low rank groups. This enables us to determine efficiently the instanton expansion up to fairly high order for these gauge groups, e. g. eighth order for SU(3). We find that after taking a derivative, the instanton expansion of the pre-potential has integer coefficients. We also postulate the form of the modular anomaly equations, the recursion relations and the form of the instanton expansions for the SO(2n) and E_n gauge groups, even though the corresponding Seiberg-Witten-Donagi curves are unknown at this time.