Low-energy spectrum of N = 4 super-Yang-Mills on T^3: flat connections, bound states at threshold, and S-duality

TL;DR

This paper analyzes the low-energy spectrum of N=4 super-Yang-Mills theory on a three-torus, exploring flat connections, bound states, and S-duality, using semi-classical methods and matrix quantum mechanics.

Contribution

It provides a semi-classical computation of the low-energy spectrum and bound states in N=4 SYM on T^3, confirming S-duality constraints and revealing underlying combinatorial structures.

Findings

Computed spectra for various gauge groups such as SU(n), Spin(2n+1), and Sp(2n)

Confirmed S-duality constraints for the bound state counts

Discovered subtle combinatorial identities related to the theory's structure

Abstract

We study (3+1)-dimensional N=4 supersymmetric Yang-Mills theory on a spatial three-torus. The low energy spectrum consists of a number of continua of states of arbitrarily low energies. Although the theory has no mass-gap, it appears that the dimensions and discrete abelian magnetic and electric 't Hooft fluxes of the continua are computable in a semi-classical approximation. The wave-functions of the low-energy states are supported on submanifolds of the moduli space of flat connections, at which various subgroups of the gauge group are left unbroken. The field theory degrees of freedom transverse to such a submanifold are approximated by supersymmetric matrix quantum mechanics with 16 supercharges, based on the semi-simple part of this unbroken group. Conjectures about the number of normalizable bound states at threshold in the latter theory play a crucial role in our analysis. In this way, we compute the low-energy spectra in the cases where the simply connected cover of the gauge group is given by SU(n), Spin(2n+1) or Sp(2n). We then show that the constraints of S-duality are obeyed for unique values of the number of bound states in the matrix quantum mechanics. In the cases based on Spin(2n+1) and Sp(2n), the proof involves surprisingly subtle combinatorial identities, which hint at a rich underlying structure.