Zero-energy states of N = 4 SYM on T^3: S-duality and the mapping class group

TL;DR

This paper investigates the zero-energy spectrum of N=4 super-Yang-Mills theory on a three-torus, exploring its invariance under S-duality and the mapping class group, and confirming dualities through detailed representation analysis.

Contribution

It extends previous work by decomposing bound states into irreducible representations of the mapping class group and verifies S-duality predictions with intricate combinatorial identities.

Findings

Confirmed invariance of zero-energy states under SL_2(Z) S-duality.

Decomposed bound states into irreducible representations of SL_3(Z).

Validated S-duality predictions for specific gauge groups.

Abstract

We continue our studies of the low-energy spectrum of N=4 super-Yang-Mills theory on a spatial three-torus. In two previous papers, we computed the spectrum of normalizable zero-energy states for all choices of gauge group and all values of the electric and magnetic 't Hooft fluxes, and checked its invariance under the SL_2(Z) S-duality group. In this paper, we refine the analysis by also decomposing the space of bound states into irreducible unitary representations of the SL_3(Z) mapping class group of the three-torus. We perform a detailed study of the S-dual pairs of theories with gauge groups Spin(2n+1) and Sp(2n). The predictions of S-duality (which commutes with the mapping class group) are fulfilled as expected, but the proof requires some surprisingly intricate combinatorial infinite product identities.