An Elliptic Superpotential for Softly Broken N=4 Supersymmetric Yang-Mills Theory

TL;DR

This paper derives an exact superpotential for softly broken N=4 supersymmetric Yang-Mills theory, revealing its connection to elliptic Calogero-Moser systems and its modular properties under S-duality.

Contribution

It introduces a novel superpotential formulation for N=1 theories from N=4 SYM deformations, linking it to elliptic integrable systems and confirming vacuum structure predictions.

Findings

Superpotential matches the complexified elliptic Calogero-Moser Hamiltonian.

Reproduces vacuum structure predicted by Donagi and Witten.

Exhibits modular properties under S-duality.

Abstract

An exact superpotential is derived for the N=1 theories which arise as massive deformations of N=4 supersymmetric Yang-Mills (SYM) theory. The superpotential of the SU(N) theory formulated on R^{3}\times S^{1} is shown to coincide with the complexified potential of the N-body elliptic Calogero-Moser Hamiltonian. This superpotential reproduces the vacuum structure predicted by Donagi and Witten for the corresponding four-dimensional theory and also transforms covariantly under the S-duality group of N=4 SYM. The analysis yields exact formulae with interesting modular properties for the condensates of gauge-invariant chiral operators in the four-dimensional theory.