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

TL;DR

The work derives an exact, R-independent superpotential for N=1 theories obtained by soft breaking N=4 SU(N) SYM on R^3 x S^1, showing it equals the complexified potential of the elliptic Calogero-Moser Hamiltonian. By combining compactification, holomorphy, modular invariance, and DW/N=2 insights, the author fixes the superpotential to W = m_1 m_2 m_3 sum_{a>b} P(X_a - X_b), with P the Weierstrass function, and demonstrates its SL(2,Z) modular weight and the correspondence between its critical points and DW vacua. The construction yields exact condensates and gluino expectations, reproduces known 3D and 4D limits, and reveals a deep link between softly broken N=4 theories and integrable systems. The results offer a unified framework for analyzing vacua, dualities, and chiral condensates in softly broken supersymmetric gauge theories, and point toward generalizations to other gauge groups and N=2–to–N=1 flows via integrable systems.

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.