Whitham Hierarchies, Instanton Corrections and Soft Supersymmetry Breaking in N=2 SU(N) Super Yang-Mills Theory

TL;DR

The paper addresses computing nonperturbative information for $\mathcal{N}=2$ SU($N$) Yang–Mills theory by embedding it into the Whitham hierarchy and linking the prepotential to a root-lattice theta function. It introduces a recursive procedure that determines the full instanton expansion from one-loop data through derivatives of the prepotential with respect to the slow times, organized by homogeneous Casimir combinations and theta-function expansions. The authors compute explicit one- and two-instanton terms for SU($N$) and show how slow times can be promoted to spurion superfields, yielding soft ${\mathcal N}=2$ to ${\mathcal N}=0$ deformations associated to higher Casimir operators, with a detailed SU(3) analysis revealing monopole/dyon condensation patterns and AD-point behavior. The work provides a powerful computational framework connecting integrable systems with low-energy QFT and illuminates how nonsupersymmetric deformations can be systematically incorporated in exact Seiberg–Witten-like setups, with potential links to D-brane constructions.

Abstract

We study N=2 super Yang-Mills theory with gauge group SU(N) from the point of view of the Whitham hierarchy. We develop a new recursive method to compute the whole instanton expansion of the prepotential using the theta function associated to the root lattice of the group. Explicit results for the one and two-instanton corrections in SU(N) are presented. Interpreting the slow times of the hierarchy as additional couplings, we show how they can be promoted to spurion superfields that softly break N=2 supersymmetry down to N=0. This provides a family of nonsupersymmetric deformations of the theory, associated to higher Casimir operators of the gauge group. The SU(3) theory is analyzed in some detail.