More Evidence for the WDVV Equations in N=2 SUSY Yang-Mills Theories

TL;DR

The paper extends evidence that 4d/5d ${ m N}=2$ SW prepotentials satisfy the WDVV equations by developing a nonperturbative framework based on associative algebras of meromorphic 1-forms on spectral curves, with a focus on hyperelliptic curves and integrable systems. It provides explicit perturbative and nonperturbative results for various gauge groups and matter contents, showing WDVV compatibility in many cases (notably for fundamental representations) and highlighting exceptions (notably adjoint matter in Calogero systems) where standard WDVV fails due to non-associativity. The approach uses residue formulas to connect $F_{IJK}$ to the algebraic structure of $dW_I$ and demonstrates that hyperelliptic cases yield a closed, associative algebra enabling the WDVV relations, while non-hyperelliptic (elliptic Calogero) configurations require deformations of the WDVV framework. Overall, the work reinforces a deep link between SW theory, integrable systems, and topological/quantum-cohomology-like structures, with implications for strings, branes, and higher-dimensional compactifications.

Abstract

We consider 4d and 5d N=2 supersymmetric theories and demonstrate that in general their Seiberg-Witten prepotentials satisfy the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. General proof for the Yang-Mills models (with matter in the first fundamental representation) makes use of the hyperelliptic curves and underlying integrable systems. A wide class of examples is discussed, it contains few understandable exceptions. In particular, in perturbative regime of 5d theories in addition to naive field theory expectations some extra terms appear, like it happens in heterotic string models. We consider also the example of the Yang-Mills theory with matter hypermultiplet in the adjoint representation (related to the elliptic Calogero-Moser system) when the standard WDVV equations do not hold.