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

TL;DR

This paper demonstrates that the Seiberg-Witten prepotentials in 4d and 5d N=2 supersymmetric theories generally satisfy the WDVV equations, with proofs for Yang-Mills models and discussions of exceptions and special cases.

Contribution

The paper provides a general proof that Seiberg-Witten prepotentials satisfy WDVV equations in N=2 SUSY theories, including cases with matter and in different dimensions.

Findings

WDVV equations hold for most N=2 SUSY Yang-Mills theories with matter.

Exceptions occur in certain perturbative regimes and specific matter representations.

Additional terms appear in 5d theories, similar to heterotic string models.

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.