On Integrable Structure behind the Generalized WDVV Equations

TL;DR

This paper explores the integrable structures underlying the generalized WDVV equations, especially in contexts like Seiberg-Witten theory where traditional equations are inconsistent, proposing alternative formulations involving matrices derived from the prepotential.

Contribution

It introduces a new perspective on the integrability of generalized WDVV equations in situations lacking a distinguished moduli space direction, extending the understanding of their structure.

Findings

Identifies limitations of traditional WDVV equations in certain theories.

Proposes alternative equations involving matrices from the third derivatives of the prepotential.

Highlights the role of these matrices in maintaining integrability.

Abstract

In the theory of quantum cohomologies the WDVV equations imply integrability of the system $(I\partial_\mu - zC_\mu)\psi = 0$. However, in generic situation -- of which an example is provided by the Seiberg-Witten theory -- there is no distinguished direction (like $t^0$) in the moduli space, and such equations for $\psi$ appear inconsistent. Instead they are substituted by $(C_\mu\partial_\nu - C_\nu\partial_\mu)\psi^{(\mu)} \sim (F_\mu\partial_\nu - F_\nu\partial_\mu)\psi^{(\mu)} = 0$, where matrices $(F_\mu)_{\alpha\beta} = \partial_\alpha \partial_\beta \partial_\mu F$.