WDVV equation and Triple-product Relation

TL;DR

This paper explores the connections between WDVV equations, the noncommutative KP hierarchy, and Seiberg-Witten theory, revealing how these integrable systems relate through tau-functions and spectral curves.

Contribution

It demonstrates that WDVV-like equations emerge from the noncommutative KP tau-function and links the Seiberg-Witten prepotential to the Whitham hierarchy, unifying several integrable systems.

Findings

WDVV equations relate to noncommutative KP tau-function

Spectral curve in SW theory matches Toda-chain hierarchy

Whitham hierarchy encompasses Toda/KP hierarchies

Abstract

We study the relation between the WDVV equations and the $\tau$-function of the noncommutative KP (NCKP) hierarchy. WDVV-like equations (Hirota triple-product relation) in the noncommutative context appear as a consequence of the non-trivial equation for $\tau$-function of the NC KP hierarchy, while the prepotential in the Seiberg-Witten (SW) theory has been identified to the $\tau$-function of the Whitham hierarchy. We show that the spectral curve for the SW theory is the same as the Toda-chain hierarchy. We also show that Whitham hierarchy includes commutative Toda/KP hierarchy as a construction. Further, we comment on the origin of the Hirota triple-product relation in the context of the SW theory.