On Associativity Equations in Dispersionless Integrable Hierarchies

TL;DR

This paper explores how associativity (WDVV) equations naturally arise in the context of dispersionless integrable hierarchies, showing their connection to tau-functions of KP and Toda hierarchies and constructing new finite-variable solutions.

Contribution

It demonstrates that associativity equations are encoded in the dispersionless limit of Hirota equations and constructs new finite-variable solutions to WDVV equations.

Findings

Tau-functions of dispersionless KP/Toda hierarchies solve associativity equations.

Associativity equations are derived from the dispersionless Hirota equations.

New finite-variable solutions to WDVV equations are constructed.

Abstract

We discuss the origin of the associativity (WDVV) equations in the context of quasiclassical or Whitham hierarchies. The associativity equations are shown to be encoded in the dispersionless limit of the Hirota equations for KP and Toda hierarchies. We show, therefore, that any tau-function of dispersionless KP or Toda hierarchy provides a solution to associativity equations. In general, they depend on infinitely many variables. We also discuss the particular solution to the dispersionless Toda hierarchy that describes conformal mappings and construct a family of new solutions to the WDVV equations depending on finite number of variables.