Solutions of the WDVV Equations and Integrable Hierarchies of KP type

TL;DR

This paper explores how reductions of KP hierarchies connected to loop algebras produce solutions to the Darboux-Egoroff system and constructs solutions to the Witten-Dijkgraaf-Verlinde-Verlinde equations using advanced Riemann-Hilbert techniques.

Contribution

It introduces a novel method linking KP hierarchy reductions to solutions of important integrable PDEs and equations in mathematical physics.

Findings

Solutions to the Darboux-Egoroff system derived from KP reductions.

Explicit construction of solutions to WDVV equations via Riemann-Hilbert dressing.

New connections established between integrable hierarchies and Frobenius manifold structures.

Abstract

We show that reductions of KP hierarchies related to the loop algebra of $SL_n$ with homogeneous gradation give solutions of the Darboux-Egoroff system of PDE's. Using explicit dressing matrices of the Riemann-Hilbert problem generalized to include a set of commuting additional symmetries, we construct solutions of the Witten--Dijkgraaf--E. Verlinde--H. Verlinde equations.