Duality Transformations for Generalized WDVV equations in Seiberg-Witten theory

TL;DR

This paper explores how electric-magnetic duality transformations act as symmetries within solution sets of generalized WDVV equations in Seiberg-Witten theory, using Picard-Fuchs equations and homology basis changes.

Contribution

It demonstrates that duality transformations preserve solution sets of the generalized WDVV equations via Picard-Fuchs equations and homology basis changes.

Findings

Duality transformations map solutions within the same set.

Solutions are connected through Picard-Fuchs equations.

Homology basis changes explain covariance of the equations.

Abstract

It is known that electric-magnetic duality transformations are symmetries of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. In Seiberg-Witten theory the solutions to these equations come in certain sets according to the gauge group. We show that the duality transformations transform solutions within a set to another solution within the same set, by using a subset of the Picard-Fuchs equations on the Seiberg-Witten family of Riemann surfaces. The electric-magnetic duality transformations can be thought of as changes of a canonical homology basis on the surfaces which in our derivation is clearly responsible for the covariance of the generalized WDVV system.