Duality in Integrable Systems and Generating Functions for New Hamiltonians

TL;DR

This paper explores the duality in integrable systems related to Seiberg-Witten theory, showing how tau-functions serve as generating functions for dual Hamiltonians, with coefficients expressed via system coordinates.

Contribution

It demonstrates that theta-function coefficients can be written solely in terms of Seiberg-Witten system coordinates, representing dual Hamiltonians.

Findings

Tau-functions act as generating functions for dual systems.

Theta-function coefficients are expressed in terms of system coordinates.

Provides a new perspective on duality in integrable systems.

Abstract

Duality in the integrable systems arising in the context of Seiberg-Witten theory shows that their tau-functions indeed can be seen as generating functions for the mutually Poisson-commuting hamiltonians of the {\em dual} systems. We demonstrate that the $\Theta$-function coefficients of their expansion can be expressed entirely in terms of the co-ordinates of the Seiberg-Witten integrable system, being, thus, some set of hamiltonians for a dual system.