Combinatorics of multisecant Fay identities

TL;DR

This paper generalizes Fay identities for theta functions on Riemann surfaces, leading to new solutions for multidimensional integrable systems like the Hirota equation and Toda systems.

Contribution

It introduces a set of multisecant Fay identities that extend classical results, enabling the construction of quasiperiodic solutions for complex multidimensional integrable equations.

Findings

Derived generalized Fay identities for theta functions.

Constructed quasiperiodic solutions for multidimensional Hirota and Toda systems.

Extended classical integrable system solutions to higher dimensions.

Abstract

We derive a set of identities for the theta functions on compact Riemann surfaces which generalize the famous trisecant Fay identity. Using these identities we obtain quasiperiodic solutions for a multidimensional generalization of the Hirota bilinear difference equation and for a multidimensional Toda-type system.