# Stieltjes-Bethe equations in higher genus and branched coverings with even ramifications

## TL;DR

This paper introduces a higher genus generalization of Stieltjes-Bethe equations, linking projective structures, branched coverings with even ramifications, and Hurwitz spaces, and explores their relation to Yang-Yang functions and tau-functions.

## Contribution

It proposes a natural higher genus analog of Stieltjes-Bethe equations and establishes connections with branched projective structures and Hurwitz spaces with even ramifications.

## Key findings

- Defined higher genus Stieltjes-Bethe equations
- Linked these equations to branched projective structures
- Identified a higher genus Yang-Yang function and its relation to tau-functions

## Abstract

We describe projective structures on a Riemann surface corresponding to monodromy groups which have trivial $SL(2)$ monodromies around singularities and trivial $PSL(2)$ monodromies along homologically non-trivial loops on a Riemann surface. We propose a natural higher genus analog of Stieltjes-Bethe equations. Links with branched projective structures and with Hurwitz spaces with ramifications of even order are established. We find a higher genus analog of the genus zero Yang-Yang function (the function generating accessory parameters) and describe its similarity and difference with Bergman tau-function on the Hurwitz spaces.