Baker-Akhiezer spinor kernel and tau-functions on moduli spaces of meromorphic differentials

TL;DR

This paper develops a comprehensive framework for Baker-Akhiezer spinor kernels and tau-functions on moduli spaces of meromorphic differentials, deriving variational formulas and exploring their geometric and analytical properties.

Contribution

It introduces the Baker-Akhiezer tau-function, relates it to Bergman and KP tau-functions, and derives variational formulas on moduli spaces with applications to divisor class relationships.

Findings

Derived Rauch-Ahlfors type variational formulas for moduli spaces

Established relationships between divisor classes on moduli spaces

Analyzed global properties of tau-functions and their geometric implications

Abstract

In this paper we study Baker-Akhiezer spinor kernel on moduli spaces of meromorphic differentials on Riemann surfaces. We introduce the Baker-Akhiezer tau-function which is related to both Bergman tau-function (which was studied before in the context of Hurwitz spaces and spaces of holomorphic and quadratic differentials) and KP tau-function on such spaces. In particular, we derive variational formulas of Rauch-Ahlfors type on moduli spaces of meromorphic differentials with prescribed singularities: we use the system of homological coordinates, consisting of absolute and relative periods of the meromorphic differential, and show how to vary the fundamental objects associated to a Riemann surface (the matrix of $b$-periods, normalized Abelian differentials, the Bergman bidifferential, the Szeg\"o kernel and the Baker-Akhiezer spinor kernel) with respect to these coordinates. The variational formulas encode dependence both on the moduli of the Riemann surface and on the choice of meromorphic differential (variation of the meromorphic differential while keeping the Riemann surface fixed corresponds to flows of KP type). Analyzing the global properties of the Bergman and Baker-Akhiezer tau-functions we establish relationships between various divisor classes on the moduli spaces.