Tau functions for the Dirac operator on the cylinder

TL;DR

This paper computes the determinant of the Dirac operator with mass on a cylinder, relating it to the isomonodromic tau-function, using boundary value spaces, Green functions, and canonical solutions.

Contribution

It introduces a method to calculate the Dirac operator's determinant on a cylinder, connecting it to the isomonodromic tau-function with explicit formulas and deformation equations.

Findings

Derived explicit formulas for the tau-function in the case n=2.

Established deformation equations for the expansion coefficients of solutions.

Connected the determinant calculation to the geometry of the Grassmannian bundle.

Abstract

The goal of the present paper is to calculate the determinant of the Dirac operator with a mass in the cylindrical geometry. The domain of this operator consists of functions that realize a unitary one-dimensional representation of the fundamental group of the cylinder with $n$ marked points. The determinant represents a version of the isomonodromic $\tau$-function, itroduced by M. Sato, T. Miwa and M. Jimbo. It is calculated by comparison of two sections of the $\mathrm{det}^*$-bundle over an infinite-dimensional grassmannian. The latter is composed of the spaces of boundary values of some local solutions to Dirac equation. The principal ingredients of the computation are the formulae for the Green function of the singular Dirac operator and for the so-called canonical basis of global solutions on the 1-punctured cylinder. We also derive a set of deformation equations satisfied by the expansion coefficients of the canonical basis in the general case and find a more explicit expression for the $\tau$-function in the simplest case $n=2$.