Sato Grassmannian and Degenerate Sigma Function

TL;DR

This paper investigates the degeneration of hyperelliptic sigma functions using the Sato Grassmannian, providing a decomposition formula that relates tau functions of different genera and deriving a corresponding formula for the degenerate sigma function.

Contribution

It introduces a novel approach to analyze degenerate hyperelliptic sigma functions via the Sato Grassmannian, linking tau functions across genera.

Findings

Decomposition of tau functions for degenerate hyperelliptic curves

Explicit formula for degenerate hyperelliptic sigma functions

Connection between Plücker coordinates and sigma function degeneration

Abstract

The degeneration of the hyperelliptic sigma function is studied. We use the Sato Grassmannian for this purpose. A simple decomposition of a rational function gives a decomposition of Pl\"ucker coordinates of a frame of the Sato Grassmannian. It then gives a decomposition of the tau function corresponding to the degeneration of a hyperelliptic curve of genus $g$ in terms of the tau functions corresponding to a hyperelliptic curve of genus $g-1$. Since the tau functions are described by sigma functions, we get the corresponding formula for the degenerate hyperelliptic sigma function.