Equations of the moduli of Higgs pairs and infinite Grassmannian

TL;DR

This paper defines the moduli space of Higgs pairs over a smooth projective curve, embeds it into a product space via a relative Krichever map, and derives explicit KP-type equations describing this moduli space, especially in the totally ramified spectral cover case.

Contribution

It introduces a scheme structure for the Higgs pairs moduli space, constructs an injective relative Krichever map, and derives explicit equations characterizing the moduli space.

Findings

The moduli space is a closed subscheme of a product space.

The relative Krichever map is injective.

Explicit KP-type equations are obtained for the moduli space.

Abstract

In this paper the moduli space of Higgs pairs over a fixed smooth projective curve with extra formal data is defined and it is endowed with a scheme structure. We introduce a relative version of the Krichever map using a fibration of Sato Grassmannians and show that this map is injective. This fact and the characterization of the points of the image of the Krichever map allow us to prove that this moduli space is a closed subscheme of the product of the moduli of vector bundles (with formal extra data) and a formal anologue of the Hitchin base. This characterization also provide us the method to compute explicitely KP-type equations which describe the moduli space of Higgs pairs. Finally, for the case where the spectral cover is totally ramified at a fixed point of the curve, these equations are given in terms of the characteristic coefficients of the Higgs field.