Remarks on the intersection of two quadrics

TL;DR

This paper explores the geometric and algebraic structures of integrable systems arising from intersections of two quadrics, linking Higgs bundles, the geometric Langlands program, and recent developments in Spin(2g) bundles on hyperelliptic curves.

Contribution

It provides a new interpretation of an integrable system via quasi parabolic Higgs bundles and connects it to the geometric Langlands program and recent research on Spin(2g) bundles.

Findings

Interpretation of the integrable system in terms of Higgs bundles

Discussion of connections to the geometric Langlands program

Linking the system to Spin(2g) bundles on hyperelliptic curves

Abstract

The article takes the formula for the integrable system defined by Beauville et al on the cotangent bundle of the intersection of two quadrics X, and interprets it in terms of rank 2 quasi parabolic Higgs bundles on the projective line. We then discuss aspects related to the geometric Langlands programme in this simple concrete context. We conclude with a description of the link with the recent paper of Benedetti et al identifying X and its integrable system in terms of invariant Spin(2g) bundles on a hyperelliptic curve of genus g.