Line bundles on spectral curves and the generalised Legendre transform construction of hyperkaehler metrics

TL;DR

This paper explores the relationship between conjugacy classes and line bundles on spectral curves, extending the Legendre transform method to construct a broad class of hyperkaehler and pseudo-hyperkaehler metrics, including new examples.

Contribution

It generalizes the correspondence between conjugacy classes and line bundles to K-conjugacy classes and advances the Legendre transform approach for constructing hyperkaehler metrics.

Findings

Many known hyperkaehler metrics are obtained via this method

The work introduces a large class of new pseudo-hyperkaehler metrics

Connections to monopole metrics are established

Abstract

An analogue of the correspondence between GL(k)-conjugacy classes of matricial polynomials and line bundles is given for K-conjugacy classes, where K is one of the following: maximal parabolic, maximal torus, GL(k-1) embedded diagonally. The generalised Legendre transform construction of hyperkaehler metrics is studied further, showing that many known hyperkaehler metrics (including the ones on coadjoint orbits) arise in this way, and giving a large class of new (pseudo-)hyperkaehler metrics, analogous to monopole metrics.