Higgs bundle

In mathematics Higgs bundles are an aid in the representation theory of space groups and fundamental groups of complex manifolds. They were introduced by Nigel Hitchin and named after Peter Higgs because of the analogy to the Higgs bosons.

Definition

A Higgs bundle is a pair consisting of a holomorphic vector bundle over a Riemann surface and a Higgs field, that is a -valued holomorphic 1-form.

Stability, Polystabilität

A Higgs bundle is called stable if for all invariant holomorphic sub-bundles, the inequality

Applies. (Must Note that this inequality is valid only for invariant sub-bundles, a stable Higgs bundle so no need to be a stable vector bundle. )

A Higgs bundle is called POLYSTABIL if there is a direct sum

Stable Higgs bundles with

For is.

Representation theory

Based on results of Corlette and Donaldson proved Hitchin and Simpson the following equivalences for Riemann surfaces:

Higher dimensional generalization

About higher dimensional complex manifolds we define a Higgs bundle over as a pair of a holomorphic vector bundle and a -valued holomorphic 1-form that satisfies the equation.

( In the case of Riemann surfaces, this equation is trivially satisfied. )