Dirac structures on the space of connections

TL;DR

This paper constructs twisted Dirac structures on the space of irreducible connections over manifolds, showing how these structures relate to flat connections and gauge transformations, with implications for geometric analysis.

Contribution

It introduces a novel twisted Dirac structure framework on connection spaces, linking three- and four-manifold cases and analyzing gauge group actions.

Findings

Twisted Dirac structures are constructed on connection spaces.

Flat connections are characterized by non-twisted Dirac structures.

Boundary restrictions relate three- and four-manifold structures.

Abstract

We shall give a twisted Dirac structure on the space of irreducible connections on a SU(n)-bundle over a three-manifold, and give a family of twisted Dirac structures on the space of irreducible connections on the trivial SU(n)-bundle over a four-manifold. The twist is described by the Cartan 3-form on the space of connections. It vanishes over the subspace of flat connections. So the spaces of flat connections are endowed with ( non-twisted ) Dirac structures. The Dirac structure on the space of flat connections over the three-manifold is obtained as the boundary restriction of a corresponding Dirac structure over the four-manifold. We discuss also the action of the group of gauge transformations over these Dirac structures.