Dirac Geometry of the Holonomy Fibration

TL;DR

This paper provides a gauge-theoretic framework for understanding the Dirac structure on a Lie group, connecting it to the space of connections and developing tools for reduction of Courant algebroids.

Contribution

It introduces a novel gauge-theoretic description of the Dirac structure, clarifies its nature as an infinite-dimensional reduction, and establishes a correspondence between Hamiltonian loop group and quasi-Hamiltonian spaces.

Findings

The Dirac structure arises as an infinite-dimensional reduction of connection spaces.

The formal Poisson structure is actually a Dirac structure due to unbounded operators.

Tools for reducing Courant algebroids are developed and applied.

Abstract

In this paper, we solve the problem of giving a gauge-theoretic description of the natural Dirac structure on a Lie Group which plays a prominent role in the theory of D- branes for the Wess-Zumino-Witten model as well as the theory of quasi-Hamiltonian spaces. We describe the structure as an infinite-dimensional reduction of the space of connections over the circle. Our insight is that the formal Poisson structure on the space of connections is not an actual Poisson structure, but is itself a Dirac structure, due to the fact that it is defined by an unbounded operator. We also develop general tools for reducing Courant algebroids and morphisms between them, allowing us to give a precise correspondence between Hamiltonian loop group spaces and quasi- Hamiltonian spaces.