6D Interpretation of 3D Gravity

TL;DR

This paper reveals a 6D geometric interpretation of 3D gravity through Hitchin theory, showing that 3D gravity can be viewed as a dimensional reduction of a 6D topological theory involving Chern-Simons forms.

Contribution

It establishes a novel 6D perspective on 3D gravity by linking it to Hitchin equations and topological forms, explaining the pure connection formulation.

Findings

3D gravity equations are equivalent to 6D Hitchin equations.

3D gravity can be derived from a 6D Hitchin theory via dimensional reduction.

The reduction involves parametrizing SU(2) invariant 3-forms with connections and 2-forms.

Abstract

We show that 3D gravity, in its pure connection formulation, admits a natural 6D interpretation. The 3D field equations for the connection are equivalent to 6D Hitchin equations for the Chern-Simons 3-form in the total space of the principal bundle over the 3-dimensional base. Turning this construction around one gets an explanation of why the pure connection formulation of 3D gravity exists. More generally, we interpret 3D gravity as the dimensional reduction of the 6D Hitchin theory. To this end, we show that any SU(2) invariant closed 3-form in the total space of the principal SU(2) bundle can be parametrised by a connection together with a 2-form field on the base. The dimensional reduction of the 6D Hitchin theory then gives rise to 3D gravity coupled to a topological 2-form field.