Variations on a theme of MacDowell-Mansouri

TL;DR

This paper explores a gauge-theoretic functional inspired by the MacDowell-Mansouri formulation, revealing that its critical points correspond to specific geometric structures like almost-Kahler and Kahler-Einstein 4-manifolds under certain conditions.

Contribution

It introduces a novel class of gauge-theoretic functionals derived from the Pontryagin density, linking their critical points to special geometric structures on 4-manifolds.

Findings

Critical points are constant scalar curvature almost-Kahler 4-manifolds.

Under additional conditions, critical points are Kahler-Einstein 4-manifolds.

Results connect gauge theory functionals with complex differential geometry.

Abstract

Inspired by the MacDowell-Mansouri formulation of four-dimensional General Relativity, we study a class of four-dimensional gauge-theoretic functionals obtained from the Pontryagin density of a G-connection by inserting, under the trace, a matrix that breaks the gauge group G to a subgroup H. Concretely, we study the model with the pair (G,H) given by (SU(3), U(2)). We show that the critical points of the resulting functional are constant scalar curvature almost-Kahler 4-manifolds. On compact 4-manifolds, a stronger conclusion holds under the additional assumption that the scalar curvature is non-negative and the first Chern class is such that an Einstein metric can exist. In this case, results in the literature imply that the critical points are Kahler-Einstein 4-manifolds.