Topological Gravity versus Supergravity on Manifolds with Special Holonomy

TL;DR

This paper develops a topological gravity theory in four dimensions by imposing self-duality on the spin connection, revealing connections to supergravity and manifolds with special holonomy.

Contribution

It constructs a topological gravity model that relates to N=2 supergravity through twisting, highlighting its relevance for manifolds with special holonomy.

Findings

Topological gravity is SU(2) invariant, with full SO(4) invariance recovered after untwisting.

The theory parallels topological Yang-Mills in eight dimensions with broken and recovered invariance.

Reveals the importance of topological theories for understanding manifolds with special holonomy.

Abstract

We construct a topological theory for euclidean gravity in four dimensions, by enforcing self-duality conditions on the spin connection. The corresponding topological symmetry is associated to the SU(2) X diffeomorphism X U(1) invariance. The action of this theory is that of d=4, N=2 supergravity, up to a twist. The topological field theory is SU(2) invariant, but the full SO(4) invariance is recovered after untwist. This suggest that the topological gravity is relevant for manifolds with special holonomy. The situation is comparable to that of the topological Yang-Mills theory in eight dimensions, for which the SO(8) invariance is broken down to Spin(7), but is recovered after untwisting the topological theory.