Homological aspects of topological gauge-gravity equivalence

TL;DR

This paper explores the homological structures arising from gauge-gravity dualities, extending previous off-shell formulations to include constraints that lead to nontrivial homology in Riemann-Cartan manifolds.

Contribution

It formalizes the gauge-gravity map using fiber bundle theory and analyzes the impact of constraints on the homology of Riemann-Cartan manifolds.

Findings

Emergence of nontrivial homology in Riemann-Cartan manifolds

Formalization of gauge-gravity correspondence via fiber bundles

Constraints induce partial breaking of diffeomorphism invariance

Abstract

In the works of A. Ach\'ucarro and P. K. Townsend and also by E. Witten, a duality between three-dimensional Chern-Simons gauge theories and gravity was established. In all cases, the results made use of the field equations. In a previous work, we were capable to generalize Witten's work to the off-shell cases, as well as to four dimensional Yang-Mills theory with de Sitter gauge symmetry. The price we paid is that curvature and torsion must obey some constraints under the action of the interior derivative. These constraints implied on the partial breaking of diffeomorphism invariance. In the present work, we, first, formalize our early results in terms of fiber bundle theory by establishing the formal aspects of the map between a principal bundle (gauge theory) and a coframe bundle (gravity) with partial breaking of diffeomorphism invariance. Then, we study the effect of the constraints on the homology defined by the interior derivative. The main result is the emergence of a nontrivial homology in Riemann-Cartan manifolds.