Homological aspects of topological gauge-gravity equivalence

Abstract

In the works of A. Achúcarro 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.

Figures (5)

• Figure 1: Schematics of the structure of the principal bundle $P$ describing a gauge theory. The blue surface represents spacetime and the lines represent the fibers $\mathbb{G}(x)$.
• Figure 2: Schematics of the structure of the coframe bundle $P^\prime$ describing a gravity theory. The blue surface represents spacetime and the lines represent the fibers $SO(n)$. The green surfaces represent the collection of all tangent spaces that can be reached from a vielbein map.
• Figure 3: Schematics of the map $P\longmapsto P^\prime_X$ in Theorem \ref{['theo.cofr1']} characterizing the map between a CS theory for the Poincaré group to an EH gravity in three dimensions.
• Figure 4: Schematics of the map $P\longmapsto P^\prime_X$ characterizing the map between a CS theory for the orthogonal group $SO(4)$ group to an EH gravity with cosmological constant in three dimensions.
• Figure 5: Schematics of the map $P\longmapsto P^\prime_X$ characterizing the map between a Pontryagin theory for the orthogonal group $SO(5)$ group to an $SO(4)$ topological gravity in four dimensions.

Theorems & Definitions (66)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Proposition 2.1
• Proposition 2.2
• Definition 2.7
• Definition 2.8