Topological field theories in n-dimensional spacetimes and Cartan's equations

TL;DR

This paper develops a general framework for topological field theories in n-dimensional spacetimes based on Cartan's equations, and demonstrates how adding constraints can produce theories with local degrees of freedom, such as 2D gravity with matter.

Contribution

It introduces a new action principle for topological field theories in arbitrary dimensions that incorporates auxiliary fields and detailed canonical analysis.

Findings

The action principles define topological field theories in n dimensions.

Adding constraints to the fields yields theories with local degrees of freedom.

Constructs 2D gravity coupled to matter with local excitations.

Abstract

Action principles of the BF type for diffeomorphism invariant topological field theories living in n-dimensional spacetime manifolds are presented. Their construction is inspired by Cuesta and Montesinos' recent paper where Cartan's first and second structure equations together with first and second Bianchi identities are treated as the equations of motion for a field theory. In opposition to that paper, the current approach involves also auxiliary fields and holds for arbitrary n-dimensional spacetimes. Dirac's canonical analysis for the actions is detailedly carried out in the generic case and it is shown that these action principles define topological field theories, as mentioned. The current formalism is a generic framework to construct geometric theories with local degrees of freedom by introducing additional constraints on the various fields involved that destroy the topological character of the original theory. The latter idea is implemented in two-dimensional spacetimes where gravity coupled to matter fields is constructed out, which has indeed local excitations.