Chern-Simons Term for BF Theory and Gravity as a Generalized Topological Field Theory in Four Dimensions

TL;DR

This paper establishes a novel connection between Chern-Simons and BF topological field theories using generalized differential calculus, enabling derivation of gravity with cosmological constant from BF theory in four dimensions.

Contribution

It introduces a generalized differential calculus framework that links Chern-Simons and BF theories, and derives gravity as a constrained BF theory in four dimensions.

Findings

BF theory derived from generalized second Chern class

Gravity with cosmological constant obtained from BF theory

Generalized Chern-Weil homomorphism established

Abstract

A direct relation between two types of topological field theories, Chern-Simons theory and BF theory, is presented by using ``Generalized Differential Calculus'', which extends an ordinary p-form to an ordered pair of p and (p+1)-form. We first establish the generalized Chern-Weil homomormism for generalized curvature invariant polynomials in general even dimensional manifolds, and then show that BF gauge theory can be obtained from the action which is the generalized second Chern class with gauge group G. Particularly when G is taken as SL(2,C) in four dimensions, general relativity with cosmological constant can be derived by constraining the topological BF theory.