Cartan’s equations define a topological field theory of the $BF$ type

Cartan’s first and second structure equations together with first and second Bianchi identities can be interpreted as equations of motion for the tetrad, the connection and a set of two-form fields $T^I$ and $R_J^I$. From this viewpoint, these equations define by themselves a field theory. Restricting the analysis to four-dimensional spacetimes (keeping gravity in mind), it is possible to give an action principle of the $BF$ type from which these equations of motion are obtained. The action turns out to be equivalent to a linear combination of the Nieh-Yan, Pontrjagin, and Euler classes, and so the field theory defined by the action is topological. Once Einstein’s equations are added, the resulting theory is general relativity. Therefore, the current results show that the relationship between general relativity and topological field theories of the $BF$ type is also present in the first-order formalism for general relativity.