Chern–Simons terms in metric-affine space-time: Bianchi identities as Euler–Lagrange equations

Abstract

The metric-affine gauge theory of gravity encompasses a space-time with the following geometrical fields: coframe ϑα, metric g, and an independent linear connection Γαβ. Within this geometrical framework, all four-forms B=dC are constructed which qualify as boundary terms for a gauge Lagrangian, that is, they are GL(4,R)-scalars as well as exact forms derived from Chern–Simons type three-forms C. The result of our search is summarized in Eq. (4.20). The translational piece dCTT is new. The boundary terms effectively serve as Lagrangians for the Bianchi identities of nonmetricity, torsion, and curvature. In the canonical formalism, the normal parts of the Chern–Simons three-forms represent generating functions that are capable of generating new Ashtekar type variables. Eventually, the Bach–Lanczos identity is generalized to the metric-affine space-time.