Quadratic Lagrangians and Topology in Gauge Theory Gravity

TL;DR

This paper investigates topological invariants in gauge theory gravity, revealing their origins, properties, and implications for quadratic Lagrangians, while simplifying existing derivations and exploring connections to Yang-Mills instantons.

Contribution

It introduces new topological invariants in gauge gravity, generalizes their identities, and simplifies the derivation process compared to prior work.

Findings

Identified two topological invariants from scalar and pseudoscalar parts.

Reduced ten quadratic Riemann tensor terms to eight independent terms.

Derived field equations for parity non-violating quadratic Lagrangians.

Abstract

We consider topological contributions to the action integral in a gauge theory formulation of gravity. Two topological invariants are found and are shown to arise from the scalar and pseudoscalar parts of a single integral. Neither of these action integrals contribute to the classical field equations. An identity is found for the invariants that is valid for non-symmetric Riemann tensors, generalizing the usual GR expression for the topological invariants. The link with Yang-Mills instantons in Euclidean gravity is also explored. Ten independent quadratic terms are constructed from the Riemann tensor, and the topological invariants reduce these to eight possible independent terms for a quadratic Lagrangian. The resulting field equations for the parity non-violating terms are presented. Our derivations of these results are considerably simpler that those found in the literature.