Beyond Einstein-Cartan gravity: Quadratic torsion and curvature invariants with even and odd parity including all boundary terms

TL;DR

This paper explores advanced gravitational gauge theories with torsion, incorporating quadratic invariants of curvature and torsion with even and odd parity, revealing new relations and emphasizing the natural role of Riemann-Cartan spacetime for chiral fermions.

Contribution

It establishes the relations between different quadratic torsion and curvature invariants and highlights the importance of torsion in parity violation and chiral matter coupling.

Findings

Inclusion of torsion square terms is essential for discussing the Barbero-Immirzi parameter.

Exact relations between different quadratic gauge Lagrangians are derived using topological forms.

Riemann-Cartan spacetime naturally accommodates parity-violating terms and chiral fermions.

Abstract

Recently, gravitational gauge theories with torsion have been discussed by an increasing number of authors from a classical as well as from a quantum field theoretical point of view. The Einstein-Cartan(-Sciama-Kibble) Lagrangian has been enriched by the parity odd pseudoscalar curvature (Hojman, Mukku, and Sayed) and by torsion square and curvature square pieces, likewise of even and odd parity. (i) We show that the inverse of the so-called Barbero-Immirzi parameter multiplying the pseudoscalar curvature, because of the topological Nieh-Yan form, can only be appropriately discussed if torsion square pieces are included. (ii) The quadratic gauge Lagrangian with both parities, proposed by Obukhov et al. and Baekler et al., emerges also in the framework of Diakonov et al.(2011). We establish the exact relations between both approaches by applying the topological Euler and Pontryagin forms in a Riemann-Cartan space expressed for the first time in terms of irreducible pieces of the curvature tensor. (iii) Only in a Riemann-Cartan spacetime, that is, in a spacetime with torsion, parity violating terms can be brought into the gravitational Lagrangian in a straightforward and natural way. Accordingly, Riemann-Cartan spacetime is a natural habitat for chiral fermionic matter fields.