Low-energy general relativity with torsion: a systematic derivative expansion

We attempt to build systematically the low-energy effective Lagrangian for the Einstein--Cartan formulation of gravity theory that generally includes the torsion field. We list all invariant action terms in certain given order; some of the invariants are new. We show that in the leading order the fermion action with torsion possesses additional U(1)_L x U(1)_R gauge symmetry, with 4+4 components of the torsion (out of the general 24) playing the role of Abelian gauge bosons. The bosonic action quadratic in torsion gives masses to those gauge bosons. Integrating out torsion one obtains a point-like 4-fermion action of a general form containing vector-vector, axial-vector and axial-axial interactions. We present a quantum field-theoretic method to average the 4-fermion interaction over the fermion medium, and perform the explicit averaging for free fermions with given chemical potential and temperature. The result is different from that following from the "spin fluid" approach used previously. On the whole, we arrive to rather pessimistic conclusions on the possibility to observe effects of the torsion-induced 4-fermion interaction, although under certain circumstances it may have cosmological consequences.Comment: 33 pages, 1 figure. A new section, discussion and references added. Final (published) versio

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

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.Comment: 12 page latex, as version 2 an old file was submitted by mistake, this is now the real corrected fil

Barbero-Immirzi parameter, manifold invariants and Euclidean path integrals

The Barbero-Immirzi parameter $\gamma$ appears in the \emph{real} connection formulation of gravity in terms of the Ashtekar variables, and gives rise to a one-parameter quantization ambiguity in Loop Quantum Gravity. In this paper we investigate the conditions under which $\gamma$ will have physical effects in Euclidean Quantum Gravity. This is done by constructing a well-defined Euclidean path integral for the Holst action with non-zero cosmological constant on a manifold with boundary. We find that two general conditions must be satisfied by the spacetime manifold in order for the Holst action and its surface integral to be non-zero: (i) the metric has to be non-diagonalizable; (ii) the Pontryagin number of the manifold has to be non-zero. The latter is a strong topological condition, and rules out many of the known solutions to the Einstein field equations. This result leads us to evaluate the on-shell first-order Holst action and corresponding Euclidean partition function on the Taub-NUT-ADS solution. We find that $\gamma$ shows up as a finite rotation of the on-shell partition function which corresponds to shifts in the energy and entropy of the NUT charge. In an appendix we also evaluate the Holst action on the Taub-NUT and Taub-bolt solutions in flat spacetime and find that in that case as well $\gamma$ shows up in the energy and entropy of the NUT and bolt charges. We also present an example whereby the Euler characteristic of the manifold has a non-trivial effect on black-hole mergers.Comment: 18 pages; v2: references added; to appear in Classical and Quantum Gravity; v3: typos corrected; minor revisions to match published versio