New look at the Riemann-Cartan theory

The geometry of torsion in the Riemann-Cartan (RC) theory can be described by an Abelian axial-vector field interacting with the axial-vector fermion current in a purely Riemannian background. On the basis of this observation we note that the Schwinger model formulated in curved spacetime can be interpreted as the two-dimensional version of the RC theory. In two dimensions as well as in four dimensions there is a one-parameter family of regulators that can be used to compute the axial anomaly. In four dimensions we set the value of the arbitrary parameter equal to zero and compute the axial anomaly, including counterterms, using Fujikawa's approach. The addition of the Wess-Zumino Lagrangian changes the original RC theory into a nonanomalous Abelian gauge theory of the torsion field. Guided by the analogy with the Schwinger model, we offer several forms of Lgravity from which one can deduce the spin content of the quanta of torsion