  It is well known that only the axial piece of the torsion couples minimally
to fermions in a Riemann-Cartan geometry, while the other ones decouple. In
this paper, we consider the Dirac field minimally coupled to a dynamical
background with torsion and compute its contribution to the fermionic one-loop
effective action. Such a contribution owns topological nature since it can be
linked with topological invariants from Riemann-Cartan spaces, like Nieh-Yan
and Chern-Simons terms. Furthermore, we propose a novel modified theory of
gravity constructed by adding the aforementioned one-loop contribution to the
Einstein-Cartan action. The modified field equations reduce to those ones of GR
under certain circumstances, providing therefore trivial solutions. However, in
particular, we find a non-trivial solution where the modified field equations
do not reduce to the GR ones.
