  We study the mechanism of chiral symmetry breaking for fermionic systems in a
gravitational background with curvature and torsion. The analysis is based on a
scale-dependent effective potential derived from a bosonized version of the
Nambu-Jona-Lasino model in a Riemann-Cartan background. We have investigated
the fate of chiral symmetry in two different regimes. First, to gain some
intuition on the combined effect of curvature and torsion, we investigate the
regime of weak curvature and torsion. However, this regime does not access the
deep infrared limit, which is essential to answer questions related to the
mechanism of gravitational catalysis in fermionic systems. Second, we look at
the regime of vanishing curvature and homogeneous torsion. In this case,
although we cannot probe the combined effects of curvature and torsion, we can
access the deep infrared contributions of the background torsion to the
mechanism of chiral symmetry breaking. Our main finding is that, in the
scenario where only torsion is present, there is no indication of a mechanism
of gravitational catalysis.
