  We derive a formula for the gravitational part of the spectral action for
Dirac operators on 4-dimensional manifolds with totally anti-symmetric torsion.
We find that the torsion becomes dynamical and couples to the traceless part of
the Riemann curvature tensor. Finally we deduce the Lagrangian for the Standard
Model of particle physics in presence of torsion from the Chamseddine-Connes
Dirac operator.
