The Spectral Action for Dirac Operators with skew-symmetric Torsion

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.