The Spectral Action for Dirac Operators with skew-symmetric Torsion

TL;DR

This paper derives a formula for the spectral action of Dirac operators with anti-symmetric torsion, revealing torsion's dynamical role and its coupling to curvature, and connects it to the Standard Model Lagrangian.

Contribution

It provides a new explicit formula for the spectral action with torsion and explores its implications for gravity and particle physics.

Findings

Torsion becomes dynamical in the spectral action.

Torsion couples to the traceless Riemann curvature tensor.

The Standard Model Lagrangian is derived with torsion included.

Abstract

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.