Supersymmetric Particle in a Spacetime with Torsion and the Index Theorem

TL;DR

This paper explores the effects of torsion on supersymmetric particles and the Dirac operator, linking torsion, the Nieh-Yan form, and the index theorem to verify chiral anomaly results in curved spacetime.

Contribution

It introduces a supersymmetric Lagrangian with torsion, connecting the Nieh-Yan form to the index theorem and providing a new method to verify chiral anomalies in torsioned spaces.

Findings

Derived the supersymmetric Lagrangian with torsion

Computed the Dirac operator index using path integrals

Confirmed the consistency with recent chiral anomaly results

Abstract

The supersymmetric Lagrangian compatible with the presence of torsion in the background spacetime requires, in addition to the minimal coupling, an interaction between the spin and the torsion of the form ${1/2} N_{\mu\nu\lambda\rho}\psi^\mu\psi^\nu\psi^\lambda\psi^\rho$, where $N$ is the Nieh-Yan 4-form. This gives rise to a coupling between helicity ($h$) and the Nieh-Yan density of the form $hN$. The classical Lagrangian allows computing the index for the Dirac operator on a four-dimensional compact manifold with curvature and torsion using the path integral representation for the index. This calculation provides an independent check for the recent result of the chiral anomaly in spaces with torsion.