Spinors in Weyl Geometry

TL;DR

This paper investigates the behavior of spinor fields in Weyl geometry, demonstrating how to maintain conformal invariance and computing the one-loop effective action in two dimensions, revealing a surprising equivalence with scalar fields.

Contribution

It introduces a method to couple spinors to Weyl geometry preserving conformal invariance and computes the effective action, showing its equivalence to scalar field cases in two dimensions.

Findings

Conformal invariance is maintained for spinors in Weyl geometry.

The one-loop effective action for spinors matches that of scalar fields in two dimensions.

The result suggests a deeper connection between spinor and scalar field theories in this setting.

Abstract

We consider the wave equation for spinors in ${\cal D}$-dimensional Weyl geometry. By appropriately coupling the Weyl vector $\phi _{\mu}$ as well as the spin connection $\omega _{\mu a b } $ to the spinor field, conformal invariance can be maintained. The one loop effective action generated by the coupling of the spinor field to an external gravitational field is computed in two dimensions. It is found to be identical to the effective action for the case of a scalar field propagating in two dimensions.