# Trace anomaly for non-relativistic fermions

## TL;DR

This paper investigates the trace anomaly of non-relativistic fermions in 2+1 dimensions coupled to curved Newton-Cartan geometry, revealing classical Weyl invariance and quantum anomalies proportional to relativistic coefficients.

## Contribution

It provides a detailed analysis of Weyl invariance and computes the trace anomaly for non-relativistic fermions using null reduction and Heat Kernel methods, extending previous scalar results.

## Key findings

- Classical Weyl invariance is preserved in curved backgrounds.
- Quantum trace anomaly coefficients are proportional to relativistic ones.
- Anomaly coefficients scale as 1/m, with m being the non-relativistic mass.

## Abstract

We study the coupling of a 2+1 dimensional non-relativistic spin 1/2 fermion to a curved Newton-Cartan geometry, using null reduction from an extra-dimensional relativistic Dirac action in curved spacetime. We analyze Weyl invariance in detail: we show that at the classical level it is preserved in an arbitrary curved background, whereas at the quantum level it is broken by anomalies. We compute the trace anomaly using the Heat Kernel method and we show that the anomaly coefficients a, c are proportional to the relativistic ones for a Dirac fermion in 3+1 dimensions. As for the previously studied scalar case, these coefficents are proportional to 1/m, where m is the non-relativistic mass of the particle.