# On the Heat Kernel and Weyl Anomaly of Schr\"odinger invariant theory

## TL;DR

This paper introduces a novel method to compute the heat kernel for Schr"odinger invariant theories from relativistic conformal theories, demonstrating the absence of Weyl anomalies in such non-relativistic models.

## Contribution

The authors develop a DLCQ-inspired approach to relate heat kernels of Schr"odinger and relativistic conformal theories, proving the vanishing of Weyl anomalies in Schr"odinger theories with complex scalars.

## Key findings

- Weyl anomaly for Schr"odinger theory is proportional to delta(m) times relativistic anomaly
- Schr"odinger scalar field theories have no Weyl anomalies
- Method applies to theories with even z>2 and one time derivative

## Abstract

We propose a method inspired from discrete light cone quantization (DLCQ) to determine the heat kernel for a Schr\"odinger field theory (Galilean boost invariant with $z=2$ anisotropic scaling symmetry) living in $d+1$ dimensions, coupled to a curved Newton-Cartan background starting from a heat kernel of a relativistic conformal field theory ($z=1$) living in $d+2$ dimensions. We use this method to show the Schr\"odinger field theory of a complex scalar field cannot have any Weyl anomalies. To be precise, we show that the Weyl anomaly $\mathcal{A}^{G}_{d+1}$ for Schr\"odinger theory is related to the Weyl anomaly of a free relativistic scalar CFT $\mathcal{A}^{R}_{d+2}$ via $\mathcal{A}^{G}_{d+1}= 2\pi \delta (m) \mathcal{A}^{R}_{d+2}$ where $m$ is the charge of the scalar field under particle number symmetry. We provide further evidence of vanishing anomaly by evaluating Feynman diagrams in all orders of perturbation theory. We present an explicit calculation of the anomaly using a regulated Schr\"odinger operator, without using the null cone reduction technique. We generalise our method to show that a similar result holds for one time derivative theories with even $z>2$.