# Weyl versus Conformal Invariance in Quantum Field Theory

## TL;DR

This paper demonstrates that conformal invariance in flat spacetime generally implies Weyl invariance in curved backgrounds for unitary theories up to 10 dimensions, analyzing curvature corrections and anomalies.

## Contribution

It establishes a general link between conformal and Weyl invariance in quantum field theories and explores curvature corrections and potential anomalies in various dimensions.

## Key findings

- Conformal invariance implies Weyl invariance in curved backgrounds for theories up to 10 dimensions.
- Curvature corrections to Weyl transformations are absent for low-dimensional, low-spin operators.
- Possible anomalies involve the Weyl (Cotton) tensor in specific dimensions.

## Abstract

We argue that conformal invariance in flat spacetime implies Weyl invariance in a general curved background metric for all unitary theories in spacetime dimensions $d \leq 10$. We also study possible curvature corrections to the Weyl transformations of operators, and show that these are absent for operators of sufficiently low dimensionality and spin. We find possible `anomalous' Weyl transformations proportional to the Weyl (Cotton) tensor for $d > 3$ ($d = 3$). The arguments are based on algebraic consistency conditions similar to the Wess-Zumino consistency conditions that classify possible local anomalies. The arguments can be straightforwardly extended to larger operator dimensions and higher $d$ with additional algebraic complexity.