# Note on Weyl versus Conformal Invariance in Field Theory

## TL;DR

This paper examines the relationship between Weyl and conformal invariance in field theories, showing that unitarity alone does not guarantee Weyl invariance and clarifying the conditions under which primary scalar operators are Weyl-covariant.

## Contribution

It demonstrates that unitarity alone is insufficient for Weyl invariance and explicitly constructs Weyl-covariant operators from primary scalars in curved backgrounds.

## Key findings

- Unitarity does not imply Weyl invariance in general.
- Primary scalar operators become Weyl-covariant when coupled to gravity in a Weyl invariant manner.
- Explicit correspondence between relevant/marginal operators and Weyl-covariant operators.

## Abstract

It was argued recently that conformal invariance in flat spacetime implies Weyl invariance in a general curved background for unitary theories and possible anomalies in the Weyl variation of scalar operators are identified. We argue that generically unitarity alone is not sufficient for a conformal field theory to be Weyl invariant. Furthermore, we show explicitly that when a unitary conformal field theory couples to gravity in a Weyl invariant way, each primary scalar operator that is either relevant or marginal in the unitary conformal field theory corresponds to a Weyl-covariant operator in the curved background.