# Conformal equations that are not Virasoro or Weyl invariant

## TL;DR

This paper demonstrates that most globally conformal invariant differential equations in two dimensions are neither Virasoro nor Weyl invariant, challenging common assumptions in conformal field theory.

## Contribution

It shows that, beyond specific exceptions, conformal invariance does not imply Virasoro or Weyl invariance in two-dimensional differential equations.

## Key findings

- Most conformally invariant equations lack Virasoro invariance.
- Higher spin laws and conformal Killing equations are exceptions.
- The Laplace equation remains conformally invariant.

## Abstract

While the argument by Zamolodchikov and Polchinski suggests global conformal invariance implies Virasoro invariance in two-dimensional unitary conformal field theories with discrete dilatation spectrum, it is not the case in more general situations without these assumptions. We indeed show that almost all the globally conformal invariant differential equations in two dimensions are neither Virasoro invariant nor Weyl invariant. The only exceptions are the higher spin conservation laws, conformal Killing tensor equations and the Laplace equation of a conformal scalar.