Conformal equations that are not Virasoro or Weyl 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.

Weyl invariance with twisted energy-momentum tensor

In this appendix, we discuss obstructions for the Weyl invariance due to a gravitational anomaly. Suppose we have a conformal field theory with U(1) current algebra generated by the conserved traceless energy-momentum tensor T and the U(1) current J. Here, we take the canonical energy-momentum tensor T (e.g., constructed by the Sugawara form).

This theory admits a one-parameter deformation of the energy-momentum tensor given by the twist

$$\begin{aligned} T' = T + s \partial J \ . \end{aligned}$$

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The twisted energy-momentum tensor is still conserved and traceless ${\bar{\partial }} {T'} = 0$ so that it generates the twisted Virasoro symmetry. Indeed, $L'_{n} = L_n + s (n+1) J_n$ satisfies the Virasoro algebra (with different central charge). Note that without changing the Hilbert space, the original representation of the Virasoro algebra (generically) does not contain the unitary representation of the twisted Virasoro algebra,⁹ but this will not be critical in the following.

Can we construct the Weyl invariant uplift with the twisted energy-momentum tensor? It is not always the case. Let us take an example of free compact boson X with a radius r (i.e., $X \sim X+ 2\pi r$) with the U(1) current $J = \partial X$. In order to realize the twisted energy-momentum tensor

$$\begin{aligned} T' = -\frac{1}{2} \partial X \partial X + s \partial ^2 X \end{aligned}$$

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in the curved background, we need to add the curvature coupling

$$\begin{aligned} \delta S = \int d^2x \sqrt{g} s X R \ . \end{aligned}$$

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However, this curvature coupling does not respect the identification $X \rightarrow X + 2\pi r$ unless s is pure imaginary and quantized in the unit of 1 / r.

Therefore, this theory with generic s does not admit a Weyl invariant uplift, or more precisely there is no diffeomorphism invariant uplift in the curved background, so it evades the argument by Zumino’s theorem in the main text. One may trace back the difficulty to the gravitational anomaly: The twisted energy-momentum tensor has the spectrum with non-integer spin $L_{0}' - {\bar{L}}_0'$. Only when s takes particular values, the twisted spin becomes integers and the theory can be put on the general curved background (at the sacrifice of unitarity).

Note that when X is non-compact, the above difficulty does not arise because the twisting does not change the spin. The theory is known as linear dilaton theory, and it has the Weyl invariant uplift. Another possibility is to restrict the spectrum so that the $L_{0}'-{\bar{L}}_0'$ is an integer: This case was covered by the Coulomb gas approach to minimal models.