Cancelling Weyl anomaly from position dependent coupling

TL;DR

This paper investigates how position-dependent couplings in conformal field theories can be used to cancel the Weyl anomaly on curved manifolds, especially under specific Weyl transformations, with results in two and four dimensions.

Contribution

It demonstrates the possibility of canceling the Weyl anomaly using position-dependent couplings for certain Weyl transformations in 2D and 4D conformal field theories with marginal deformations.

Findings

Weyl anomaly cancellation is achievable for constant and infinitesimal Weyl transformations.

The cancellation persists at finite order when the Weyl scaling factor is annihilated by conformal powers of Laplacian.

Mathematical properties of Q-curvature facilitate the anomaly cancellation under specific conditions.

Abstract

Once we put a quantum field theory on a curved manifold, it is natural to further assume that coupling constants are position dependent. The position dependent coupling constants then provide an extra contribution to the Weyl anomaly so that we may attempt to cancel the entire Weyl anomaly on the curved manifold. We show that such a cancellation is possible for constant Weyl transformation or infinitesimal but generic Weyl transformation in two and four dimensional conformal field theories with exactly marginal deformations. When the Weyl scaling factor is annihilated by conformal powers of Laplacian (e.g. by Fradkin-Tseytlin-Riegert-Paneitz operator in four dimensions), the cancellation persists even at the finite order thanks to a nice mathematical property of the $Q$-curvature under the Weyl transformation.