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 the 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.
Highlights
The Weyl anomaly in quantum field theory has a long history. It states that the Weyl transformation of the metric in the curved space-time may not be a symmetry of the system even though the quantum field theory under consideration has the conformal symmetry in the flat space-time
In most conformal field theories, the Weyl anomaly is nonvanishing, and we say that the Weyl symmetry is quantum mechanically broken in curved space-time
What we would like to study in this paper is to find a way to cancel the Weyl anomaly from the other source, e.g., from the position-dependent coupling constant
Summary
The Weyl anomaly in quantum field theory has a long history (see, e.g., Ref. [1] for a historical overview). [1] for a historical overview) It states that the Weyl transformation of the metric in the curved space-time may not be a symmetry of the system even though the quantum field theory under consideration has the conformal symmetry in the flat space-time. In most conformal field theories, the Weyl anomaly is nonvanishing, and we say that the Weyl symmetry is quantum mechanically broken in curved space-time. What we would like to study in this paper is to find a way to cancel the Weyl anomaly from the other source, e.g., from the position-dependent coupling constant.. One way to avoid this is to cancel the anomaly of the background gauge field from the space-time curvature and vice versa.
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