A Sum Rule for Boundary Contributions to the Trace Anomaly

TL;DR

This paper derives a sum rule connecting boundary contributions to the trace anomaly in boundary conformal field theories, validated through free and weakly interacting models.

Contribution

It introduces a new sum rule linking boundary correlators to the trace anomaly, enhancing understanding of boundary effects in conformal field theories.

Findings

Sum rule relates two and three point functions of the displacement operator.

Validation of the sum rule in free theories and a weakly interacting model.

Boundary contributions to the trace anomaly are explicitly connected to boundary correlators.

Abstract

In the context of boundary conformal field theory, we derive a sum rule that relates two and three point functions of the displacement operator. For four dimensional conformal field theory with a three dimensional boundary, this sum rule in turn relates the two boundary contributions to the anomaly in the trace of the stress tensor. We check our sum rule for a variety of free theories and also for a weakly interacting theory, where a free scalar in the bulk couples marginally to a generalized free field on the boundary.