Is boundary conformal in CFT?

TL;DR

This paper investigates conditions under which boundary scale invariance in conformal field theories implies boundary conformal invariance, providing proofs, counterexamples, and holographic arguments across different dimensions.

Contribution

It offers new insights into boundary conformal invariance, including proofs, counterexamples, and holographic methods, extending understanding beyond previous assumptions.

Findings

Cardy's condition is necessary but not sufficient for boundary conformal invariance in 1+1 dimensions.

In 1+2 dimensions, Cardy's condition is sufficient for boundary conformal invariance.

Holographic proof supports boundary conformal invariance under the boundary null energy condition.

Abstract

We discuss boundary conditions for conformal field theories that preserve the boundary Poincare invariance. As in the bulk field theories, a question arises whether boundary scale invariance leads to boundary conformal invariance. With unitarity, Cardy's condition of vanishing momentum flow is necessary for the boundary conformal invariance, but it is not sufficient in general. We show both a proof and a counterexample of the enhancement of boundary conformal invariance in (1+1) dimension, which depends on the extra assumption we make. In (1+2) dimension, Cardy's condition is shown to be sufficient. In higher dimensions, we give a perturbative argument in favor of the enhancement based on the boundary g-theorem. With the help of the holographic dual recently proposed, we show a holographic proof of the boundary conformal invariance under the assumption of the boundary strict null energy condition, which also gives a sufficient condition for the strong boundary g-theorem.