Is boundary conformal in CFT?

We discuss boundary conditions for conformal field theories that preserve the boundary Poincaré 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.