Boundary conformal anomalies on hyperbolic spaces and Euclidean balls

Abstract

We compute conformal anomalies for conformal field theories with free conformal scalars and massless spin 1/2 fields in hyperbolic space ℍd and in the ball Bd, for 2≤d≤7. These spaces are related by a conformal transformation. In even dimensional spaces, the conformal anomalies on ℍ2n and B2n are shown to be identical. In odd dimensional spaces, the conformal anomaly on B2n+1 comes from a boundary contribution, which exactly coincides with that of ℍ2n + 1 provided one identifies the UV short-distance cutoff on B2n+1 with the inverse large distance IR cutoff on ℍ2n + 1, just as prescribed by the conformal map. As an application, we determine, for the first time, the conformal anomaly coefficients multiplying the Euler characteristic of the boundary for scalars and half-spin fields with various boundary conditions in d = 5 and d = 7.