Boundary Conformal Anomalies on Hyperbolic Spaces and Euclidean Balls

TL;DR

This paper calculates conformal anomalies for free scalar and spinor fields in hyperbolic and ball geometries across dimensions 2 to 7, revealing their relationship via conformal transformations and boundary contributions.

Contribution

It provides the first detailed computation of boundary-related conformal anomaly coefficients in 5 and 7 dimensions for various boundary conditions.

Findings

Anomalies in even dimensions are identical on hyperbolic space and ball.

Boundary contributions determine anomalies in odd dimensions.

Explicit anomaly coefficients for scalars and spinors in 5D and 7D are obtained.

Abstract

We compute conformal anomalies for conformal field theories with free conformal scalars and massless spin $1/2$ fields in hyperbolic space $\mathbb{H}^d$ and in the ball $\mathbb{B}^d$, for $2\leq d\leq 7$. These spaces are related by a conformal transformation. In even dimensional spaces, the conformal anomalies on $\mathbb{H}^{2n}$ and $\mathbb{B}^{2n}$ are shown to be identical. In odd dimensional spaces, the conformal anomaly on $\mathbb{B}^{2n+1}$ comes from a boundary contribution, which exactly coincides with that of $\mathbb{H}^{2n+1}$ provided one identifies the UV short-distance cutoff on $\mathbb{B}^{2n+1}$ with the inverse large distance IR cutoff on $\mathbb{H}^{2n+1}$, 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 $d=7$.