Free energy and boundary anomalies on $\mathbb{S}^a\times \mathbb{H}^b$ spaces

TL;DR

This paper calculates free energies and boundary anomalies for conformal scalar fields on spaces $\

Contribution

It introduces a new family of spaces $\\mathbb{S}^a\times \\mathbb{H}^b$ and computes their free energies and anomalies, including explicit entanglement entropy results and holographic analyses.

Findings

Exact match of free energy on $\\mathbb{S}^{2n+1}\times \\mathbb{H}^{2k}$ and $\\mathbb{S}^{2n+2k+1}$.

Identification of conformal anomalies from boundary terms in specific spaces.

Holographic computations align with free field results.

Abstract

We compute free energies as well as conformal anomalies associated with boundaries for a conformal free scalar field. To that matter, we introduce the family of spaces of the form $\mathbb{S}^a\times \mathbb{H}^b$, which are conformally related to $\mathbb{S}^{a+b}$. For the case of $a=1$, related to the entanglement entropy across $\mathbb{S}^{b-1}$, we provide some new explicit computations of entanglement entropies at weak coupling. We then compute the free energy for spaces $\mathbb{S}^a\times \mathbb{H}^b$ for different values of $a$ and $b$. For spaces $\mathbb{S}^{2n+1}\times \mathbb{H}^{2k}$ we find an exact match with the free energy on $\mathbb{S}^{2n+2k+1}$. For $\mathbb{H}^{2k+1}$ and $\mathbb{S}^{3}\times \mathbb{H}^{3}$ we find conformal anomalies originating from boundary terms. We also compute the free energy for strongly coupled theories through holography, obtaining similar results.