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

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.

36 pages, no figures. V2: refs. added

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