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

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 $$\mathbb{S}$^{2n+1}\times $\mathbb{H}$^{2k}$wefindanexactmatchwiththefreeenergyon$$\mathbb{S}$^{2n+2k+1}$\mathrm{.}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.