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

Abstract

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 S a × ℍ b $$ {\mathbb{S}}^a \times {\mathrm{\mathbb{H}}}^b $$ , which are conformally related to S a + b $$ {{\mathbb{S}}^a}^{+b} $$ . For the case of a = 1, related to the entanglement entropy across S b − 1 $$ {{\mathbb{S}}^b}^{-1} $$ , we provide some new explicit computations of entanglement entropies at weak coupling. We then compute the free energy for spaces S a × ℍ b $$ {\mathbb{S}}^a \times {\mathrm{\mathbb{H}}}^b $$ for different values of a and b. For spaces S 2 n + 1 × ℍ 2 k $$ {\mathbb{S}}^{2n+1}\times {\mathrm{\mathbb{H}}}^{2k} $$ we find an exact match with the free energy on S 2 n + 2 k + 1 $$ {\mathbb{S}}^{2n+2k+1} $$ . For ℍ2k + 1 and S 3 × ℍ 3 $$ {\mathbb{S}}^3 \times {\mathrm{\mathbb{H}}}^3 $$ we find conformal anomalies originating from boundary terms. We also compute the free energy for strongly coupled theories through holography, obtaining similar results.

Keywords

• AdS-CFT Correspondence
• Anomalies in Field and String Theories
• Conformal Field Theory