The joy of factorization at large N: five-dimensional indices and AdS black holes

Abstract

Abstract We discuss the large N factorization properties of five-dimensional supersymmetric partition functions for CFT with a holographic dual. We consider partition functions on manifolds of the form ℳ = ℳ 3 × S ϵ 2 $$ \mathrm{\mathcal{M}}={\mathrm{\mathcal{M}}}_3\times {S}_{\upepsilon}^2 $$ , where ϵ is an equivariant parameter for rotation. We show that, when ℳ3 is a squashed three-sphere, the large N partition functions can be obtained by gluing elementary blocks associated with simple physical quantities. The same is true for various observables of the theories on ℳ 3 = Σ g × S 1 $$ {\mathrm{\mathcal{M}}}_3={\Sigma}_{\mathfrak{g}}\times {S}^1 $$ , where Σ g $$ {\Sigma}_{\mathfrak{g}} $$ is a Riemann surface of genus 𝔤, and, with a natural assumption on the form of the saddle point, also for the partition function, corresponding to either the topologically twisted index or a mixed one. This generalizes results in three and four dimensions and correctly reproduces the entropy of known black objects in AdS6 × w S 4 and AdS7 × S 4. We also provide the supersymmetric background and explicitly perform localization for the mixed index on Σ g × S 1 × S ϵ 2 $$ {\Sigma}_{\mathfrak{g}}\times {S}^1\times {S}_{\upepsilon}^2 $$ , filling a gap in the literature.

Keywords

• AdS-CFT Correspondence
• Field Theories in Higher Dimensions
• Supersymmetric Gauge Theory
• Black Holes in String Theory