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

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 $\mathcal{M}={\mathcal{M}}_3\times S_{\epsilon }^2$, where $\epsilon$ is an equivariant parameter for rotation. We show that, when ${\mathcal{M}}_3$ 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 ${\mathcal{M}}_3={\Sigma }_{\mathfrak{g}}\times S^1$, where ${\Sigma }_{\mathfrak{g}}$ is a Riemann surface of genus $\mathfrak{g}$, 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 AdS${}_6{\times }_w{S}^4$ and AdS${}_7\times S^4$. We also provide the supersymmetric background and explicitly perform localization for the mixed index on ${\Sigma }_{\mathfrak{g}}\times S^1\times S_{\epsilon }^2$, filling a gap in the literature.

arXiv comment76 pages

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