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

TL;DR

This paper explores the large N factorization of five-dimensional supersymmetric partition functions with holographic duals, revealing how they can be constructed from basic building blocks and matching black hole entropy in AdS spaces.

Contribution

It extends large N factorization results to five-dimensional theories, providing new localization computations and connecting to black hole entropy in higher-dimensional AdS spaces.

Findings

Partition functions factorize into elementary blocks.

Reproduces entropy of AdS black objects.

Provides explicit localization for mixed indices.

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 $\mathcal{M}= \mathcal{M}_3 \times S^2_\epsilon$, 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 $\Sigma_\mathfrak{g} \times S^1 \times S^2_\epsilon$, filling a gap in the literature.