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

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=M3×Sϵ2, 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× S1, where Σ 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×wS4 and AdS7× S4. We also provide the supersymmetric background and explicitly perform localization for the mixed index on Σg×S1×Sϵ2, filling a gap in the literature.