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 $$ \mathrm{\mathcal{M}}={\mathrm{\mathcal{M}}}_3\times {S}_{\upepsilon}^2 $$ ℳ = ℳ 3 × S ϵ 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 $$ {\mathrm{\mathcal{M}}}_3={\Sigma}_{\mathfrak{g}}\times {S}^1 $$ ℳ 3 = Σ g × S 1 , where $$ {\Sigma}_{\mathfrak{g}} $$ 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 $$ {\Sigma}_{\mathfrak{g}}\times {S}^1\times {S}_{\upepsilon}^2 $$ Σ g × S 1 × S ϵ 2 , filling a gap in the literature.

The large N limit of topologically twisted indices: a direct approach

“…The result (1.2) is in complete agreement with the general holographic expectations based on gravitational blocks [13] and correctly reproduce the entropy of the rotating AdS 4 × S 7 black holes found in [5]. Similar large N factorizations hold in higher dimensions too [8]. In this paper we work to all orders but perturbatively in .…”

“…Then we can study the saddle point approximation of the refined index matrix model (2.1) with respect to u i and m i . The method has been efficiently used to study five-dimensional indices [6][7][8].…”

“…We will assume that in the large N limit the fluxes can be treated as continuous variables and we will take a saddle point approximation with respect to both holonomies and fluxes. The method has been already efficiently used to study five-dimensional indices [6][7][8]. We will of course reproduce the results in [2] and their generalizations [9,10] for = 0 and we will provide the general result for generic .…”

Gravitational Blocks, Spindles and GK Geometry

“…The main result of this section will be that the supersymmetric action (2.9), together with the associated constraint (2.10) and flux quantization conditions (2.11), localize to integrals over the fibres X ± over the poles of the spindle. 9 We refer to these contributions as gravitational blocks, since they generalize similar (conjectured) formulas that have already appeared in the literature -see [23,25,30,45,48,51,52].…”

“…There it was shown that the entropy can be obtained by extremizing certain entropy functions that are obtained by summing ("gluing") basic building blocks. This observation was inspired by the factorization of partition functions of N = 2 field theories in d = 3, proven in [46] (for later developments in field theory see also [47][48][49]). In subsequent work [30] it was shown that for the class of explicit supergravity solutions associated with M2 and D3-branes 3 wrapping spindles, obtained by uplifting solutions in D = 4, 5 STU gauged supergravity theories on S 7 , S 5 , respectively, it is possible to write the off-shell trial entropy and trial central charge in the form of gravitational blocks, both for the twist and the anti-twist [30,31,[34][35][36][37].…”