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 M = M 3 × S 2 , where is an equivariant parameter for rotation. We show that, when 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 M 3 = Σ g × S 1 , 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 AdS 6 × w S 4 and AdS 7 × S 4 . We also provide the supersymmetric background and explicitly perform localization for the mixed index on Σ g × S 1 × S 2 , filling a gap in the literature.

Notes on 5d Partition Functions - II

“…The holographic prediction was verified at large N for certain classes of theories, including Seiberg theories [1] and related quiver theories [2], by explicit field theoretical computation of S 3 b × Σ g partition function in [3]. Similar results for different 5d manifolds have been obtained in the large N limit [4,5] and in Cardy limit [6].…”

“…We can follow the approach of extremizing the twisted superpotential first and then evaluating the free energy on those solutions but for the purposes of swiftness, we follow the approach of extremizing the free energy with respect to both the gauge variable ũ and gauge flux m following [6]. From past experiences, we expect the extremum values for ũ to be imaginary so we substitute ũi → iσ i .…”

D4-branes wrapped on a spindle

“…A More details on the AdS 4 × Σ g solutions A.1 Relation with the Lagrangian of [1] Here we make contact between the D = 6 Lagrangian (2.1), that we use in the paper, and the Lagrangian used in [1], given explicitly in [51]. In order to minimise confusion we have relabelled some of the quantities in [51] as follows ϕ 1,2 → φ1,2 , F 3,6 → 1 2 F 3,6 , m → m 1 , (A.1) 16 Again, the recent paper [41] discusses related setups, for compactifications of d = 5 and d = 6 SCFTs on various smooth manifolds, including S 2 ǫ × Σ g and S 2 ǫ1 × S 2 ǫ2 .…”

“…For g = 0,(5.38) can also be viewed as the off-shell free energy on S 3 × S 2 ǫ of N = 2 SYM in d = 5. Using this, the form (5.38) has been recently proved in[41] for S 3 b × S 2 ǫ .…”

Modular factorization of superconformal indices

“…• The holographic duals of holomorphic blocks were dubbed gravitational blocks [91] and have played a role in a number of follow-ups [92][93][94]. It will be interesting to see how if our modular family of holomorphic blocks can add to their analysis.…”