We compute the superconformal index of the $$ \mathcal{N} $$ N = 4 SU(N) Yang-Mills theory through a residue calculation. The method is similar in spirit to the Bethe Ansatz formalism, except that all poles are explicitly known, and we do not require specialization of any of the chemical potentials. Our expression for the index allows us to revisit the Cardy limit using modular properties of four-dimensional supersymmetric partition functions. We find that all residues contribute at leading order in the Cardy limit. In a specific region of flavour chemical potential space, close to the two unrefined points, in fact all residues contribute universally. These universal residues precisely agree with the entropy functions of the asymptotically AdS5 black hole and its “twin saddle” respectively. Finally, we discuss how our formula is suited to study the implications of four-dimensional modularity for the index beyond the Cardy limit.

The microscopic description of AdS space obeys the holographic principle in the sense that the number of microscopic degrees of freedom is given by the area of the holographic boundary. We assume the same applies to the microscopic holographic theories for non-AdS spacetimes, specifically for Minkowski, de Sitter, and AdS below its curvature radius. By taking general lessons from AdS/CFT we derive the cut-off energy of the holographic theories for these non-AdS geometries. Contrary to AdS/CFT, the excitation energy decreases towards the IR in the bulk, which is related to the negative specific heat of black holes. We construct a conformal mapping between the non-AdS geometries and AdS 3 × S q spacetimes, and relate the microscopic properties of the holographic theories for non-AdS spaces to those of symmetric product CFTs. We find that the mechanism responsible for the inversion of the energy-distance relation corresponds to the long string phenomenon. This same mechanism naturally explains the negative specific heat for nonAdS black holes and the value of the vacuum energy in (A)dS spacetimes.

Extremal black holes in general dimensions are well known to contain an AdS 2 factor in their near-horizon geometries. If the extremal limit is taken in conjunction with a specific vanishing horizon limit, the so-called Extremal Vanishing Horizon (EVH) limit, the AdS 2 factor lifts to a locally AdS 3 factor with a pinching angular direction. In this paper, we study the EVH limit of asymptotically AdS black holes which preserve some supersymmetry. The primary example we consider is the 1/16 th BPS asymptotically AdS 5 black hole, whose EVH limit has an AdS 3 factor in its near-horizon geometry. We also consider the near-EVH limit of this black hole, in which the near-horizon geometry instead contains an extremal BTZ factor. We employ recent results on the large-N limit of the superconformal index of the dual CFT 4 to understand the emergence of a CFT 2 in the IR of the CFT 4 , which is the field theory dual to the emergence of the locally AdS 3 factor in the near-horizon geometry. In particular, we show that the inverse Laplace transform of the superconformal index, yielding the black hole entropy, becomes equivalent to the derivation of a Cardy formula for the dual CFT 2 . Finally, we examine the EVH limit of supersymmetric black holes in other dimensions.