$SL(3,\mathbb{Z})$ Modularity and New Cardy Limits of the $\mathcal{N}=4$ Superconformal Index

Abstract

The entropy of $1/16$-th BPS AdS${}_5$ black holes can be microscopically accounted for by the superconformal index of the $\mathcal{N}=4$ super-Yang-Mills theory. One way to compute this is through a Cardy-like limit of a formula for the index obtained in [1] using the "$S$-transformation" of the elliptic $\Gamma$ function. In this paper, we derive more general $SL(3,\mathbb{Z})$ modular properties of the elliptic $\Gamma$ function. We then use these properties to obtain a three integer parameter family of generalized Cardy-like limits of the $\mathcal{N}=4$ superconformal index. From these limits, we obtain entropy formulae that have a similar form as that of the original AdS${}_5$ black hole, up to an overall rescaling of the entropy. We interpret this both on the field theory and the gravitational side. Finally, we comment on how our work suggests a generalization of the Farey tail to four dimensions.

arXiv comment63 pages + appendices. v2: refs added, various corrections and improvements, discussion section rewritten. v3: corrected defn normalized partition function, added tau=sigma limit modular property in section 3.5 + its physical interpretation in section 5.2. (published version)

References in corpus (67)

• Gauge Theory Correlators from Non-Critical String Theory
• Microscopic Origin of the Bekenstein-Hawking Entropy
• An Index for 4 dimensional Super Conformal Theories
• AdS3 Black Holes and a Stringy Exclusion Principle
• Counting chiral primaries in N=1 d=4 superconformal field theories
• General supersymmetric AdS5 black holes
• General Non-Extremal Rotating Black Holes in Minimal Five-Dimensional Gauged Supergravity
• Supersymmetric AdS5 black holes
• Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories
• Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS${}_5$ black holes
• Supersymmetric multi-charge AdS_5 black holes
• A Black Hole Farey Tail
• Five-Dimensional Gauged Supergravity Black Holes with Independent Rotation Parameters
• The Library of Babel: On the origin of gravitational thermodynamics
• Localization on Hopf surfaces
• 4d Index to 3d Index and 2d TQFT
• An extremization principle for the entropy of rotating BPS black holes in AdS${}_5$
• Superstars and Giant Gravitons
• A Farey Tail for Attractor Black Holes
• Black holes in 4d $\mathcal{N}=4$ Super-Yang-Mills
• The elliptic gamma function and SL(3,Z) x Z^3
• Higgs branch localization in three dimensions
• Elliptic Genera and N=2 Superconformal Field Theory
• The asymptotic growth of states of the 4d N=1 superconformal index
• Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula
• The $\mathcal{N}=1$ Chiral Multiplet on $T^2\times S^2$ and Supersymmetric Localization
• A 4d N=1 Cardy Formula
• Cardy-like asymptotics of the 4d $\mathcal{N}=4$ index and AdS${}_5$ blackholes
• EFT and the SUSY Index on the 2nd Sheet
• Gluing gravitational blocks for AdS black holes
• Factorisation and holomorphic blocks in 4d
• High-temperature asymptotics of supersymmetric partition functions
• Supersymmetric Casimir Energy and the Anomaly Polynomial
• Supersymmetric phases of 4d N=4 SYM at large N
• Asymptotic growth of the 4d $\mathcal{N}=4$ index and partially deconfined phases
• A gravity interpretation for the Bethe Ansatz expansion of the $\mathcal{N}=4$ SYM index
• AdS black holes and finite N indices
• Sub-leading Structures in Superconformal Indices: Subdominant Saddles and Logarithmic Contributions
• Global Properties of Supersymmetric Theories and the Lens Space
• Superconformal indices at large $N$ and the entropy of AdS${}_5$ $\times$ SE${}_5$ black holes
• Entropy of near-extremal black holes in AdS_5
• tt* Geometry in 3 and 4 Dimensions
• Higgs branch localization of $\mathcal{N}=1$ theories on $S^3\times S^1$
• S-duality and the N=2 Lens Space Index
• Residues, modularity, and the Cardy limit of the 4d $\mathcal{N}=4$ superconformal index
• Delayed Deconfinement and the Hawking-Page Transition
• The large-$N$ limit of the 4d $\mathcal{N}=1$ superconformal index
• Black hole entropy function for toric theories via Bethe Ansatz
• Entropy function from toric geometry
• Modular invariance on $S^1\times S^3$ and circle fibrations
• AdS${}_5$ Black Hole Entropy near the BPS Limit
• Gravitational Cardy Limit and AdS Black Hole Entropy
• Localization of 4d $\mathcal{N}=1$ theories on ${\mathbb{D}}^2\times {\mathbb{T}}^2$
• Superconformal index of low-rank gauge theories via the Bethe Ansatz
• Comments on deconfinement in AdS/CFT
• Factorization of 4d N=1 superconformal index
• Probing the EVH limit of supersymmetric AdS black holes
• Modularity of supersymmetric partition functions
• Supersymmetric Casimir Energy and $\mathrm{SL}(3,\mathbb{Z})$ Transformations
• Microscopic Entropy of AdS${}_3$ Black Holes Revisited
• Toward an Effective CFT${}_2$ from $\mathcal{N}=4$ Super Yang-Mills and Aspects of Hawking Radiation
• The growth of the $\frac{1}{16}$-BPS index in 4d $\mathcal{N}=4$ SYM
• From multi-gravitons to Black holes: The role of complex saddles
• The superconformal index of $\mathcal{N}=4$ $USp(2N_c)$ and $SO(N_c)$ SYM as a matrix integral
• Expanding on the Cardy-like limit of the superconformal index of 4d $\mathcal{N}=1$ ABCD SCFTs
• Microstate Counting via Bethe Ansätze in the 4d $\mathcal{N}=1$ Superconformal Index
• The Bethe-Ansatz approach to the $\mathcal{N}=4$ superconformal index at finite rank

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• Words to describe a black hole
• Exact Stringy Microstates from Gauge Theories
• Decomposition of BPS Moduli Spaces and Asymptotics of Supersymmetric Partition Functions
• Supersymmetric phases of AdS${}_4$/CFT${}_3$
• The Panorama of Spin Matrix Theory
• Quantum Phases of $4d$ $SU(N)$ $\mathcal{N}=4$ SYM
• Supersymmetric Grey Galaxies, Dual Dressed Black Holes and the Superconformal Index
• The large-$N$ limit of 4d superconformal indices for general BPS charges
• Root of unity asymptotics for Schur indices of 4d Lagrangian theories
• S-duality in the Cardy-like limit of the superconformal index
• Off-shell phase diagram of BPS black holes in AdS${}_5$