Emergent space-time and the supersymmetric index

Abstract

It is of interest to find criteria on a 2d CFT which indicate that it gives rise to emergent gravity in a macroscopic 3d AdS space via holography. Symmetric orbifolds in the large $N$ limit have partition functions which are consistent with an emergent space-time string theory with $L_{\mathrm{string}}\sim L_{\mathrm{AdS}}$. For supersymmetric CFTs, the elliptic genus can serve as a sensitive probe of whether the SCFT admits a large radius gravity description with $L_{\mathrm{string}}\ll L_{\mathrm{AdS}}$ after one deforms away from the symmetric orbifold point in moduli space. We discuss several classes of constructions whose elliptic genera strongly hint that gravity with $L_{\mathrm{Planck}}\ll L_{\mathrm{string}}\ll L_{\mathrm{AdS}}$ can emerge at suitable points in moduli space.

29 pages, 2 figures; v2, very minor changes; v3, very minor additions to sections 3.2 and 4.2

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• Entanglement Entropy at Large Central Charge
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• Permutation orbifolds and holography
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• $\mathcal{N}=2$ Minimal Models: A Holographic Needle in a Symmetric Orbifold Haystack
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• Small Black Holes and Near-Extremal CFTs
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• Siegel Paramodular Forms and Sparseness in AdS${}_3$/CFT${}_2$
• Twisted holography on AdS${}_3\times S^3\times K3$ & the planar chiral algebra
• Siegel Paramodular Forms from Exponential Lifts: Slow versus Fast Growth