Emergent space-time and the supersymmetric index

TL;DR

The work proposes that the supersymmetric elliptic genus of 2d CFTs encodes a moduli-invariant signal for emergent AdS$_3$ gravity, using slow growth in the NS-sector genus as a diagnostic for a gravity-dual with $L_{ m string}\ll L_{ m AdS}$. By leveraging the DMVV formula for symmetric products and analyzing seed weak Jacobi forms, the authors derive precise growth controls, showing that seed theories like ${\rm K3}$ yield sub-Hagedorn growth in Sym$^N(X)$, while generic seeds lead to Hagedorn behavior; they further explore how Hecke operators and very special Jacobi forms can produce cancellations that slow growth, and extend the analysis to wreath products, revealing multiple regimes where gravity-like behavior can emerge or be suppressed. The results provide a concrete, modularly controlled criterion to identify points in CFT moduli space compatible with weakly curved gravity and outline extensions to more complex constructions and refined indices. Overall, the paper strengthens the link between modular invariants in 2d SCFTs and the existence of gravitational duals, offering a practical framework to test emergent spacetime scenarios via the elliptic genus.

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 $L_{\rm string} \sim L_{\rm AdS}$. For supersymmetric CFTs, the elliptic genus can serve as a sensitive probe of whether the SCFT admits a large radius gravity description with $L_{\rm string} \ll L_{\rm AdS}$ 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 $L_{\rm Planck} \ll L_{\rm string} \ll L_{\rm AdS}$ can emerge at suitable points in moduli space.

Figures (2)

• Figure 1: The moduli space of an SCFT, with the symmetric orbifold point and the small region dual to large radius gravity labeled. The elliptic genus computed at any point in the moduli space will match the sparsest spectrum arising anywhere, and so is sensitive to existence of a weakly curved gravity region.
• Figure 2: Growth of coefficients in NS sector of $\text{Sym}^2(\text{Sym}^{40}(\phi_{0,1}))$. Note the three very distinctive regions of $a < \Delta < \frac{b}{4}$, $\frac{b}{4} < \Delta < \frac{ab}{4}$, and $\frac{ab}{4}<\Delta$. (The region $\Delta < a$ is too small here to notice.)