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.
Highlights
At large N, dual to AdS gravity theories with sequentially smaller curvatures, the simple condition that the KK spectrum converge at large N is already nontrivial
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
For supersymmetric CFTs, the elliptic genus can serve as a sensitive probe of whether the SCFT admits a large radius gravity description with Lstring LAdS after one deforms away from the symmetric orbifold point in moduli space
Summary
We discuss some basic facts about elliptic genera of 2d SCFTs that we will use throughout the rest of the paper. The elliptic genera we will study have vanishing weight, and their index is related to the central charge of the CFT by c = 6M. It is an important fact that weak Jacobi terms are determined by the terms of negative polarity, which implies that for a given weight and index the space of such objects is finite dimensional. The space of weak Jacobi forms of even weight and integral index is a bi-graded ring with four generators. Two of these are the Eisenstein series E4. With (w, M ) = (0, 1) and (−2, 1) respectively Given these generators, it is a simple matter to write down a basis of weight 0 and index M for any desired M
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