Since 1997 physicists have understood that anti-de Sitter quantum gravity in d+1 dimensions can emerge from suitable d dimensional conformally invariant quantum field theories. This is a primary example of holography. But the question of precisely which quantum field theories give rise to emergent space-time geometry has remained enigmatic.
Perhaps the simplest setting where this question might be answered is that of 2d quantum field theory and 3d gravity. On the one hand, 3d gravity is simpler than its 4d cousin because there are no propagating gravitons. On the other hand, it does contain black holes with many of the familiar properties of their higher-dimensional cousins. Similarly, 2d conformal theories are simpler than their higher dimensional cousins while exhibiting many similar features.
These two recent papers use a special property of state-counting partition functions of 2d supersymmetric conformal theories -- modular invariance -- to find a criterion distinguishing theories with weakly curved supergravity duals, from those without. Supergravity theories have supersymmetric (so-called "BPS") black hole solutions, whose entropy is precisely captured by a supersymmetric analogue of the partition function (the elliptic genus). Using modularity, the authors show that only theories with a precisely bounded growth in the low-energy density of states in the elliptic genus can correctly capture the black hole entropy, and hence can exhibit weakly curved gravitational duals. In the simplest and most common class of constructions of 3d AdS gravity / 2d CFT dual pairs, they are able to put stringent constraints on the possible elliptic genera of theories with weakly curved duals.
Other considerations -- related to entanglement entropy, and chaotic behavior of correlation functions -- are also thought to be relevant to emergence of weakly curved gravity from quantum field theories. The authors are exploring connections between the criteria in the recent papers, and other potential criteria based on those considerations, in ongoing work.