AdS$_3$ Gravity and RCFT Ensembles with Multiple Invariants

TL;DR

This work demonstrates that AdS$_3$ gravity partition functions derived via Poincaré series for $SU(N)_1$ WZW models can be expressed as a positive-weight ensemble average over a large family of modular invariants. The authors show that the Gram-type matrices governing the invariant coefficients decompose into products of Toeplitz blocks, enabling explicit inversion for arbitrary numbers of invariants and yielding non-negative weights for the vacuum seed across all $N$. They develop a general scheme for arbitrary $m$ with prime factorization $m=\prod_i p_i^{n_i}$, establishing that the weights factorize and remain positive, thus producing a physically sensible ensemble average over RCFTs. The results extend previous two- and three-invariant cases to arbitrarily many invariants and align with the idea that certain AdS$_3$ gravity theories correspond to averages over boundary CFTs. This broadens the class of models for which Poincaré-sum constructions yield unitary, positive ensembles, and invites a deeper exploration of the criteria governing when such sums “work.”

Abstract

We use the Poincaré series method to compute gravity partition functions associated to SU(N) level 1 WZW models with arbitrarily large numbers of modular invariants. The result is an average over these invariants, with the weights being given by inverting a matrix whose size is of order the number of invariants. For the chosen models, this matrix takes a special form that allows us to invert it for arbitrary size and thereby explicitly calculate the weights of this average. For the identity seed we find that the weights are positive for all N, consistent with each model being dual to an ensemble average over CFT's.