A note on ensemble holography for rational tori

TL;DR

This paper investigates ensemble-averaged holography in rational free boson CFTs with extended chiral algebras, exploring modular invariants, weights, and potential dual gravity theories involving Chern-Simons actions and wormhole contributions.

Contribution

It introduces a detailed analysis of ensemble averages over rational CFTs with extended chiral algebras, linking modular invariants to holographic dualities and soliton sectors.

Findings

Ensemble weights are equal in primitive chiral algebra cases.

Non-primitive cases involve a semigroup structure on modular invariants.

Evidence for a duality with a gravity theory based on U(1) Chern-Simons action.

Abstract

We study simple examples of ensemble-averaged holography in free compact boson CFTs with rational values of the radius squared. These well-known rational CFTs have an extended chiral algebra generated by three currents. We consider the modular average of the vacuum character in these theories, which results in a weighted average over all modular invariants. In the simplest case, when the chiral algebra is primitive (in a sense we explain), the weights in this ensemble average are all equal. In the non-primitive case the ensemble weights are governed by a semigroup structure on the space of modular invariants. These observations can be viewed as evidence for a holographic duality between the ensemble of CFTs and an exotic gravity theory based on a compact $U(1) \times U(1)$ Chern-Simons action. In the bulk description, the extended chiral algebra arises from soliton sectors, and including these in the path integral on thermal AdS$_3$ leads to the vacuum character of the chiral algebra. We also comment on wormhole-like contributions to the multi-boundary path integral.