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) × 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 AdS3 leads to the vacuum character of the chiral algebra. We also comment on wormhole-like contributions to the multi-boundary path integral.
Highlights
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) × U(1) ChernSimons action
We consider the modular average of the vacuum character in these theories, which results in a weighted average over all modular invariants
The extended chiral algebra arises from soliton sectors, and including these in the path integral on thermal AdS3 leads to the vacuum character of the chiral algebra
Summary
We start by reviewing (see [22, 27] for more details) rational CFTs which possess an extended chiral symmetry algebra Ak characterized by a positive integer ‘level’ k. The algebra is generated by three chiral currents: a spin-1 current J and two spin-k currents W + and W − It is most described by its free field realization in terms of a compact free boson at a rational value of the radius squared. Choosing a pair of relatively prime positive integers p, q such that k = pq (p and q are allowed to be one), we consider a compact boson at radius R2 = p/q. This theory contains primaries of the form. In the case that k is prime does Ak α2 have a unique compact boson realization
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