Narain to Narnia

TL;DR

This paper extends the holographic correspondence between 3D topological gravity with Chern-Simons theory and 2D Narain CFTs, exploring its applicability to orbifolds and correlation functions, and proposing a conjecture on the central charge requirement.

Contribution

It generalizes the holographic correspondence to toroidal orbifolds and correlation functions, and introduces a conjecture relating the central charge to the asymptotic density of states.

Findings

The correspondence applies to toroidal orbifolds but not to K3 or Calabi-Yau models.

It extends to correlation functions of twist operators using topological properties of rational tangles.

The conjecture links the central charge to the asymptotic density of states of the chiral algebra.

Abstract

We generalize the holographic correspondence between topological gravity coupled to an abelian Chern-Simons theory in three dimensions and an ensemble average of Narain's family of massless free bosons in two dimensions, discovered by Afkhami-Jeddi et al. and by Maloney and Witten. We find that the correspondence also works for toroidal orbifolds but not for K3 or Calabi-Yau sigma-models and not always for the minimal models. We conjecture that the correspondence requires that the central charge is equal to the critical central charge defined by the asymptotic density of states of the chiral algebra. For toroidal orbifolds, we extend the holographic correspondence to correlation functions of twist operators by using topological properties of rational tangles in the three-dimensional ball, which represent configurations of vortices associated to a discrete gauge symmetry.