Limits of minimal models and continuous orbifolds

TL;DR

This work identifies the $\lambda=0$ limit of ${\cal W}_N$ minimal models with the singlet sector of a free boson theory, then completes this sector via a continuous orbifold by $G={\rm SU}(N)/\mathbb{Z}_N$ to restore modular invariance. The untwisted sector reproduces the singlet spectrum, while twisted sectors account for the previously neglected light states; for $N=2$ the twisted sector precisely matches the Runkel-Watts $c\to1$ limit and yields crossing symmetry. The analysis connects to a higher-spin AdS$_3$ dual perspective and provides a concrete CFT construction for the lambda=0 regime, with clear identifications between orbifold twists, light states, and coset-limit representations. The results suggest a robust framework potentially extendable to finite $\lambda$ and related to Liouville-type limit theories, offering insight into the structure of light states in higher-spin holography.

Abstract

The lambda=0 't Hooft limit of the 2d W_N minimal models is shown to be equivalent to the singlet sector of a free boson theory, thus paralleling exactly the structure of the free theory in the Klebanov-Polyakov proposal. In 2d, the singlet sector does not describe a consistent theory by itself since the corresponding partition function is not modular invariant. However, it can be interpreted as the untwisted sector of a continuous orbifold, and this point of view suggests that it can be made consistent by adding in the appropriate twisted sectors. We show that these twisted sectors account for the `light states' that were not included in the original 't Hooft limit. We also show that, for the Virasoro minimal models (N=2), the twisted sector of our orbifold agrees precisely with the limit theory of Runkel & Watts. In particular, this implies that our construction satisfies crossing symmetry.