On the Spectrum of Pure Higher Spin Gravity

TL;DR

This paper analyzes the spectrum of pure higher spin gravity in AdS3, revealing the structure of light states, modular invariance constraints, and the necessity of conical defect states for unitarity.

Contribution

It provides a detailed spectral analysis of higher spin theories in AdS3 and introduces conical defect states to restore unitarity in the spectrum.

Findings

Light spectrum includes massless particles of spins 2 to N

Modular invariance constrains the heavy spectrum and indicates negative norm states

Inclusion of conical defect states with specific deficit angles restores unitarity

Abstract

We study the spectrum of pure massless higher spin theories in $AdS_3$. The light spectrum is given by a tower of massless particles of spin $s=2,\cdots,N$ and their multi-particles states. Their contribution to the torus partition function organises into the vacuum character of the ${\cal W}_N$ algebra. Modular invariance puts constraints on the heavy spectrum of the theory, and in particular leads to negative norm states, which would be inconsistent with unitarity. This negativity can be cured by including additional light states, below the black hole threshold but whose mass grows with the central charge. We show that these states can be interpreted as conical defects with deficit angle $2\pi(1-1/M)$. Unitarity allows the inclusion of such defects into the path integral provided $M \geq N$.