On the Spectrum of Pure Higher Spin Gravity

TL;DR

The work addresses the spectrum of pure massless higher spin gravity in AdS$_3$ with ${\cal W}_N$ symmetry, showing that modular invariance forces a heavy-spectrum that inherently contains negative-norm states unless a minimal set of light operators is added. The authors develop two modular-density constructions (Poincaré and Rademacher) to characterize the heavy spectrum from a given light spectrum, uncovering continuous, non-positive densities and ambiguities in the heavy sector. They propose a gravitational resolution by interpreting the required light states as conical defects in higher-spin gravity, showing that unitarity selects defects with deficit angle $2\pi(1-1/M)$ and $M\ge N$, and that the one-loop gravity computation reproduces a central-charge relation $c=c_g+2N^3-N-1$ (up to subleading terms). The results support a bulk picture where consistency of HS gravity with ${\cal W}_N$ symmetry hinges on including specific conical-defect sectors, linking modular bootstrap to bulk topology and one-loop corrections. The study highlights both the constraints on putative unitary ${\cal W}_N$ CFTs at large central charge and the precise gravitational interpretation of necessary light states.

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 $2π(1-1/M)$. Unitarity allows the inclusion of such defects into the path integral provided $M \geq N$.