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Higher Spin Theories in AdS_3 and a Gravitational Exclusion Principle

Alejandra Castro, Arnaud Lepage-Jutier, Alexander Maloney

TL;DR

This work analyzes 3D AdS gravity with massless higher-spin fields, showing that modular invariance enforces a gravitational exclusion bound $c \ge N-1$ linking higher-spin gauge symmetry to the central charge. By examining the growth of ${\cal W}_N$ descendants against Cardy’s density-of-states bound, it reveals that the perturbative spectrum would overcount states unless non-perturbative effects remove them or the theory renormalizes, a phenomenon tied to linearization instability. The ${\cal W}_N$ minimal models provide explicit examples where $c<N-1$ and null vectors alter the high-energy spectrum, suppressing large BTZ black holes at finite $N,k$. The results illustrate a gravitational exclusion principle in AdS$_3$ and illuminate the delicate balance between higher-spin symmetry, black-hole states, and holographic duals.

Abstract

We consider theories of three dimensional quantum gravity in Anti-de Sitter space which possess massless higher-spin gauge symmetry. The perturbative spectrum of the theory includes higher spin excitations which can be organized into vacuum representations of the W_N algebra; these are higher spin versions of the boundary gravitons. We describe a fundamental bound which relates the value of the cosmological constant to the amount of gauge symmetry present. In the dual CFT language, this is the statement that modular invariance implies that the theory can not be quantized unless the central charge is sufficiently large, i.e. if c is greater than or equal to N-1. This bound relies on the assumption that all of the perturbative excitations exist as full states in the quantum theory, and can be circumvented if the theory possesses a linearization instability. The W_N minimal models -- recently conjectured to be dual to certain higher spin AdS theories by Gaberdiel and Gopakumar - provide an example of this phenomenon. This result can be regarded as an example of a "gravitational exclusion principle" in Anti-de Sitter space, where a non-perturbative quantum gravity mechanism involving black holes places a limit on the number of light degrees of freedom present.

Higher Spin Theories in AdS_3 and a Gravitational Exclusion Principle

TL;DR

This work analyzes 3D AdS gravity with massless higher-spin fields, showing that modular invariance enforces a gravitational exclusion bound linking higher-spin gauge symmetry to the central charge. By examining the growth of descendants against Cardy’s density-of-states bound, it reveals that the perturbative spectrum would overcount states unless non-perturbative effects remove them or the theory renormalizes, a phenomenon tied to linearization instability. The minimal models provide explicit examples where and null vectors alter the high-energy spectrum, suppressing large BTZ black holes at finite . The results illustrate a gravitational exclusion principle in AdS and illuminate the delicate balance between higher-spin symmetry, black-hole states, and holographic duals.

Abstract

We consider theories of three dimensional quantum gravity in Anti-de Sitter space which possess massless higher-spin gauge symmetry. The perturbative spectrum of the theory includes higher spin excitations which can be organized into vacuum representations of the W_N algebra; these are higher spin versions of the boundary gravitons. We describe a fundamental bound which relates the value of the cosmological constant to the amount of gauge symmetry present. In the dual CFT language, this is the statement that modular invariance implies that the theory can not be quantized unless the central charge is sufficiently large, i.e. if c is greater than or equal to N-1. This bound relies on the assumption that all of the perturbative excitations exist as full states in the quantum theory, and can be circumvented if the theory possesses a linearization instability. The W_N minimal models -- recently conjectured to be dual to certain higher spin AdS theories by Gaberdiel and Gopakumar - provide an example of this phenomenon. This result can be regarded as an example of a "gravitational exclusion principle" in Anti-de Sitter space, where a non-perturbative quantum gravity mechanism involving black holes places a limit on the number of light degrees of freedom present.

Paper Structure

This paper contains 8 sections, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Higher spin fields in AdS$_3$
  3. Partition function and growth of states
  4. Minimal Models and Black Holes
  5. Asymptotic Behaviour of $\chi_N$ and $\chi_\infty$
  6. Asymptotics of the Partition Function $F(q)$
  7. Asymptotics of the ${\cal W}_N$ character $\chi_N$
  8. Asymptotics of the ${\cal W}_\infty$ character $\chi_\infty$ and the MacMahon Function

Figures (2)

  • Figure 1: For $N=2$ (straight line), $N=3$ (long dash) and $N=6$ (short dash), we plot the ratio of numerical value of $\log(p_n^N)$ over the approximated value given by \ref{['AC:da']}. As $N$ increases, we require larger values of $n$ to reach the Cardy regime.
  • Figure 2: Ratio of $\log p^\infty_n$ over the saddle point approximation \ref{['AB:z']}.