Constraints on Higher Spin CFT$_2$

TL;DR

This paper derives constraints on 2D higher spin conformal field theories using unitarity, modular invariance, and causality, revealing bounds on operator dimensions and implications for holographic duals in AdS$_3$.

Contribution

It establishes new bounds on higher-spin primary dimensions in irrational 2D CFTs with $ ext{W}_N$ symmetry, especially at large central charge, impacting holographic duality.

Findings

Positivity of the Kac matrix constrains higher-spin operator dimensions.

Lower bounds on non-vacuum higher-spin primary dimensions are linear in central charge.

Implication that large-$c$ dual theories lack local perturbative degrees of freedom.

Abstract

We derive constraints on two-dimensional conformal field theories with higher spin symmetry due to unitarity, modular invariance, and causality. We focus on CFTs with $\mathcal{W}_N$ symmetry in the "irrational" regime, where $c>N-1$ and the theories have an infinite number of higher-spin primaries. The most powerful constraints come from positivity of the Kac matrix, which (unlike the Virasoro case) is non-trivial even when $c>N-1$. This places a lower bound on the dimension of any non-vacuum higher-spin primary state, which is linear in the central charge. At large $c$, this implies that the dual holographic theories of gravity in AdS$_3$, if they exist, have no local, perturbative degrees of freedom in the semi-classical limit.