Constraints on Higher Spin CFT$_2$

TL;DR

The paper investigates 2d CFTs with higher spin symmetry in the irrational regime c>N-1, where an infinite tower of higher-spin primaries is expected. It develops strong spectral constraints from three pillars: unitarity via the W_N Kac matrix, modular invariance (including a spin-3 charged modular bootstrap), and causality arguments in the lightcone limit. A universal lower bound h_min = h_{crit}(N,c) is derived (verified for N≤6 and conjectured for all N) and shown to linearize in c, with large-c implications that preclude local perturbative bulk degrees in a putative AdS_3 dual unless N_currents is infinite. The results illuminate the difficulty of realizing irrational higher-spin CFT_2s and suggest that, in the large-c limit, viable theories are extremal or require N to scale with c; modular constraints further sharpen the allowed spectrum, notably for W_3 theories. Together, these findings constrain the landscape of higher-spin holography in AdS_3 and point to a deep tension between higher-spin symmetry and semiclassical gravity in two dimensions.

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.

Figures (4)

• Figure 1: Exclusion plot for $\mathcal{W}_3$, with $c = 10^4$. The weight and charge $(h,q_3)$ of all primaries must fall in the shaded region, where $\lambda_2 > 0$.
• Figure 2: Exclusion plot for $\mathcal{W}_4$ charge $q_4$ and conformal weight $h$, with $c=10^4, q_3=0$. The shaded region is allowed, i.e. has all eigenvalues positive. One eigenvalue of the level-one Kac matrix is non-negative everywhere in the plotted region, and the pluses/minuses show the signs of the other two eigenvalues. The dashed curve shows the null states referred to in footnote \ref{['footnote:null']}.
• Figure 3: The red region is where $\Psi^{(1,3)}>0$; modular invariance implies the existence of an operator in this region. The blue region is the unitarity bound set by the positivity of the Kac matrix.
• Figure 4: The red regions are where $\Psi^{(1,3)}-c \gamma \Psi^{(0,1)}>0$; modular invariance implies the existence of an operator in each of these regions for any value of $\gamma$. The regions become narrower as $\gamma$ is increased. The blue region is the unitarity bound set by the positivity of the Kac matrix. This is a stronger constraint than the one in Figure \ref{['fig1']}.