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 AdS3, if they exist, have no local, perturbative degrees of freedom in the semi-classical limit.
Highlights
Theories with higher spin symmetry have been extensively investigated in two and higher dimensions
We focus on CFTs with WN 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 is non-trivial even when c > N − 1
Summary
There is an infinite number of states arbitrarily close to (2.1) This claim does not assume large c, and it is true and non-trivial even with just Virasoro symmetry.. The crossing equation, G(z, z) = G(1 − z, 1 − z), is expanded in the double lightcone limit, z 1 − z 1 In this limit, one channel is dominated by the disconnected product of two-point functions — i.e., the identity conformal block — and to reproduce this singular behavior in the other channel, there must be an infinite sum over operators with twist accumulating at 2∆. The vacuum in one channel can be reproduced by an individual state in the other channel, and there is no need for an infinite sum
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