Abstract
We study the spectrum of pure massless higher spin theories in AdS3. The light spectrum is given by a tower of massless particles of spin s = 2, ⋯ , N and their multi-particles states. Their contribution to the torus partition function organises into the vacuum character of the {mathcal{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 ≥ N.
Highlights
Possess a rich spectrum of classical solutions, see for instance [6], it is natural to ask whether they provide consistent theories of quantum gravity, or they suffer from pathologies at the quantum level
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
The partition function is highly constrained by consistency conditions, including unitarity, modular invariance, and WN symmetry
Summary
Consider a unitary irrational 2d CFT with WN symmetry and central charge c > N − 1. The generic characters for WN algebras are not known They are known in the case where the chemical potential for the higher spin currents is turned off. Note that this condition implies r, s are coprime: (r, s) = 1 Since this transformation results in an equivalent torus, the partition function should be invariant. Our last assumption is some spectrum of light operators, where by light we denote operators below the BTZ threshold at h = h = c With these assumptions, we would like to construct the resulting partition function, or equivalently, the density ρ(h, h) of WN primaries. We would like to construct the resulting partition function, or equivalently, the density ρ(h, h) of WN primaries To this end it is convenient to define a new partition function, modular invariant, given by.
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