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 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.

In this paper we consider a class of isospectral deformations of the inhomogeneous string boundary value problem. The deformations considered are generalizations of the isospectral deformation that has arisen in connection with the Camassa-Holm equation for the shallow water waves. It is proved that these new isospectral deformations result in evolution equations on the mass density whose form depends on how the string is tied at the endpoints. Moreover, it is shown that the evolution equations in this class linearize on the spectral side and hence can be solved by the inverse spectral method. In particular, the problem involving a mass density given by a discrete finite measure and arbitrary boundary conditions is shown to be solvable by Stieltjes' continued fractions.

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