The scaling dimension of the first excited state in two-dimensional conformal field theories (CFTs) satisfies a universal upper bound. Using the modular bootstrap, we extend this result to CFTs with W-3 algebras which are generically dual to higher spin theories in AdS(3). Assuming unitarity and modular invariance, we show that the conformal weights h, (h) over bar of the lightest charged state satisfy h < c/12+O(1) and _ <(h)over bar> < <(c)over bar>/12+O(1) in the limit where the central charges c, (c) over bar are large. Furthermore, we show that in this limit any consistent CFT with W-3 currents must contain at least one state whose W-3 charge w obeys vertical bar w vertical bar > 4 vertical bar h -c/24 vertical bar/root 10 pi c + O(1). We discuss hints on the existence of stronger bounds and comment on the interpretation of our results in the dual higher spin theory.

We construct a non-linear theory of interacting spin-2 fields that is invariant under the partially massless (PM) symmetry to all orders. This theory is based on the SO(1, 5) group, in analogy with the SO(2, 4) formulation of conformal gravity, but has a quadratic spectrum free of ghost instabilities. The action contains a vector field associated with a local SO(2) symmetry which is manifest in the vielbein formulation of the theory. We show that, in a perturbative expansion, the SO(2) symmetry transmutes into the PM transformations of a massive spin-2 field. In this context, the vector field is crucial to circumvent earlier obstructions to an order-by-order construction of PM symmetry. Although the non-linear theory lacks enough first class constraints to remove all helicity-0 modes from the spectrum, the PM transformations survive to all orders. The absence of ghosts and strong coupling effects at the non-linear level are not addressed here.

In this paper we investigate a particular ghost-free bimetric theory that exhibits the partially massless ( PM) symmetry at quadratic order. At this order the global SO(1,4) symmetry of the theory is enhanced to SO(1,5). We show that this global symmetry becomes inconsistent at cubic order, in agreement with a previous calculation. Furthermore, we find that the PM symmetry of this theory cannot be extended beyond cubic order in the PM field. More importantly, it is shown that the PM symmetry cannot be extended to quartic order in any theory with one massless and one massive spin-2 fields.

We propose a set of boundary terms for higher spin theories in AdS(3) that lead to a well-defined variational principle compatible with Dirichlet boundary conditions for the metric and higher spin fields. These boundary terms are valid for higher spin theories in the Fefferman-Graham gauge and they allow us to compute the canonical free energy of higher spin black holes directly from the Euclidean, covariant, on-shell action. Using these results we reproduce the thermodynamics of the higher spin black hole of Ammon, Gutperle, Kraus, and Perlmutter and comment on the corresponding theory of induced W-gravity at the boundary.

Pure three-dimensional gravity in anti-de Sitter space can be formulated as an SL(2, R) x SL(2, R) Chern-Simons theory, and the latter can be reduced to a WZW theory at the boundary. In this paper we show that AdS(3) gravity with free boundary conditions is described by a string at the boundary whose target spacetime is also AdS(3). While boundary conditions in the standard construction of Coussaert, Henneaux, and van Driel are enforced through constraints on the WZW currents, we find that free boundary conditions are partially enforced through the string Virasoro constraints.