The dynamical consistency of the non-projectable version of Hořava gravity is investigated by focusing on the asymptotically flat case. It is argued that for generic solutions of the constraint equations the lapse must vanish asymptotically. We then consider particular values of the coupling constants for which the equations are tractable and in that case we prove that the lapse must vanish everywhere -and not only at infinity. Put differently, the Hamiltonian constraints are generically all second-class. We then argue that the same feature holds for generic values of the couplings, thus revealing a physical inconsistency of the theory. In order to cure this pathology, one might want to introduce further constraints but the resulting theory would then lose much of the appeal of the original proposal by Hořava. We also show that there is no contradiction with the time reparametrization invariance of the action, as this invariance is shown to be a so-called "trivial gauge symmetry" in Hořava gravity, hence with no associated first-class constraints.

We consider (2 + 1)-dimensional (N, M)-extended higher-spin anti-de Sitter supergravity and study its asymptotic symmetries. The theory is described by the Chern-Simons action based on the real, infinite-dimensional higher-spin superalgebra shs E (N|2, R) ⊕ shs E (M|2, R). We specify consistent boundary conditions on the higher-spin super-gauge connection corresponding to asymptotically anti-de Sitter spacetimes. We then determine the residual gauge transformations that preserve these asymptotic conditions and compute their Poisson bracket algebra. We find that the asymptotic symmetry is enhanced from the higher-spin superalgebra to some (N, M)-extended super-W ∞ nonlinear superalgebra. The latter has the same classical central charge as pure Einstein gravity. Special attention is paid to the (1, 1) case. Truncation to the bosonic sector yields the previously found W ∞ algebra, while truncation to the osp(N|2, R) sector reproduces the N-extended superconformal algebra (in its nonlinear version for N > 2). We discuss string theory realization of these higher-spin anti-de Sitter supergravity theories as well as relations to previous treatments of super-W ∞ in the literature.

The spectrum of Prokushkin-Vasiliev Theory is puzzling in light of the Gaberdiel-Gopakumar conjecture because it generically contains an additional sector besides higher-spin gauge and scalar fields. We find the unique truncation of the theory avoiding this problem to order 2 in perturbations around AdS 3 . The second-order backreaction on the physical gauge sector induced by the scalars is computed explicitly. The cubic action for the physical fields is determined completely. The subtle issue of the allowed class of pseudo-local field redefinitions is discussed.