Two Virasoro symmetries in stringy warped AdS$_3$

We study three-dimensional consistent truncations of type IIB supergravity which admit warped AdS$_3$ solutions. These theories contain subsectors that have no bulk dynamics. We show that the symplectic form for these theories, when restricted to the non-dynamical subsectors, equals the symplectic form for pure Einstein gravity in AdS$_3$. Consequently, for each consistent choice of boundary conditions in AdS$_3$, we can define a consistent phase space in warped AdS$_3$ with identical conserved charges. This way, we easily obtain a Virasoro $\times$ Virasoro asymptotic symmetry algebra in warped AdS$_3$; two different types of Virasoro $\times$ Ka\v{c}-Moody symmetries are also consistent alternatives. Next, we study the phase space of these theories when propagating modes are included. We show that, as long as one can define a conserved symplectic form without introducing instabilities, the Virasoro $\times$ Virasoro asymptotic symmetries can be extended to the entire (linearized) phase space. This implies that, at least at semi-classical level, consistent theories of gravity in warped AdS$_3$ are described by a two-dimensional conformal field theory, as long as stability is not an issue.