  We calculate the spectrum of a two dimensional CFT on a cylinder, perturbed
by a general linear combination of $T\bar{T}$, $J\bar{T}$ and $T\bar{J}$, by
utilizing the relation of this problem to certain solvable single trace
current-current deformations of the worldsheet theory of strings on $AdS_3$. We
show that this spectrum is well behaved (i.e. the energies are real) if and
only if the dual bulk geometry is well behaved (i.e. the signature of the bulk
spacetime is $(1,2)$ and there are no closed timelike curves). We also comment
on the relations of string theory on $AdS_3$ and its deformations to symmetric
product CFT's, matrix string theory, and a black hole description of highly
excited fundamental strings.
