Liouville strings in AdS$_3$: the worldsheet story

The worldsheet $\sigma$-model of strings on AdS$_3$ with NS-NS fluxes can be equivalently described in terms of a Liouville field theory coupled to a timelike field with background charge, plus a marginal deformation. The operator that produces the deformation is non-normalizable and, from the string theory perspective, is associated to the spectral flow sector $\omega =2$. Its role is to control the winding number violation in scattering amplitudes. In the semiclassical (large $k$) limit, the central charge of the Liouville factor tends to that of the dual CFT$_2$, namely $c\simeq 6k$. This may appear very similar to what occurs in the Liouville-type description of the dual deformed orbifold CFT$_2$, where a twist-2 deformation operator dressed with a non-normalized field also appears. Indeed, there are some similarities to that; however, there are also significant differences, which we discuss.