Deforming and dissecting AdS$_3$ with matter

We study deformations of the model by Henneaux, Mart\'inez, Troncoso and Zanelli [arXiv:hep-th/0201170] which features asymptotically AdS$_3$ black hole solutions that incorporate the exact backreaction of a scalar field. The presence of bulk matter causes the $T \overline T$ deformation of the (putative) dual CFT$_2$ to differ from the deformation defined in the bulk by imposing Dirichlet boundary conditions at finite radius. We work out both of these deformations explicitly and verify that $T \overline T$-deforming the boundary theory corresponds to imposing mixed boundary conditions on the metric at the conformal boundary, whereas the bulk "Dirichlet deformation" gives rise to a field theory deforming operator that includes $T \overline T$ as well as other irrelevant terms. We check our results by calculating the deformed energy spectrum for either case using both the bulk and boundary prescriptions, finding agreement after taking into account additional terms coming from the flow of the scalar source. We interpret our explicit results and compare them with the predictions of similar proposals in the literature.