$T\\bar T$ and the mirage of a bulk cutoff
Guica, Monica, Monten, Ruben
Abstract
We use the variational principle approach to derive the large\nNN\nholographic dictionary for two-dimen-sional\nT\\bar TTT‾-deformed\nCFTs, for both signs of the deformation parameter. The resulting dual\ngravitational theory has mixed boundary conditions for the non-dynamical\ngraviton; the boundary conditions for matter fields are undeformed. When\nthe matter fields are turned off and the deformation parameter is\nnegative, the mixed boundary conditions for the metric at infinity can\nbe reinterpreted on-shell as Dirichlet boundary conditions at finite\nbulk radius, in agreement with a previous proposal by McGough, Mezei and\nVerlinde. The holographic stress tensor of the deformed CFT is fixed by\nthe variational principle, and in pure gravity it coincides with the\nBrown-York stress tensor on the radial bulk slice with a particular\ncosmological constant counterterm contribution. In presence of matter\nfields, the connection between the mixed boundary conditions and the\nradial ``bulk cutoff’’ is lost. Only the former correctly reproduce the\nenergy of the bulk configuration, as expected from the fact that a\nuniversal formula for the deformed energy can only depend on the\nuniversal asymptotics of the bulk solution, rather than the details of\nits interior. The asymptotic symmetry group associated with the mixed\nboundary conditions consists of two commuting copies of a\nstate-dependent Virasoro algebra, with the same central extension as in\nthe original CFT.