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The T T ¯ $$ T\overline{T} $$ deformation is a solvable irrelevant deformation whose properties depend on the sign of the deformation parameter μ. In particular, T T ¯ $$ T\overline{T} $$ -deformed CFTs with μ < 0 have been proposed to be holographically dual to Einstein gravity where the metric satisfies Dirichlet boundary conditions at a finite cutoff surface. In this paper, we put forward a holographic proposal for T T ¯ $$ T\overline{T} $$ -deformed CFTs with μ > 0, in which case the bulk geometry is constructed by gluing a patch of AdS3 to the original spacetime. As evidence, we show that the T T ¯ $$ T\overline{T} $$ trace flow equation, the spectrum on the cylinder, and the partition function on the torus and the sphere, among other results, can all be reproduced from bulk calculations in glue-on AdS3.

We generate a class of string backgrounds by a sequence of TsT transformations of the NS1-NS5 system that we argue are holographically dual to states in a single-trace T T ¯ $$ T\overline{T} $$ + J T ¯ $$ J\overline{T} $$ + T J ¯ $$ T\overline{J} $$ -deformed CFT2. The new string backgrounds include general rotating black hole solutions with two U(1) charges as well as a smooth solution without a horizon that is interpreted as the ground state. As evidence for the correspondence we (i) derive the long string spectrum and relate it to the spectrum of the dual field theory; (ii) show that the black hole thermodynamics can be reproduced from single-trace T T ¯ $$ T\overline{T} $$ + J T ¯ $$ J\overline{T} $$ + T J ¯ $$ T\overline{J} $$ -deformed CFTs; and (iii) show that the energy of the ground state matches the energy of the vacuum in the dual theory. We also study geometric properties of these new spacetimes and find that for some choices of the parameters the three-dimensional Ricci scalar in the Einstein frame can become positive in a region outside the horizon before reaching closed timelike curves and singularities.