We develop the holographic dictionary for pure $\mathrm{ AdS}_3$ gravity where the Lagrangian of the dual $2d$ conformal field theory has been deformed by an arbitrary function of the energy-momentum tensor. In addition to the $T \overline{ T}$ deformation, examples of such functions include a class of marginal stress tensor deformations which are special because they leave the generating functional of connected correlators unchanged up to a redefinition of the source and expectation value. Within this marginal class, we identify the unique deformation that commutes with the $T \overline{ T}$ flow, which is the root-$T \overline{ T}$ operator, and write down the modified boundary conditions corresponding to this root-$T \overline{ T}$ deformation. We also identify the unique marginal stress tensor flow for the cylinder spectrum of the dual CFT which commutes with the inviscid Burgers' flow driven by $T \overline{ T}$, and we propose this unique flow as a candidate root-$T \overline{ T}$ deformation of the energy levels. We study BTZ black holes in $\mathrm{ AdS}_3$ subject to root-$T \overline{ T}$ deformed boundary conditions, and find that their masses flow in a way which is identical to that of our candidate root-$T \overline{ T}$ energy flow equation, which offers evidence that this flow is the correct one. Finally, we also obtain the root-$T \overline{ T}$ deformed boundary conditions for the gauge field in the Chern-Simons formulation of $\mathrm{ AdS}_3$ gravity.
Categories: hep-th gr-qc math-ph math.MP