Root-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>T</mml:mi><mml:mover accent="true"><mml:mi>T</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml:mover></mml:math> deformed boundary conditions in holography
Stephen Ebert, Christian Ferko, Zhengdi Sun
Abstract
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.