Refining the Cutoff 3d Gravity / $T\bar{T}$ Correspondence

Pure gravity in AdS$_3$ is a theory of boundary excitations, most simply expressed as a constrained free scalar with an improved stress tensor that is needed to match the Brown--Henneaux central charge. Excising a finite part of AdS gives rise to a static gauge Nambu--Goto action for the boundary graviton. We show that this is the $T\bar{T}$ deformation of the infinite volume theory, as the effect of the improvement term on the deformed action can be absorbed into a field redefinition. The classical gravitational stress tensor is reproduced order by order by the $T\bar{T}$ trace equation. We calculate the finite volume energy spectrum in static gauge and find that the trace equation imposes sufficient constraints on the ordering ambiguities to guarantee agreement with the light-cone gauge prediction. The correlation functions, however, are not completely fixed by the trace equation. We show how both the gravitational action and the $T\bar{T}$ deformation allow for finite improvement terms, and we match these to the undetermined total derivative terms in Zamolodchikov's point splitting definition of the $T\bar{T}$ operator.