Canonical analysis of the gravitational description of the $T\bar{T}$ deformation

TL;DR

This work develops an ADM-like Hamiltonian analysis of the 2D gravity description of the $T\bar{T}$ deformation, showing that classical gravitational constraints enforce relations between target-space energy-momentum and undeformed quantities that reproduce the finite-volume $T\bar{T}$ flow. The construction identifies Dirac observables $P^a$ and demonstrates that the resulting implicit equations encode the Burgers-type flow for the deformed spectrum, with special attention to gauge fixing and the reduced phase space. The approach is explicit for free massless scalars, where the theory exhibits an ISO$(1,N{+}1)$ symmetry and connects to non-critical string interpretations in the winding sector, and is extended to $J\bar{T}$ deformations, which introduce boundary twists. The paper then shows how canonical quantization leads to a covariant torus path integral that matches the known $T\bar{T}$-gravity path-integral results of Dubovsky et al., clarifying measure and one-loop exactness aspects. Overall, the work unifies constrained Hamiltonian methods with covariant quantization in the TTbar context and outlines several avenues for future exploration, including non-winding sectors and higher-spin or current deformations.

Abstract

The description of the $T\bar{T}$ deformation in terms of two-dimensional gravity is analyzed from the Hamiltonian point of view, in a manner analogous to the ADM description of general relativity. We find that the Hamiltonian constraints of the theory imply relations between target-space momentum at finite volume which are equivalent to the $T\bar{T}$ finite volume flow equations. This fully-quantum $T\bar{T}$ result emerges already at the classical level within the gravitational theory. We exemplify the analysis for the case when the undeformed sector is a collection of $D-2$ free massless scalars, where it is shown that -- somewhat non-trivially -- the target-space two-dimensional Poincaré symmetry is extended to $D$ dimensions. The connection between canonical quantization of this constrained Hamiltonian system and previous path integral quantizations is also discussed. We extend our analysis to the ``gravitational'' description of $J\bar{T}$-type deformations, where it is found that the flow equations obtained involve deformations that twist the spatial boundary conditions.