$T \bar T$ Deformations, Massive Gravity and Non-Critical Strings

TL;DR

The paper shows that the TTbar deformation of a 2D QFT on curved spacetime is classically equivalent to coupling to a 2D ghost-free massive gravity, and that the CFT case further reduces to a non-critical string theory with a B-field via Stückelberg fields. It provides a classical derivation, a metric/zweibein formulation, and a stochastic path integral framework (random geometries) that mirrors holographic cutoff ideas. A key result is a closed-form classical TTbar Hamiltonian for any CFT on curved spacetime, and a Polyakov-string interpretation that ties TTbar to non-critical string theory, with generalizations to TTbar+J T̄+T J̄ deformations. The work also extends these insights to special non-conformal theories, showing how the target-space metric and B-field are modified, and demonstrates consistency with recent path-integral proposals in the literature, offering a coherent gravitational and string-theoretic picture of TTbar deformations.

Abstract

The $T \bar T$ deformation of a 2 dimensional field theory living on a curved spacetime is equivalent to coupling the undeformed field theory to 2 dimensional `ghost-free' massive gravity. We derive the equivalence classically, and using a path integral formulation of the random geometries proposal, which mirrors the holographic bulk cutoff picture. We emphasize the role of the massive gravity \stu fields which describe the diffeomorphism between the two metrics. For a general field theory, the dynamics of the \stu fields is non-trivial, however for a CFT it trivializes and becomes equivalent to an additional pair of target space dimensions with associated curved target space geometry and dynamical worldsheet metric. That is, the $T \bar T$ deformation of a CFT on curved spacetime is equivalent to a non-critical string theory in Polyakov form, with a non-zero $B$-field. We give a direct proof of the equivalence classically without relying on gauge fixing, and determine the explicit form for the classical Hamiltonian of the $T\bar T$ deformation of an arbitrary CFT on a curved spacetime. When the QFT action is a sum of a CFT plus an operator of fixed scaling dimension, as for example in the sine-Gordon model, the equivalence to a non-critical theory string holds with a modified target space metric and modified $B$-field. Finally we give a stochastic path integral formulation for the general $T \bar T+J \bar T+T \bar J$ deformation of a general QFT, and show that it reproduces a recent path integral proposal in the literature.