The
TˉT deformation of a 2 dimensional field theory living on a curved
spacetime is classically 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 St\"uckelberg fields which describe the
diffeomorphism between the two metrics. For a general field theory, the
dynamics of the St\"uckelberg 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
TˉT deformation of a CFT on curved
spacetime is equivalent to a non-critical string theory in Polyakov form. 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
TˉT 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 an additional potential.
Finally we give a stochastic path integral formulation for the general
TˉT+JˉT+TˉJ deformation of a general QFT, and show that it reproduces
a recent path integral proposal in the literature.