  We recast the joint $J\bar{T}$, $T\bar{J}$ and $T\bar{T}$ deformations as
coupling the original theory to a mixture of topological gravity and gauge
theory. This geometrizes the general flow triggered by irrelevant deformations
built out of conserved currents and the stress-energy tensor, by means of a
path integral kernel. The partition function of the deformed theory satisfies a
diffusion-like flow equation similar to that found in the pure $T\bar{T}$ case.
Our proposal passes two stringent tests. Firstly, we recover the classical
deformed actions from the kernel, reproducing the known expressions for the
free boson and fermion. Secondly, we explicitly compute the torus path integral
along the flow and show it localizes to a finite-dimensional, one-loop exact
integral over base space torus moduli. The dressed energy levels so obtained
match exactly onto those previously reported in the literature.
