Modular covariance and uniqueness of JT deformed CFTs

We study families of two dimensional quantum field theories, labeled by a dimensionful parameter , that contain a holomorphic conserved U(1) current J(z). We assume that these theories can be consistently defined on a torus, so their partition sum, with a chemical potential for the charge that couples to J, is modular covariant. We further require that in these theories, the energy of a state at finite is a function only of , and of the energy, momentum and charge of the corresponding state at = 0, where the theory becomes conformal. We show that under these conditions, the torus partition sum of the theory at = 0 uniquely determines the partition sum (and thus the spectrum) of the perturbed theory, to all orders in , to be that of a JT deformed conformal field theory (CFT). We derive a flow equation for the J deformed partition sum, and use it to study non-perturbative effects. We find non-perturbative ambiguities for any non-zero value of , and comment on their possible relations to holography.