Modular covariance and uniqueness of $J\bar{T}$ deformed CFTs

Abstract

We study families of two dimensional quantum field theories, labeled by a dimensionful parameter $μ$, 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 $μ$ is a function only of $μ$, and of the energy, momentum and charge of the corresponding state at $μ=0$, where the theory becomes conformal. We show that under these conditions, the torus partition sum of the theory at $μ=0$ uniquely determines the partition sum (and thus the spectrum) of the perturbed theory, to all orders in $μ$, to be that of a $μJ\bar T$ deformed conformal field theory (CFT). We derive a flow equation for the $J\bar{T}$ deformed partition sum, and use it to study non-perturbative effects. We find non-perturbative ambiguities for any non-zero value of $μ$, and comment on their possible relations to holography.

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