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

TL;DR

This paper proves that under certain conditions, the spectrum of $J\bar{T}$ deformed CFTs is uniquely determined by the undeformed theory, and derives a flow equation for their partition sum, exploring non-perturbative effects and ambiguities.

Contribution

The authors establish the uniqueness of $J\bar{T}$ deformed spectra based on modular covariance and state energy dependence, and derive a flow equation for the deformed partition sum.

Findings

The partition sum at zero deformation uniquely determines the deformed spectrum.

A flow equation for the $J\bar{T}$ deformed partition sum is derived.

Non-perturbative ambiguities are identified for non-zero deformation values.

Abstract

We study families of two dimensional quantum field theories, labeled by a dimensionful parameter $\mu$, 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 $\mu$ is a function only of $\mu$, and of the energy, momentum and charge of the corresponding state at $\mu=0$, where the theory becomes conformal. We show that under these conditions, the torus partition sum of the theory at $\mu=0$ uniquely determines the partition sum (and thus the spectrum) of the perturbed theory, to all orders in $\mu$, to be that of a $\mu J\bar T$ 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 $\mu$, and comment on their possible relations to holography.