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

TL;DR

The paper extends the modular covariance approach used for $tT\\bar{T}$ deformations to theories with a holomorphic $U(1)$ current, showing that a modular-covariant torus partition sum with a charge chemical potential uniquely yields the $\mu J\\bar{T}$-deformed CFT spectrum to all orders in the dimensionless coupling ${\widehat{\mu}}=\mu/R$. It derives a recursion for the deformed partition sum, promotes it to a flow equation for the full partition function, and presents Burgers-like evolution equations for state energies and charges, reproducing known spectra. Non-perturbative ambiguities are analyzed via a negative-branch energy solution and a modularly invariant completion $\mathcal{Z}_{np}$, with a notable degeneracy at $i\pi k {\widehat{\mu}}\nu+2\tau_2=0$. The formalism is illustrated with charged free boson and fermion examples, where exact closed-form results and covariant-derivative structures clarify the perturbative and non-perturbative content. The work also discusses holographic perspectives and motivates future work on UV completions and bulk interpretations 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.