Solving a family of $T\bar{T}$-like theories

TL;DR

This work formulates a universal transport framework for a broad family of TT-like deformations built from antisymmetric combinations of conserved currents in 2D QFTs. Starting from a CFT, it couples linear background fields and then evolves under quadratic, irrelevantly-deforming operators, yielding a closed-form flow for energy levels that recovers the Burgers-type equation for TT$\bar{T}$ and extends to $J\bar{T}$ and related deformations. The authors introduce a spectrum- generating operator formalism, perform perturbative quantum checks, and obtain a compact quadratic equation for the spectrum that encapsulates the effects of multiple deformations and background fields, with explicit checks against AdS/CFT and string-theoretic constructions. The framework provides a systematic route to explore coupling-space structures, initial-condition fixing, and potential generalizations to higher-spin (KdV) currents, offering a coherent path to understanding UV behavior and holographic interpretations of TT-like theories.

Abstract

We deform two-dimensional quantum field theories by antisymmetric combinations of their conserved currents that generalize Smirnov and Zamolodchikov's $T\bar{T}$ deformation. We obtain that energy levels on a circle obey a transport equation analogous to the Burgers equation found in the $T\bar{T}$ case. This equation relates charges at any value of the deformation parameter to charges in the presence of a (generalized) Wilson line. We determine the initial data and solve the transport equations for antisymmetric combinations of flavor symmetry currents and the stress tensor starting from conformal field theories. Among the theories we solve is a conformal field theory deformed by $J\bar{T}$ and $T\bar{T}$ simultaneously. We check our answer against results from AdS/CFT.

Figures (2)

• Figure 1: Graphical representation of the strategy solving deformations of CFTs by quadratic composite operators. Turning on the background fields $a,b$ determines the initial value surface, drawn here as a bright orange plane. These are the directions corresponding to linear deformations. The $\lambda$ direction in coupling space represents the deformation by the quadratic composite operator ${{\mathcal{O}}}$. We erect a coordinate system by first deforming by $\int dx \ {{\mathcal{O}}}_\lambda$ as in \ref{['DeformedTheory']} and going $\lambda$ distance. Subsequently we implement the linear deformations. Hence, deforming a generic point in coupling space by ${{\delta}} \lambda \int dx\ {{\mathcal{O}}}$ (indicated by blue arrow) does not in general agree with ${{\delta}} \lambda\times \partial_\lambda H (\lambda,a,b)$.
• Figure 2: Left: Graphical representation of \ref{['CouplingSpace2']}. Independent of which order we evolve the spectrum of the CFT with $V$ and $W$, we get the same spectrum. The result also agrees by the simultaneous irrelevant deformation by $V$ and $W$ represented by the diagonal orange arrow. equation Right: For the special case of $V=J{\bar{T}},\, W=J\Theta$, the structure of the coupling space is more complicated, as explained in \ref{['CouplingSpace1']}.