$T\bar T$ deformation of correlation functions

TL;DR

<3-5 sentence high-level summary> The paper develops a comprehensive framework for understanding how the solvable $T\overline T$ deformation (and a related $J^1\wedge J^2$ case) acts on correlation functions in 2d QFT. It shows that the deformation behaves as a derivation on the operator algebra, expressible via field- or state-dependent string insertions, and derives explicit evolution equations that allow UV divergences to be renormalized, yielding finite correlators and a deformed OPE with a Callan–Symanzik RG structure. A key result is that correlators can be resummed to a diffusion-like form in a suitable scaling limit, and that deformed currents and the stress tensor can be consistently defined and kept finite with preserved Ward identities. The work also provides multiple perspectives, including a Green-function approach and a field-valued diffeomorphism interpretation, and discusses implications for geometry, entanglement, and holography, while outlining important open problems such as curved-space generalization and Virasoro algebra fate.

Abstract

We study the evolution of correlation functions of local fields in a two-dimensional quantum field theory under the $λT\bar T$ deformation, suitably regularized. We show that this may be viewed in terms of the evolution of each field, with a Dirac-like string being attached at each infinitesimal step. The deformation then acts as a derivation on the whole operator algebra, satisfying the Leibniz rule. We derive an explicit equation which allows for the analysis of UV divergences, which may be absorbed into a non-local field renormalization to give correlation functions which are UV finite to all orders, satisfying a (deformed) operator product expansion and a Callan-Symanzik equation. We solve this in the case of a deformed CFT, showing that the Fourier-transformed renormalized two-point functions behave as $k^{2Δ+2λk^2}$, where $Δ$ is their IR conformal dimension. We discuss in detail deformed Noether currents, including the energy-momentum tensor, and show that, although they also become non-local, when suitably improved they remain finite, conserved and satisfy the expected Ward identities. Finally, we discuss how the equivalence of the $T\bar T$ deformation to a state-dependent coordinate transformation emerges in this picture.

Figures (2)

• Figure 1: The potential $\chi^b$ is made single-valued by removing the union of paths $S_n$ from each argument $x_n$ to an arbitrarily chosen point $X$.
• Figure 2: The discontinuity $[\chi^b(x)]$ across $S_n$ is given by the flux of $J^b$ across a contour $C$ surrounding $x_n$.