Modular invariance and uniqueness of $T\bar{T}$ deformed CFT

TL;DR

The paper investigates 2D QFTs on a torus with a single-scale irrelevant deformation whose energies depend only on undeformed energies and momenta, showing that modular invariance fixes the perturbative torus partition sum to all orders as that of a $T\bar{T}$-deformed CFT. It derives a unique all-orders recursion for the perturbed partition function and identifies the deformed spectrum, equivalent to the inviscid Burgers equation, while revealing a non-perturbative split: unique completion for λ>0 and non-perturbative ambiguities for λ<0, which are tied to asymptotic series and states with diverging energies. The discussion connects these results to holographic pictures in AdS3 and outlines potential UV completions and extensions to related deformations. Overall, the work provides a rigorous modular-invariance-based characterization and non-perturbative structure of $T\bar{T}$-like theories and their holographic interpretations.

Abstract

Any two dimensional quantum field theory that can be consistently defined on a torus is invariant under modular transformations. In this paper we study families of quantum field theories labeled by a dimensionful parameter $t$, that have the additional property that the energy of a state at finite $t$ is a function only of $t$ and of the energy and momentum of the corresponding state at $t=0$, where the theory becomes conformal. We show that under this requirement, the partition sum of the theory at $t=0$ uniquely determines the partition sum (and thus the spectrum) of the perturbed theory, to all orders in $t$, to be that of a $T\bar T$ deformed CFT. Non-perturbatively, we find that for one sign of $t$ (for which the energies are real) the partition sum is uniquely determined, while for the other sign we find non-perturbative ambiguities. We characterize these ambiguities and comment on their possible relations to holography.