Locality and Conserved Charges in $T\overline{T}$-Deformed CFTs

TL;DR

This work provides a perturbative construction of $T\overline{T}$-deformed CFTs that preserves (quasi-)locality by mapping a nonlocal fake Hamiltonian to a local deformed Hamiltonian through a unitary transformation. The authors derive the deformed Hamiltonian and stress tensor up to third order in the deformation parameter, revealing a one-parameter family of local Hamiltonians that reproduce the Zamolodchikov spectrum, with essential $c$-dependent terms, and they explicitly connect these ambiguities to the $\det T_\lambda$ operator. They verify the deformed energy spectrum for primary states, establish current conservation and the $T\overline{T}$ flow for densities, and demonstrate that the infinite set of KdV charges remains integrable under the deformation, with the possibility of constructing local combinations of conjugated charges at least to leading order. Furthermore, they explore generalized deformations built from KdV charges, showing locality constraints limit the allowed deformations, thereby clarifying the structure of TTbar deformations and their quasi-locality in the IR. Overall, the paper clarifies how locality, spectrum, and conserved charges are organized under TTbar and related generalized deformations, and it sheds light on the regulator/amb ambiguity in defining the deforming operator.

Abstract

We investigate the locality properties of $T \overline T$-deformed CFTs within perturbation theory. Up to third order in the deformation parameter, we find a Hamiltonian operator which solves the flow equation, reproduces the Zamolodchikov energy spectrum, and is consistent with quasi-locality of the theory. This Hamiltonian includes terms proportional to the central charge which have not appeared before and which are necessary to reproduce the correct spectrum. We show that the Hamiltonian is not uniquely defined since it contains free parameters, starting at second order, which do not spoil the above properties. We then use it to determine the full conserved stress tensor. In our approach, the KdV charges are automatically conserved to all orders but are not a priori local. Nevertheless, we show that they can be made local to first order. Our techniques allow us to further comment on the space of Hamiltonians constructed from products of KdV charges which also flow to local charges in the deformed theory in the IR.