Conserved currents and T T ¯s irrelevant deformations of 2D integrable field theories

The TT ¯ deformation of 2-dimensional QFTs is closely-related to Jackiw- Teitelboim gravity. It has been shown that, at the classical level, this perturbation induces an interaction between the stress-energy tensor and space-time and the equations of motion of the deformed theory map onto the original ones through a field-dependent coordinate transformation. At the quantum level, instead, the perturbation is induced by a modification of the original S-matrix by a specific CDD factor and, correspondingly, the quantised energy levels evolve according to a Burgers-type equation. In this paper, we point out that, in the framework of integrable field theories, there exist infinite families of perturbations characterised by a coupling between space-time and local conserved currents, labelled by the Lorentz spin. Similarly to the TT ¯ case, the deformed models emerge through a field-dependent coordinate transformation involving conserved currents with higher Lorentz spin. Furthermore, using a geometric construction, we present a general method to derive the integrable hierarchy of the corresponding deformed models. The resulting expressions of the conserved currents turn out to be essential for the identification of the scattering phase factors which generate the deformations of the S-matrix, at the quantum level. Finally, the effect of the perturbations on the finite-volume spectrum is investigated using a non-linear integral equation. Exact spectral flow equations are derived, and links with previous literature, in particular on the JT ¯ model, are discussed. While the classical setup is very general, the sine-Gordon model and its CFT limit are used as illustrative quantum examples. Most of the final equations and considerations are, however, of broader validity, or easily generalisable to more complicated systems.