Two-dimensional CFTs and integrable models have an infinite set of conserved KdV higher spin currents. These currents can be argued to remain conserved under the T\bar{T} T T ‾ deformation and its generalizations. We determine the flow equations the KdV charges obey under the T\bar{T} T T ‾ deformation: they behave as probes “riding the Burgers flow” of the energy eigenvalues. We also study a Lorentz-breaking T_{s+1}\bar{T} T s + 1 T ‾ deformation built from a KdV current and the stress tensor, and find a super-Hagedorn growth of the density of states.
Two-dimensional CFTs and integrable models have an infinite set of conserved KdV higher spin currents. These currents can be argued to remain conserved under the T\bar{T} T T ‾ deformation and its generalizations. We determine the flow equations the KdV charges obey under the T\bar{T} T T ‾ deformation: they behave as probes “riding the Burgers flow” of the energy eigenvalues. We also study a Lorentz-breaking T_{s+1}\bar{T} T s + 1 T ‾ deformation built from a KdV current and the stress tensor, and find a super-Hagedorn growth of the density of states.
We consider the most general set of integrable deformations extending the TT[over ¯] deformation of two-dimensional relativistic QFTs. They are CDD deformations of the theory's factorized S matrix related to the higher-spin conserved charges. Using a mirror version of the generalized Gibbs ensemble, we write down the finite-volume expectation value of the higher-spin charges, and derive a generalized flow equation that every charge must obey under a generalized TT[over ¯] deformation. This also reproduces the known flow equations on the nose.
Locality and conserved charges in TT-deformed CFTs
We investigate the locality properties of T \overline T T 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 T\overline T T T ¯ 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.
Solving a family of TTˉ-like theories
We deform two-dimensional quantum field theories by antisymmetric combinations of their conserved currents that generalize Smirnov and Zamolodchikov's TTˉ deformation. We obtain that energy levels on a circle obey a transport equation analogous to the Burgers equation found in the TTˉ 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 JTˉ and TTˉ simultaneously. We check our answer against results from AdS/CFT.
It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator T \bar T T T ‾ , built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that possess a conserved U(1) U ( 1 ) current, J J . The deformation takes the schematic form J \bar T J T ‾ and is interesting because it preserves an SL(2,\mathbb{R}) \times U(1) S L ( 2 , ℝ ) × U ( 1 ) subgroup of the original global conformal symmetries. For the case of a purely (anti)chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions.
We construct a set of conserved charges which are deformations of quantum KdV charges. They are associated with the modular transformation of a GGE with the spin-σ KdV charge inserted with chemical potential ασ. The charges are formal power series in the chemical potential, with coefficients given by local integrals of motion. We define local recursion relations and equivalent flow equations, prove commutativity to all orders, and establish uniqueness of the currents modulo total derivatives for a fixed generating series. We show that the eigenvalues of these deformed charges can be obtained from the thermodynamic Bethe Ansatz in two models. In the c=1/2 free fermion theory, the deformed charges reduce to explicit linear combinations of undeformed KdV charges, and their ground-state eigenvalues agree with the TBA expansion to all orders. In the Lee-Yang model, we find numerical agreement through cubic order in the spin-5 deformation for the first few charges evaluated on low-lying states.
On the flow of states under TT
We study the TT deformation of two dimensional quantum field theories from a Hamiltonian point of view, focusing on aspects of the theory in Lorentzian signature. Our starting point is a simple rewriting of the spatial integral of the TT operator, which directly implies the deformed energy spectrum of the theory. Using this rewriting, we then derive flow equations for various quantities in the deformed theory, such as energy eigenstates, operators, and correlation functions. On the plane, we find that the deformation merely has the effect of implementing successive canonical/Bogoliubov transformations along the flow. This leads us to define a class of non-local, “dressed” operators (including a dressed stress tensor) which satisfy the same commutation relations as in the undeformed theory. This further implies that on the plane, the deformed theory retains its symmetry algebra, including conformal symmetry, if the original theory is a CFT. On the cylinder the TT deformation is much more non-trivial, but even so, correlation functions of certain dressed operators are integral transforms of the original ones. Finally, we propose a tensor network interpretation of our results in the context of AdS/CFT.