Abstract
We show that T\bar{T}, J\bar{T}TT‾,JT‾ and JT_aJTa - deformed classical CFTs posses an infinite set of symmetries that take the form of a field-dependent generalization of two-dimensional conformal transformations. If, in addition, the seed CFTs possess an affine U(1)U(1) symmetry, we show that it also survives in the deformed theories, again in a field-dependent form. These symmetries can be understood as the infinitely-extended conformal and U(1)U(1) symmetries of the underlying two-dimensional CFT, seen through the prism of the ``dynamical coordinates’’ that characterise each of these deformations. We also compute the Poisson bracket algebra of the associated conserved charges, using the Hamiltonian formalism. In the case of the J\bar{T}JT‾ and JT_{a}JTa deformations, we find two copies of a functional Witt - Kac-Moody algebra. In the case of the T\bar{T}TT‾ deformation, we show that it is also possible to obtain two commuting copies of the Witt algebra.
Highlights
We will show that an infinite-dimensional set of symmetries is preserved, and we exemplify how the Poisson brackets of their conserved charges furnish Witt and Kac-Moody algebras in the deformed theory. If this structure survives at quantum level, it would represent a non-local generalization of two-dimensional CFTs, with a possibly generalized symmetry structure that is yet to be understood
This is precisely what was found in the holographic analysis of [70], which provided a holographic dual to the J Tdeformation in terms of AdS3 gravity coupled to a U(1) Chern–Simons gauge field with mixed boundary conditions at infinity and spelled out the asymptotic symmetries associated with these boundary conditions
We studied the symmetries of classical T T, J Tand J Ta - deformed CFTs
Summary
Two-dimensional quantum field theories universally contain a composite irrelevant operator denoted as T T, with a number of remarkable properties [1]. The holographic analyses of [28] for the case of T Tand [64] for the case of J Tsuggest that the answer is affirmative These papers studied the asymptotic symmetries of AdS3 associated to the mixed boundary conditions induced by each of these deformations, and found that all the Virasoro and Kac-Moody symmetries are still present, though their generators become nonlocal due to their dependence on a certain field-dependent modification of the coordinates. We will show that an infinite-dimensional set of symmetries is preserved, and we exemplify how the Poisson brackets of their conserved charges furnish Witt and Kac-Moody algebras in the deformed theory If this structure survives at quantum level, it would represent a non-local generalization of two-dimensional CFTs, with a possibly generalized symmetry structure that is yet to be understood. Various details of the rather technical calculations are relegated to the appendices
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