Abstract
Very special $T\bar{J}$ deformations of a conformal field theory are irrelevant deformations that break the Lorentz symmetry but preserve the twisted Lorentz symmetry. We construct a holographic description of very special $T\bar{J}$ deformations. We give a holographic recipe to study the double trace as well as single trace deformations. The former is obtained from the change of the boundary condition while the latter is obtained from the change of the supergravity background.
Highlights
With a nonzero weight for JðzÞ under K, we may define the twisted Lorentz transformation Jtwisted 1⁄4 L0 − L 0 þ K such that the deformation is invariant under the twisted Lorentz symmetry
We just emphasize here that the existence of twisted Lorentz symmetry drastically simplifies the computation when we study the perturbation theory with respect to μ
In the holographic description of the double trace deformations corresponding to the very special TJdeformations, we change the boundary conditions for the metric and a bulk vector field so that they break the full isometry of the AdS space-time but preserve the twisted Lorentz symmetry
Summary
Inspired by exactly marginal and integrable JJdeformations (e.g., see [1,2,3,4,5]) and irrelevant but integrable TTdeformations of two-dimensional conformal field theories [6,7,8,9], the Lorentz breaking, irrelevant but integrable TJdeformations of conformal field theories, which break Lorentz symmetry, have been vigorously studied in recent works [10,11,12,13,14,15,16,17,18,19,20,21,22]. The first is given by the double trace TJdeformations [11] This is a natural realization of the original field theoretic idea of TJdeformations in the sense that we study the composite operators of T and Jas a double trace operator in the large N setup. This is given by the operator which has the same quantum number as the double trace TJbut is a single trace operator The existence of such operators is not necessarily guaranteed in generic holographic conformal field theories with a conserved current, but in many concrete examples of the AdS=CFT setups, the Kaluza-Klein towers of metric play the role of such operators. We propose holographic descriptions of double trace as well as single trace very special TJdeformations from effective field theory approaches in bulk gravity.
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