Abstract
Interesting deformations of AdS5 × S5 such as the gravity dual of noncommutative SYM and Schödinger spacetimes have recently been shown to be integrable. We clarify questions regarding the reality and integrability properties of the associated construction based on R matrices that solve the classical Yang-Baxter equation, and present an overview of manifestly real R matrices associated to the various deformations. We also discuss when these R matrices should correspond to TsT transformations, which not all do, and briefly analyze the symmetries preserved by these deformations, for example finding Schrödinger superalgebras that were previously obtained as subalgebras of $$ \mathfrak{p}\mathfrak{s}\mathfrak{u} $$ (2, 2|4). Our results contain a (singular) generalization of an apparently non-TsT deformation of AdS5 × S5, whose status as a string background is an interesting open question.
Highlights
We clarify questions regarding the reality and integrability properties of the associated construction based on R matrices that solve the classical Yang-Baxter equation, and present an overview of manifestly real R matrices associated to the various deformations. We discuss when these R matrices should correspond to TsT transformations, which not all do, and briefly analyze the symmetries preserved by these deformations, for example finding Schrodinger superalgebras that were previously obtained as subalgebras of psu(2, 2|4)
Realized [20] that this construction can be conveniently adapted to yield integrable deformations based on the classical Yang Baxter equation (CYBE), which has a large space of solutions
Many of these solutions have a nice interpretation in terms of string theory and AdS/CFT, including cases such as the Lunin-Maldacena background [21], the gravity dual of noncommutative supersymmetry Yang-Mills theory (SYM) [22], as well as for example certain Schrodinger spacetimes [23], which were hereby shown to be integrable
Summary
A null Melvin twist can be applied to a background that has a time translation isometry, a spatial translational isometry, and another rotational or translational isometry [35]. While in general the background need not have an SO(1, 1) isometry, in our cases it will. Under this boost t cosh α sinh α t. In the infinite boost limit the y field involved in the intermediate steps scales as yint eα(−t + y) where t and y are the original t and y coordinates. We see that in the limit we are considering, we are doing nothing but a TsT transformation where we shift z by the null direction x−. Since a TsT transformation is based on two commuting isometries, it is clear that we can associate a solution of the CYBE to any TsT transformation, which should both produce the same deformation. The converse does not appear to be the case as already indicated by the results of [25]
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