Yang-Baxter deformations beyond coset spaces (a slick way to do TsT)

Abstract

Yang-Baxter string sigma-models provide a systematic way to deform coset geometries, such as AdS p × S p , while retaining the σ-model integrability. It has been shown that the Yang-Baxter deformation in target space is simply an openclosed string map that can be defined for any geometry, not just coset spaces. Given a geometry with an isometry group and a bivector that is assumed to be a linear combination of antisymmetric products of Killing vectors, we show the equations of motion of (generalized) supergravity reduce to the Classical Yang-Baxter Equation associated with the isometry group, proving the statement made in [1]. These results bring us closer to the proof of the "YB solution generating technique" for (generalized) supergravity advertised in [1] and in particular provide an economical way to perform TsT transformations.

Classical Yang-Baxter equation from supergravity

“…In short, the generalized supergravity EOMs yield the CYBE. In support of this claim, in companion papers [34,35], a perturbative proof of the statement for the NS sector, new examples and generalizations to the RR sector and modified CYBE can be found. This paper serves to summarize this direction.…”

“…This example has been chosen in order to illustrate the applicability of our method to noncoset geometries. Note, however, that nontrivial new solutions to generalized supergravity can be obtained as well, as has been shown in [34,35] for deformations of flat space and Bianchi cosmologies, neither of which are quite as striking as Schwarzschild, so we opted to present the latter.…”

Gauged supergravity from the Lunin-Maldacena background

“…(Note that this can also be performed by a TsT transformation [16] or a Yang-Baxter deformation [17].) In addition to the pure supergravity fields, this would allow the retention of several additional multiplets in the consistent truncation.…”

Tri-vector deformations in d = 11 supergravity

“…Part of motivation for constructing the non-abelian tri-vector deformations was to test the proposals for generalised Yang-Baxter equation that have appeared recently. In [25] it has been shown using techniques of Double Field Theory and β-supergravity that for a bi-vector deformation β = 1 2 r αβ k α ∧k β to generate a solution to the field equations of d = 10 supergravity, it is sufficient that the matrix r αβ satisfy the classical Yang-Baxter equation. The same condition is imposed by assuming that the R-flux vanishes.…”

“…is sufficient for the deformed fields G, B to be a solution of supergravity [22,23]. This allows to view the transformation (1.1) as a supergravity solution generating method, valid for generic spacetimes with isometries [24,25]. The reason that the classical Yang-Baxter equation (1.2) is instrumental in the d = 10 deformation prescription is ultimately that the two-dimensional string worldsheet theory exists behind the scenes of the supergravity approximation.…”