Poisson-Lie plurals of Bianchi cosmologies and Generalized Supergravity Equations

Abstract

Poisson-Lie T-duality and plurality are important solution generating techniques in string theory and (generalized) supergravity. Since duality/plurality does not preserve conformal invariance, the usual beta function equations are replaced by Generalized Supergravity Equations containing vector J . In this paper we apply Poisson-Lie Tplurality on Bianchi cosmologies. We present a formula for the vector J as well as transformation rule for dilaton, and show that plural backgrounds together with this dilaton and J satisfy the Generalized Supergravity Equations. The procedure is valid also for non-local dilaton and non-constant J . We also show that Div Θ of the noncommutative structure Θ used for non-Abelian T-duality or integrable deformations does not give correct J for Poisson-Lie T-plurality. * hlavaty@fjfi.cvut.cz † ivo.petr@fit.cvut.cz ContentsG andḠ that may depend on coordinates of the dual group leading to puzzle discussed in [23]. Inspired by [21] we show how to deal with this issue and present formulas that allow us to calculate vector field J appearing in Generalized Supergravity Equations. The plural backgrounds, dilatons and J then satisfy these equations.Authors of [24,25,26] consider Yang-Baxter deformations of AdS 5 × S 5 and calculate vector field J as DivΘ from non-commutative structure Θ in the open string picture. This works for "full" non-Abelian dualities, but not for "factorized" dualities (e.g. some backgrounds obtained in [16]) or more general pluralities.Plan of the paper is following. In next Section we summarize basics of Poisson-Lie T-plurality including transformation of dilaton and write down formulas for vector field J . In Sections 3.1, 3.2 we derive metrics, B-fields and dilatons obtained by Poisson-Lie T-plurality of the flat background, and in Sections 4.2, 4.1 we present metrics, B-fields and dilatons obtained by Poisson-Lie T-plurality of curved Bianchi cosmologies with nontrivial dilaton. It turns out that for appropriately defined vector field J all of them satisfy Generalized Supergravity Equations.

T-folds as Poisson–Lie plurals

“…Matrices C relating Manin triples g g and g ḡ are not unique and different choices may lead to backgrounds with different curvature or torsion properties [47]. However, in [37,38] we have shown that many parameters appearing in general C are irrelevant as the resulting backgrounds differ only by a coordinate or gauge transformation. In such case we choose a representative of the class of C matrices leading to such "equivalent" backgrounds.…”

“…As this has been done in many preceeding papers, e.g. [38], we shall not go into details and restrict to a summary of necessary formulas.…”

“…This procedure was used in [37,38] to construct new solutions of Generalized Supergravity Equations from Bianchi cosmologies. Matrices C relating Manin triples g g and g ḡ are not unique and different choices may lead to backgrounds with different curvature or torsion properties [47].…”

“…Transformation formulas for and J in the context of Poisson-Lie T-plurality were given in [17,31,33,38]. In this paper we shall use these formulas in the form that take into account possible non-locality of the resulting dilaton .…”

“…In [37,38] we have presented many purely bosonic solutions of Generalized Supergravity Equations that were obtained by Poisson-Lie T-duality or plurality transformation of flat and Bianchi cosmologies [39,40]. The purpose of this paper is to present solutions that can be interpreted as T-folds.…”