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 mathcal{J} . In this paper we apply Poisson-Lie T-plurality on Bianchi cosmologies. We present a formula for the vector mathcal{J} as well as transformation rule for dilaton, and show that plural backgrounds together with this dilaton and mathcal{J} satisfy the Generalized Supergravity Equations. The procedure is valid also for non-local dilaton and non-constant mathcal{J} . We also show that Div Θ of the non-commutative structure Θ used for non-Abelian T-duality or integrable deformations does not give correct mathcal{J} for Poisson-Lie T-plurality.
Highlights
Sigma modelsLet M be (n + d)-dimensional (pseudo-)Riemannian target manifold and consider sigma model on M given by Lagrangian
C10 0 4e2x1 t2 e4x1 t4+4 0 − 2e4x1 t4 2e4x1 t4 4e2x1 t2 (3.16)obtained by this β-shift can be brought to the Brinkmann form of a plane parallel wave ds2 = 2u2u4 − 5 z32 + z42 (u4 + 1)2du2 + 2du dv + dz32 + dz42 (3.17)
Since duality/plurality does not preserve conformal invariance, the usual beta function equations are replaced by Generalized Supergravity Equations containing vector J
Summary
Let M be (n + d)-dimensional (pseudo-)Riemannian target manifold and consider sigma model on M given by Lagrangian. Assume that there is a d-dimensional Lie group G with free action on M that leaves the tensor invariant. Dualizable sigma model on N ×G is given by tensor field F defined by spectator-dependent (n + d) × (n + d) matrix E(s) and group-dependent E(x) as. In this paper the groups G will be non-semisimple Bianchi groups. Their elements will be parametrized as g = ex1T1 ex2T2 ex3T3 where ex2T2 ex3T3 and ex3T3 parametrize their normal subgroups. Bianchi cosmologies are defined on four-dimensional manifolds, dim N = 1 and we denote the spectator s1 as t
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