Abstract
In this paper we investigate Poisson–Lie transformation of dilaton and vector field {mathcal {J}} appearing in generalized supergravity equations. While the formulas appearing in literature work well for isometric sigma models, we present examples for which generalized supergravity equations are not preserved. Therefore, we suggest modification of these formulas.
Highlights
Poisson–Lie duality/plurality is based on the possibility to pass between various decompositions of Drinfel’d double D, which is a 2d-dimensional Lie group whose Lie algebra d can be decomposed into double cross sum [14] of Lie subalgebras g and gthat are maximally isotropic with respect to non-degenerate symmetric bilinear ad-invariant form
They work properly for transformations of isometric sigma models based on semiabelian Manin triples (d, g, a) but not in other cases
We propose modification (26) of formula (21) giving vector fields J which together with dilatons given by formula (13) (when (16) holds) satisfy generalized supergravity equations for all presented examples
Summary
The difference between formulas (21) and (26) can be shown in examples of Poisson–Lie transformations in Drinfel’d doubles (A |G ) where groups G are non-semisimple Bianchi groups. We present transformations in Drinfel’d doubles denoted as D D11 and D D12 so the examples are given by Poisson–Lie pluralities that follow from series of decompositions. Specifying sigma model on Abelian group G with corresponding Drinfel’d double D = (G |G ) = (1|5) whose nontrivial commutation relations read [T 1, T 2] = T 2, [T 1, T 3] = T 3, [T 1, T2] = −T2, [T 1, T3] = −T3, [T 2, T2] = T1, [T 3, T3] = T1. We recover the original background and dilaton, but vector field J = ∂y1 obtained from (21) is different from the initial one and generalized supergravity equations are not satisfied even in this simple case. Vanishing vector J obtained from formulas (13) and (21) do not satisfy generalized supergravity equations. Vector field J = ∂x2 obtained from formula (21) does not satisfy generalized supergravity equations.
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