T-Duality and Doubling of the Isotropic Rigid Rotator

Abstract

After reviewing some of the fundamental aspects of Drinfel'd doubles and Poisson-Lie T-duality, we describe the three-dimensional isotropic rigid rotator on SL(2, C) starting from a non-Abelian deformation of the natural carrier space of its Hamiltonian description on T * SU(2) ≃ SU(2) ⋉ R 3 . A new model is then introduced on the dual group SB(2, C), within the Drinfel'd double description of SL(2, C) = SU(2) ⊲⊳ SB(2, C). The two models are analyzed from the Poisson-Lie duality point of view, and a doubled generalized action is built with T SL(2, C) as carrier space. The aim is to explore within a simple case the relations between Poisson-Lie symmetry, Doubled Geometry and Generalized Geometry. In fact, all the mentioned structures are discussed, such as a Poisson realization of the C-brackets for the generalized bundle T ⊕ T * over SU(2) from the Poisson algebra of the generalized model. The two dual models exhibit many features of Poisson-Lie duals and from the generalized action both of them can be respectively recovered by gauging one of its symmetries.

Topological and dynamical aspects of Jacobi sigma models

“…It is interesting to note the form of the background metric h in (6.13). This metric has been already obtained in the context of Poisson-Lie duality of SU (2) sigma models [32,[34][35][36][37][38] as a non-degenerate metric for the group manifold of SB(2, C), the Borel subgroup of SL(2, C) of upper triangular matrices with complex elements with real diagonal and unit determinant. The latter plays the role of the Poisson-Lie dual of SU (2) in the Manin triple decomposition of the group SL(2, C).…”

A note on Faddeev-Popov action for doubled-yet-gauged particle and graded Poisson geometry

“…Using this gauged one-form, we can define a gauge invariant and (arguably) physically meaningful proper length in doubled spacetime as a path integral over the gauge connection [76], recover the doubled (and gauged) string action by Hull [29] [31], and extend to Green-Schwarz superstring [34], U-duality covariant exceptional string actions [35,36] as well as point-like particle actions [33,[37][38][39] (see (2.12) later).…”

“…The section condition has been argued to imply that the doubled coordinates are actually gauged: a gauge orbit or an equivalence class in the doubled coordinate space corresponds to a single physical point [28]. This idea of 'coordinate gauge symmetry' is naturally realized in sigma models where the doubled target spacetime coordinates are dynamical fields and thus can be genuinely gauged [29][30][31][32][33][34][35][36][37][38][39].…”

Principal Chiral Model without and with WZ term: Symmetries and Poisson-Lie T-Duality

“…If a given symmetry of the dynamics under the Lie group G is not a symmetry for the auxiliary geometric structures, that is not an isometry, but the violation is governed by the Maurer-Cartan structure equation of the Drinfel'd double associated to G , the symmetry of the dynamics is a Poisson-Lie symmetry [62].…”