D-branes in the Euclidean AdS3AdS_3 and T-duality

We show that D-branes in the Euclidean $AdS_3$ can be naturally associated to the maximally isotropic subgroups of the Lu-Weinstein double of SU(2). This picture makes very transparent the residual loop group symmetry of the D-brane configurations and gives also immediately the D-branes shapes and the $\sigma$-model boundary conditions in the de Sitter T-dual of the $SL(2,C)/SU(2)$ WZW model.Comment: 29 pages, LaTeX, references adde

On moment maps associated to a twisted Heisenberg double

We review the concept of the (anomalous) Poisson-Lie symmetry in a way that emphasises the notion of Poisson-Lie Hamiltonian. The language that we develop turns out to be very useful for several applications: we prove that the left and the right actions of a group $G$ on its twisted Heisenberg double $(D,\kappa)$ realize the (anomalous) Poisson-Lie symmetries and we explain in a very transparent way the concept of the Poisson-Lie subsymmetry and that of Poisson-Lie symplectic reduction. Under some additional conditions, we construct also a non-anomalous moment map corresponding to a sort of quasi-adjoint action of $G$ on $(D,\kappa)$. The absence of the anomaly of this "quasi-adjoint" moment map permits to perform the gauging of deformed WZW models.Comment: 52 pages, LaTeX, introduction substantially enlarged, several explanatory remarks added, final published versio

On supermatrix models, Poisson geometry and noncommutative supersymmetric gauge theories

We construct a new supermatrix model which represents a manifestly supersymmetric noncommutative regularisation of the $UOSp(2\vert 1)$ supersymmetric Schwinger model on the supersphere. Our construction is much simpler than those already existing in the literature and it was found by using Poisson geometry in a substantial way.Comment: 29 pages, we enlarge Section 3.3 by adding a comparison with older results on the subject of the component expansion

Open Strings and D-branes in WZNW model

An abundance of the Poisson-Lie symmetries of the WZNW models is uncovered. They give rise, via the Poisson-Lie $T$-duality, to a rich structure of the dual pairs of $D$-branes configurations in group manifolds. The $D$-branes are characterized by their shapes and certain two-forms living on them. The WZNW path integral for the interacting $D$-branes diagrams is unambiguously defined if the two-form on the $D$-brane and the WZNW three-form on the group form an integer-valued cocycle in the relative singular cohomology of the group manifold with respect to its $D$-brane submanifold. An example of the $SU(N)$ WZNW model is studied in some detail.Comment: 28 pages, LaTe

Dressing Cosets

The account of the Poisson-Lie T-duality is presented for the case when the action of the duality group on a target is not free. At the same time a generalization of the picture is given when the duality group does not even act on \si-model targets but only on their phase spaces. The outcome is a huge class of dualizable targets generically having no local isometries or Poisson-Lie symmetries whatsoever.Comment: 11 pages, LaTe

q-deformation of z→az+bcz+dz\to {az+b\over cz+d}

We construct the action of the quantum double of \uq on the standard Podle\'s sphere and interpret it as the quantum projective formula generalizing to the q-deformed setting the action of the Lorentz group of global conformal transformations on the ordinary Riemann sphere.Comment: LaTeX, 16 pages, we add a reference where an alternative construction of the q-Lorentz group action on the Podles sphere is give

Poisson-Lie T-duality

A description of dual non-Abelian duality is given, based on the notion of the Drinfeld double. The presentation basically follows the original paper \cite{KS2}, written in collaboration with P. \v Severa, but here the emphasis is put on the algebraic rather than the geometric aspect of the construction and a concrete example of the Borelian double is worked out in detail.Comment: 11 pages, LaTeX, Lecture given at Trieste conference on S-duality and mirror symmetry, June 1995, (signs in Eqs. (10,11) corrected, 1 reference added

u-Deformed WZW model and its gauging

We review the description of a particular deformation of the WZW model. The resulting theory exhibits a Poisson-Lie symmetry with a non-Abelian cosymmetry group and can be vectorially gauged

Poisson-Lie T-duality: Open Strings and D-branes

Global issues of the Poisson-Lie T-duality are addressed. It is shown that oriented open strings propagating on a group manifold $G$ are dual to $D$-brane - anti-$D$-brane pairs propagating on the dual group manifold \ti G. The $D$-branes coincide with the symplectic leaves of the standard Poisson structure induced on the dual group \ti G by the dressing action of the group $G$. T-duality maps the momentum of the open string into the mutual distance of the $D$-branes in the pair. The whole picture is then extended to the full modular space $M(D)$ of the Poisson-Lie equivalent \si-models which is the space of all Manin triples of a given Drinfeld double.T-duality rotates the zero modes of pairs of $D$-branes living on targets belonging to $M(D)$. In this more general case the $D$-branes are preimages of symplectic leaves in certain Poisson homogeneous spaces of their targets and, as such, they are either all even or all odd dimensional.Comment: 15 pages, LaTeX (references added

On the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models

We consider two families of commuting Hamiltonians on the cotangent bundle of the group GL(n,C), and show that upon an appropriate single symplectic reduction they descend to the spectral invariants of the hyperbolic Sutherland and of the rational Ruijsenaars-Schneider Lax matrices, respectively. The duality symplectomorphism between these two integrable models, that was constructed by Ruijsenaars using direct methods, can be then interpreted geometrically simply as a gauge transformation connecting two cross sections of the orbits of the reduction group.Comment: 16 pages, v2: comments and references added at the end of the tex