The standard notion of the non-Abelian duality in string theory is generalized to the class of σ-models admitting 'non-commutative conserved charges'. Such σ-models can be associated with every Lie bialgebra (G,G) and they possess an isometry group iff the commutant [G,G] is not equal toG. Within the enlarged class of the backgrounds the non-Abelian duality is a duality transformation in the proper sense of the word. It exchanges the roles of G andG and it can be interpreted as a symplectomorphism of the phase spaces of the mutually dual theories. We give explicit formulas for the non-Abelian duality transformation for any (G,G). The non-Abelian analogue of the Abelian modular space O(d, d; Z) consists of all maximally isotropic decompositions of the corresponding Drinfeld double.
We study a modification of Poisson geometry by a closed 3-form. Just as for ordinary Poisson structures, these "twisted" Poisson structures are conveniently described as Dirac structures in suitable Courant algebroids. The additive group of 2-forms acts on twisted Poisson structures and permits them to be seen as glued from ordinary Poisson structures defined on local patches. We conclude with remarks on deformation quantization and twisted symplectic groupoids.The ideas presented in this note grew out of an attempt to understand how Poisson geometry on a manifold is affected by the presence of a closed 3-form "field". Such forms are playing an important role in contemporary string theory. We refer, for example, to Park 14) as well as to Cornalba and Sciappa 5) and Klimčík and Ströbl 9) . Our aim here is to show that the notions of Courant algebroid and Dirac structure provide a framework in which one can easily carry out computations in Poisson geometry in the presence of a background 3-form. It seems clear that a proper understanding of the global effect of such a 3-form involves gerbes (see for example Brylinski 3) ); our work here should at least partially substantiate the claim that Courant algebroids are appropriate infinitesimal objects to associate with gerbes.Our work was stimulated in part by the many talks at the Workshop on Deformation Quantization and String Theory at Keio University (March, 2001) in which such 3-forms played an essential role. It is essentially an application of some of the ideas contained in a series of letters fromŠevera to Weinstein written in 1998.
A duality invariant first order action is constructed on the loop group of a Drinfeld double. It gives at the same time the description of both of the pair of σ-models related by Poisson-Lie T-duality. Remarkably, the action contains a WZW-term on the Drinfeld double not only for conformally invariant σ-models. The resulting actions of the models from the dual pair differ just by a total derivative corresponding to an ambiguity in specifying a two-form whose exterior derivative is the WZW three-form. This total derivative is nothing but the Semenov-Tian-Shansky symplectic form on the Drinfeld double and it gives directly a generating function of the canonical transformation relating the σ-models from the dual pair.
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 twoform 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.The Poisson-Lie (PL) T -duality [1] is a generalization of the traditional non-Abelian T -duality [2]- [5] and it proved to enjoy [1], [7]-[13], at least at the classical level, all of the structural features of the traditional Abelian T -duality [14] and [15]. In particular, our so far last paper on the subject [13] has settled (at the classical level) the remaining big issue of the PL generalization: the momentum-winding exchange.It is now of an obvious interest to promote the PL T -duality to the quantum world. Strictly speaking, a consistent quantum picture does not necessarilly imply that mutually dual quantum models have to be conformally invariant. However, we do wish to have conformal examples in order to apply the PL T -duality in string theory. In this paper we shall show that such conformal examples of PL dualizable σ-models are the standard WZNW models and we shall give the detailed classical account of the PL T -duality for them. The treatment of the first quantized strings we postpone to a forth-coming publication, where an emergence of a proliferation of quantum group structures seems unavoidable.In what follows, we shall demonstrate that a PL dualizable σ-model satisfying only a certain mild algebraic condition is necessarily a WZNW model. This means that the WZNW models are not only 'some' conformal examples of the dualizable models but, in a sence, they are very characteristic for the structure of the PL T -duality. Moreover, for various Drinfeld doubles underlying the structure of PL T -duality one recovers the same WZNW model! Hence, there are many (in fact infinitely many) Poisson-Lie symmetries in WZNW models.It turns out that the dual to the WZNW model is again the same WZNW model. This should not be interpreted as a drawback. After all, what really matters is the fact that this (self)-duality induces a non-trivial non-local map on the phase space of the model which, in particular, reshuffles zero modes of the string, much in the same way as in the Abelian T -duality. The fundamental groups of the compact non-Abelian groups 2 are rather small therefore the momentum-winding exchange for closed strings may be rather modest (cf. [13]). On the other hand, the duality transformation of the zero modes of open strings gives the rich and spectacular structure in the dual: the celebrated D-branes [16].We have devoted one paper in our series to the PL T -duality between 2 ...
A simple geometric description of T-duality is given by identifying the cotangent bundles of the original and the dual manifold. Strings propagate naturally in the cotangent bundle and the original and the dual string phase spaces are obtained by different projections. Buscher's transformation follows readily and it is literally projective. As an application of the formalism, we prove that the duality is a symplectomorphism of the string phase spaces.
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 σ-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.
A non-Abelian analogue of the Abelian T -duality momentum-winding exchange is described. The non-Abelian T -duality relates σ-models living on the cosets of a Drinfeld double with respect to its isotropic subgroups. The role of the Abelian momentum-winding lattice is in general played by the fundamental group of the Drinfeld double.
We study Hamiltonian spaces associated with pairs (E,A), where E is a Courant algebroid and Asubset E is a Dirac structure. These spaces are defined in terms of morphisms of Courant algebroids with suitable compatibility conditions. Several of their properties are discussed, including a reduction procedure. This set-up encompasses familiar moment map theories, such as group-valued moment maps, and it provides an intrinsic approach from which different geometrical descriptions of moment maps can be naturally derived. As an application, we discuss the relationship between quasi-Poisson and presymplectic groupoids
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