Open strings and D-branes in WZNW models

Abstract

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 ...

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