We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of Hamiltonian reductions, the Guillemin-Sternberg symplectic cross-section theorem and of convexity properties of the moment map. As an application we obtain moduli spaces of flat connections on an oriented compact 2-manifold with boundary as quasi-Hamiltonian quotients of the space G 2 × · · · × G 2 .

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It is stated in the literature that D-branes in the WZW-model associated with the gluing condition J = −J along the boundary correspond to branes filling out the whole group volume. We show instead that the end-points of open strings are rather bound to stay on 'integer' conjugacy classes. In the case of SU (2) level k WZW model we obtain k − 1 two dimensional Euclidean D-branes and two D particles sitting at the points e and −e.PACS numbers: 11.25. HF; 11.25. Sq. String theory on a group manifold can be described by the world-sheet Wess-Zumino-Witten (WZW) action,This theory possesses chiral currents ( with ∂ ± = ∂ t ±∂ x ),Let us perform our analysis of branes in the closed string picture where D-branes are described as special 'initial conditions' for closed strings rather than by boundary conditions in a theory of open strings. We consider Dbranes corresponding to the standard gluing condition J = −J at the initial time t = t 0 . The same gluing condition was used in . D-branes of this type were studied in the literature. For instance, Kato and Okada [3] suggest that they correspond to Neumann boundary conditions in all directions and, hence, that they fill the whole group manifold G. The same assertion is implicitly contained in [1] where the gluing condition J = −J is considered as a generalization of Neumann boundary conditions for a free bosonic string. This is clearly not the case: If we insert the parametrization g = exp(X) of the group valued field g near the group unit into the gluing conditions we obtain ∂ x X = 0, i.e. the derivative of X along the boundary vanishes. Hence, one should rather view the relation J = −J as a generalization of Dirichlet boundary conditions along the boundary. Using this argument, Stanciu and Tseytlin [4] (in the context of Nappi-Witten backgrounds) see a rather point-like structure of the associated D-branes. Our findings fit well with the analysis of Klimcik and Severa [8]: they identify D-branes in the WZW model with orbits of dressing transformations. If the 'double' (used in [8]) is chosen as G × G, the dressing orbits coincide with conjugacy classes (see [5] for details). Note, however, that no gluing conditions are specified in [8].The analysis below will show that the end-points of open strings with gluing conditions J = −J (in the closed string picture) are localized on special 'integer' conjugacy classes ghg −1 for some fixed h. In particular, for the SU (2) level k WZW model we obtain two D-particles at the points ±e and k − 1 two dimensional Euclidean D-branes.In terms of ∂ t , ∂ x , the gluing condition J = −J readsIt is convenient to introduce a special notation for the adjoint action of G on its Lie algebra, Ad(g)y = gyg −1 . Then, equation (3) can be rewritten asWe split the tangent space to the group G at the point g into an orthogonal (with respect to the Killing metric) sum,where T g G consists of vectors tangential to the orbit of Ad through g. Observe that on T ⊥ g G the operator 1 − Ad(g) vanishes whereas 1 + Ad(g) = 2. Hence, we conclude thatand...

Abstract. A quasi-Poisson manifold is a G-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in ∧ 3 g associated to an invariant inner product. We introduce the concept of the fusion of such manifolds, and we relate the quasi-Poisson manifolds to the previously introduced quasi-Hamiltonian manifolds with groupvalued moment maps.

Motivated by a recent paper of Fock and Rosly [6] we describe a mathematically precise quantization of the Hamiltonian Chern-Simons theory. We introduce the Chern-Simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly. The lattice model enjoys the symmetry with respect to a quantum gauge group. Using this fact we construct the algebra of observables of the Hamiltonian Chern-Simons theory equipped with a *-operation and a positive inner product.HUTMP 94-B336 ESI 79 (1994) UUITP 5/94 UWThPh-1994-8 hep-th 9403066 * On leave of absence from Steklov Mathematical Institute,

Branes in non-trivial backgrounds are expected to exhibit interesting dynamical properties. We use the boundary conformal field theory approach to study branes in a curved background with non-vanishing Neveu-Schwarz 3-form field strength. For branes on an S 3 , the low-energy effective action is computed to leading order in the string tension. It turns out to be a field theory on a non-commutative 'fuzzy 2-sphere' which consists of a Yang-Mills and a Chern-Simons term. We find a certain set of classical solutions that have no analogue for flat branes in Euclidean space. These solutions show, in particular, how a spherical brane can arise as bound state from a stack of D0-branes.

The geometry of D-branes can be probed by open string scattering. If the background carries a non-vanishing B-field, the world-volume becomes noncommutative. Here we explore the quantization of world-volume geometries in a curved background with non-zero Neveu-Schwarz 3-form field strength H = dB. Using exact and generally applicable methods from boundary conformal field theory, we study the example of open strings in the SU(2) Wess-Zumino-Witten model, and establish a relation with fuzzy spheres or certain (non-associative) deformations thereof. These findings could be of direct relevance for D-branes in the presence of Neveu-Schwarz 5-branes; more importantly, they provide insight into a completely new class of worldvolume geometries.DESY 99-104 AEI 1999-11 ESI 755 (1999 hep-th/9908040

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