In this note we explain how world-volume geometries of D-branes can be reconstructed within the microscopic framework where D-branes are described through boundary conformal field theory. We extract the (non-commutative) world-volume algebras from the operator product expansions of open string vertex operators. For branes in a flat background with constant non-vanishing B-field, the operator products are computed perturbatively to all orders in the field strength. The resulting series coincides with Kontsevich's presentation of the Moyal product. After extending these considerations to fermionic fields we conclude with some remarks on the generalization of our approach to curved backgrounds. * E-mail address: vschomer@x4u2.desy.deHere,f (k) denotes the usual Fourier transform of the function f : R d → C. The operator product expansions (2.2) become

We discuss D-branes from a conformal field theory point of view. In this approach, branes are described by boundary states providing sources for closed string modes, independently of classical notions. The boundary states must satisfy constraints which fall into two classes: The first consists of gluing conditions between leftand right-moving Virasoro or further symmetry generators, whereas the second encompasses non-linear consistency conditions from world sheet duality, which severely restrict the allowed boundary states. We exploit these conditions to give explicit formulas for boundary states in Gepner models, thereby computing excitation spectra of brane configurations. From the boundary states, brane tensions and RR charges can also be read off directly.

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...

Boundary conformal field theory is the suitable framework for a microscopic treatment of D-branes in arbitrary CFT backgrounds. In this work, we develop boundary deformation theory in order to study the changes of boundary conditions generated by marginal boundary fields. The deformation parameters may be regarded as continuous moduli of D-branes. We identify a large class of boundary fields which are shown to be truly marginal, and we derive closed formulas describing the associated deformations to all orders in perturbation theory. This allows us to study the global topology properties of the moduli space rather than local aspects only. As an example, we analyse in detail the moduli space of c = 1 theories, which displays various stringy phenomena.

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.

We present a comprehensive analysis of branes in the Euclidean 2D black hole (cigar). In particular, exact boundary states and annulus amplitudes are provided for D0-branes which are localized at the tip of the cigar as well as for two families of extended D1 and D2-branes. Our results are based on closely related studies for the Euclidean AdS 3 model [1] and, as predicted by the conjectured duality between the 2D black hole and the sine-Liouville model, they share many features with branes in Liouville theory. New features arise here due to the presence of closed string modes which are localized near the tip of the cigar. The paper concludes with some remarks on possible applications.

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