Anton Alekseev

D-branes in the WZW model

It is stated in the literature that D-branes in the WZW-model associated with the gluing condition J = - bar{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.

Non-commutative worldvolume geometries: D-branes on SU(2) and fuzzy spheres

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 non-commutative. 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 world-volume geometries.

Brane dynamics in background fluxes and non-commutative geometry

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-vanis ...

Combinatorial quantization of the Hamiltonian Chern-Simons theory. II

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.

Quasi-Poisson manifolds

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 ∧3g 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.

The non-commutative Weil algebra

Abstract.For any compact Lie group G, together with an invariant inner product on its Lie algebra ?, we define the non-commutative Weil algebra ?G as a tensor product of the universal enveloping algebra U(?) and the Clifford algebra Cl(?). Just like the usual Weil algebra WG=S(?*)⊗∧?*, ?G carries the structure of an acyclic, locally free G-differential algebra and can be used to define equivariant cohomology ℋG(B) for any G-differential algebra B. We construct an explicit isomorphism ?:u2009WG→?G of the two Weil algebras as G-differential spaces, and prove that their multiplication maps are G-chain homotopic. This implies that the map in cohomology HG(B)→ℋG(B) induced by ? is a ring isomorphism. For the trivial G-differential algebra B=ℝ, this reduces to the Duflo isomorphism S(?)G≅U(?)G between the ring of invariant polynomials and the ring of Casimir elements.

Current Algebras and Differential Geometry

We show that symmetries and gauge symmetries of a large class of 2-dimensional σ-models are described by a new type of a current algebra. The currents are labeled by pairs of a vector field and a 1-form on the target space of the σ-model. We compute the current-current commutator and analyse the anomaly cancellation condition, which can be interpreted geometrically in terms of Dirac structures, previously studied in the mathematical literature. Generalized complex structures correspond to decompositions of the current algebra into pairs of anomaly free subalgebras. σ-models that we can treat with our method include both physical and topological examples, with and without Wess-Zumino type terms.

UNIVERSALITY OF TRANSPORT PROPERTIES IN EQUILIBRIUM, THE GOLDSTONE THEOREM, AND CHIRAL ANOMALY

We study transport in a class of physical systems possessing two conserved chiral charges. We describe a relation between universality of transport properties of such systems and the chiral anomaly. We show that the nonvanishing of a current expectation v

Representation theory of Chern-Simons observables

Recently we suggested a new quantum algebra, the moduli algebra, which was conjectured to be a quantum algebra of observables of the Hamiltonian Chern Simons theory. This algebra provides the quantization of the algebra of functions on the moduli space of flat connections on a 2-dimensional surface. In this paper we classify unitary representations of this new algebra and identify the corresponding representation spaces with the spaces of conformal blocks of the WZW model. The mapping class group of the surface is proved to act on the moduli algebra by inner automorphisms. The generators of these automorphisms are unitary elements of the moduli algebra. They are constructed explicitly and proved to satisfy the relations of the (unique) central extension of the mapping class group.

Non-commutative gauge theory of twisted D-branes

In this work we propose new non-commutative gauge theories that describe the dynamics of branes localized along twisted conjugacy classes on group manifolds. Our proposal is based on a careful analysis of the exact microscopic solution and it generalizes the matrix models (“fuzzy gauge theories”) that are used to study e.g. the bound state formation of point-like branes in a curved background. We also construct a large number of classical solutions and interpret them in terms of condensation processes on branes localized along twisted conjugacy classes. AEI-2002-027 hep-th/0205123