Anton Alekseev

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The chern-simons theory and quantized moduli spaces of flat connections

Lecture Notes in Physics, 1996

... Page 8. 8 Anton Yu. Alekseev and Volker Schomerus In the following we make several additional... more

Non-commutative gauge theory of twisted D-branes

Nuclear Physics B, 2002

In this work we propose new non-commutative gauge theories that describe the dynamics of branes l... more 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.

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Non-Commutative Gauge Fields from Quantum Groups

NATO ASI Series, 1997

Current Algebra and Differential Geometry

Journal of High Energy Physics, 2005

We dedicate this article to Ludwig Faddeev on the occasion of his 70th birthday.

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Duistermaat-Heckman distributions for group valued moment maps

We introduce equivariant Liouville forms and Duistermaat-Heckman distributions for Hamiltonian gr... more

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

Journal of High Energy Physics, 1999

The geometry of D-branes can be probed by open string scattering. If the background carries a non... more 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.

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Quantization of Wilson loops in Wess-Zumino-Witten models

Journal of High Energy Physics, 2007

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Brane dynamics in background fluxes and non-commutative geometry

Journal of High Energy Physics, 2000

Branes in non-trivial backgrounds are expected to exhibit interesting dynamical properties. We us... more 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.

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On rational Drinfeld associators

Selecta Mathematica, 2011

... Anton Alekseev · Masha Podkopaeva · Pavol Ševera ... Similarly, by putting xi, j = ln(Xi, j )... more ... Anton Alekseev · Masha Podkopaeva · Pavol Ševera ... Similarly, by putting xi, j = ln(Xi, j ), one obtains the (filtered) Lie algebra pbn(K) with generators xi, j for 1 ≤ i < j ≤ n. The associated graded Lie algebra is the Lie algebra tn(K) of infinitesimal braids with generators ti, j = tj,i for ...

Combinatorial quantization of the Hamiltonian Chern-Simons theory I

Communications in Mathematical Physics, 1995

Motivated by a recent paper of Fock and Rosly [6] we describe a mathematically precise quantizati... more 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.

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Wilson lines on noncommutative tori

Physics Letters B, 2000

We introduce the notion of a monodromy for gauge fields with vanishing curvature on the noncommut... more We introduce the notion of a monodromy for gauge fields with vanishing curvature on the noncommutative torus. Similar to the ordinary gauge theory, traces of the monodromies define noncommutative Wilson lines. Our main result is that these Wilson lines are invariant under the Seiberg-Witten map changing the deformation parameter of the noncommutative torus.

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Universality of Transport Properties in Equilibrium, the Goldstone Theorem, and Chiral Anomaly

Physical Review Letters, 1998

We study transport in a class of physical systems possessing two conserved chiral charges. We des... more 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 non-vanishing of a current expectation value implies the presence of gapless modes, in analogy to the Goldstone theorem. Our main tool is a new formula expressing currents in terms of anomalous commutators. Universality of conductance arises as a natural consequence of the nonrenormalization of anomalies. To illustrate our formalism we examine transport properties of a quantum wire in (1+1) dimensions and of massless QED in background magnetic field in (3+1) dimensions.

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D-branes in the WZW model

Physical Review D, 1999

It is stated in the literature that D-branes in the WZW-model associated with the gluing conditio... more 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.

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Poisson Geometry and Applications

Oberwolfach Reports, 2000

This workshop concentrated on some of the major areas of Poisson geometry and its applications.

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Manin Pairs and Moment Maps

Journal of Differential Geometry - J DIFFEREN GEOM, 2000

A Lie group G in a group pair (D, G), integrating the Lie algebra \frak g in a Manin pair (\frak ... more A Lie group G in a group pair (D, G), integrating the Lie algebra \frak g in a Manin pair (\frak d,g), has a quasi-Poisson structure. We define the quasi-Poisson actions of such Lie groups G, and show that they generalize the Poisson actions of Poisson Lie groups. We define and study the moment maps for those quasi-Poisson actions which are hamiltonian. These moment maps take values in the homogeneous space D/G. We prove an analogue of the hamiltonian reduction theorem for quasi-Poisson group actions, and we study the symplectic leaves of the orbit spaces of hamiltonian quasi-Poisson spaces.

Quadratic brackets from symplectic forms

Nuclear Physics B, 1994

We give a physicist oriented survey of Poisson-Lie symmetries of classical systems. We consider f... more We give a physicist oriented survey of Poisson-Lie symmetries of classical systems. We consider finite-dimensional geometric actions and the chiral WZNW model as examples for the general construction. An essential point is the appearance of quadratic Poisson brackets for group-like variables. It is believed that upon quantization they lead to quadratic exchange algebras.

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OPEN STRINGS AND NONCOMMUTATIVE GEOMETRY OF BRANES ON GROUP MANIFOLDS

Modern Physics Letters A, 2001

In this contribution we review some recent work on the non-commutative geometry of branes on grou... more In this contribution we review some recent work on the non-commutative geometry of branes on group manifolds. In particular, we show how fuzzy spaces arise in this context from an exact world-sheet description and we sketch the construction of a low-energy effective action for massless open string modes. The latter is given by a combination of a Yang-Mills and a Chern-Simons like functional on the fuzzy world-volume. It can be used to study condensation on various brane configurations in curved backgrounds.

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On SLE Martingales in Boundary WZW Models

Letters in Mathematical Physics, 2011

Following [3], we consider the boundary WZW model on a half-plane with a cut growing according to... more Following [3], we consider the boundary WZW model on a half-plane with a cut growing according to the Schramm-Loewner stochastic evolution and the boundary fields inserted at the tip of the cut and at infinity. We study necessary and sufficient conditions for boundary correlation functions to be SLE martingales. Necessary conditions come from the requirement for the boundary field at the tip of the cut to have a depth two null vector. Sufficient conditions are established using Knizhnik-Zamolodchikov equations for boundary correlators. Combining these two approaches, we show that in the case of G = SU (2) the boundary correlator is an SLE martingale if and only if the boundary field carries spin 1/2. In the case of G = SU (n) and k = 1 there are several situations when boundary one-point correlators are SLE κ -martingales. If the boundary field is labelled by the defining n-dimensional representation of SU (n), we obtain κ = 2. For n even, by choosing the boundary field labelled by the (unique) self-adjoint fundamental representation, we get κ = 8/(n+2). We also study the situation when the distance between the two boundary fields is finite, and we show that in this case the SLE κ evolution is replaced by SLE κ,ρ with ρ = κ − 6.

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Postface

Letters in Mathematical Physics, 2009

Preface

Letters in Mathematical Physics, 2009

Anton Alekseev

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