Twisted K-Theory and K-Theory of Bundle Gerbes

In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twisted K-theory. We provide some examples. Possible applications of bundle gerbe K-theory to the classification of K-brane charges in nontrivial backgrounds are briefly discussed. Received: 29 June 2001 / Accepted: 15 October 2001     

Hitchin–Kobayashi Correspondence, Quivers, and Vortices

 A twisted quiver bundle is a set of holomorphic vector bundles over a complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles, labelled by the arrows. When the manifold is K?hler, quiver bundles admit natural gauge-theoretic equations, which unify many known equations for bundles with extra structure. In this paper we prove a Hitchin–Kobayashi correspondence for twisted quiver bundles over a compact K?hler manifold, relating the existence of solutions to the gauge equations to a stability criterion, and consider its application to a number of situations related to Higgs bundles and dimensional reductions of the Hermitian–Einstein equations. Received: 10 December 2001 / Accepted: 10 November 2002 Published online: 28 May 2003 RID="⋆" ID="⋆" Current address: Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK. E-mail:L.Alvarez-Consul@maths.bath.ac.uk RID="⋆⋆" ID="⋆⋆" Current address: Instituto de Matemáticas y Física Fundamental, CSIC, Serrano 113 bis, 28006 Madrid, Spain. E-mail:oscar.garcia-prada@uam.es Communicated by R.H. Dijkgraaf     

Chern-Weil Construction for Twisted K-Theory

We give a finite-dimensional and geometric construction of a Chern character for twisted K-theory, introducing a notion of connection on a twisted vectorial bundle which can be considered as a finite-dimensional approximation of a twisted family of Fredholm operators. Our construction is applicable to the case of any classes giving the twisting, and agrees with the Chern character of bundle gerbe modules in the case of torsion classes.     

Algebraic Quantization of the Closed Bosonic String

 The gauge invariant observables of the closed bosonic string are quantized in four space-time dimensions by constructing their quantum algebra in a manifestly covariant approach, respecting all symmetries of the classical observables. The quantum algebra is the kernel of a derivation on the universal enveloping algebra of an infinite-dimensional Lie algebra. The search for Hilbert space representations of this algebra is separated from its construction, and postponed. Received: 26 February 2002 / Accepted: 5 September 2002 Published online: 28 March 2003 RID="⋆" ID="⋆" The article is based on the diploma thesis of the first author (C.M.) [1] under the supervision of K. Pohlmeyer, Universit?t Freiburg, completing a project by the second author (K.-H.R.) lying dormant since around 1987. RID="⋆⋆" ID="⋆⋆" Present address: Dept. of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK. E-mail:C.Meusburger@ma.hw.ac.uk Communicated by H. Araki, D. Buchholz and K. Fredenhagen     

On Unitary Representations of Nilpotent Gauge Groups

 Groups of smooth maps from spheres to appropriate nilpotent Lie groups exhibit some peculiar properties of the unitary duals of infinite-dimensional groups. Received: 22 October 2000 / Accepted: 22 November 2002 Published online: 21 February 2003 RID="⋆" ID="⋆" This research was supported in part by CONICET, FONCYT, CONICOR and UNC. Communicated by H. Araki     

Twisted Index Theory on Good Orbifolds, II:¶Fractional Quantum Numbers

This paper uses techniques in noncommutative geometry as developed by Alain Connes [Co2], in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group, continuing our earlier work [MM]. We also compute the range of the higher cyclic traces on K-theory for cocompact Fuchsian groups, which is then applied to determine the range of values of the Connes–Kubo Hall conductance in the discrete model of the quantum Hall effect on the hyperbolic plane, generalizing earlier results in [Bel+E+S], [CHMM]. The new phenomenon that we observe in our case is that the Connes–Kubo Hall conductance has plateaux at integral multiples of a fractional valued topological invariant, namely the orbifold Euler characteristic. Moreover the set of possible fractions has been determined, and is compared with recently available experimental data. It is plausible that this might shed some light on the mathematical mechanism responsible for fractional quantum numbers. Received: 4 November 1999 / Accepted: 22 September 2000     

Torus Chiral <Emphasis Type="Italic">n</Emphasis>-Point Functions for Free Boson and Lattice Vertex Operator Algebras

 We obtain explicit expressions for all genus one chiral n-point functions for free bosonic and lattice vertex operator algebras. We also consider the elliptic properties of these functions. Received: 6 May 2002 / Accepted: 4 October 2002 Published online: 24 January 2003 RID="*" ID="*" Partial support provided by NSF DMS-9709820 and the Committee on Research, University of California, Santa Cruz RID="⋆⋆" ID="⋆⋆" Supported by an Enterprise Ireland Basic Research Grant and the Millenium Fund, National University of Ireland, Galway Communicated by L. Takhtajan     

Deformation Quantization and the Baum–Connes Conjecture

 Alternative titles of this paper would have been ``Index theory without index' or ``The Baum–Connes conjecture without Baum.' In 1989, Rieffel introduced an analytic version of deformation quantization based on the use of continuous fields of C ^* -algebras. We review how a wide variety of examples of such quantizations can be understood on the basis of a single lemma involving amenable groupoids. These include Weyl–Moyal quantization on manifolds, C ^* -algebras of Lie groups and Lie groupoids, and the E-theoretic version of the Baum–Connes conjecture for smooth groupoids as described by Connes in his book Noncommutative Geometry. Concerning the latter, we use a different semidirect product construction from Connes. This enables one to formulate the Baum–Connes conjecture in terms of twisted Weyl–Moyal quantization. The underlying mechanical system is a noncommutative desingularization of a stratified Poisson space, and the Baum–Connes Conjecture actually suggests a strategy for quantizing such singular spaces. Received: 30 April 2002 / Accepted: 2 October 2002 Published online: 17 April 2003 RID="⋆" ID="⋆" Supported by a Fellowship from the Royal Netherlands Academy of Arts and Sciences (KNAW). Communicated by H. Araki, D. Buchholz and K. Fredenhagen     

Holonomy for gerbes over orbifolds

In this paper we compute explicit formulas for the holonomy map for a gerbe with connection over an orbifold. We show that the holonomy descends to a transgression map in Deligne cohomology. We prove that this recovers both the inner local systems in Ruan’s theory of twisted orbifold cohomology [1] and the local system of Freed–Hopkins–Teleman in their work in twisted K-theory [2]. In the case in which the orbifold is simply a manifold we recover previous results of Gawȩdzki [3] and Brylinski[4].     

Quantum Spheres and Projective Spaces as Graph Algebras

 The C ^* -algebras of continuous functions on quantum spheres, quantum real projective spaces, and quantum complex projective spaces are realized as Cuntz-Krieger algebras corresponding to suitable directed graphs. Structural results about these quantum spaces, especially about their ideals and K-theory, are then derived from the general theory of graph algebras. It is shown that the quantum even and odd dimensional spheres are produced by repeated application of a quantum double suspension to two points and the circle, respectively. Received: 31 January 2001 / Accepted: 29 July 2002 Published online: 7 November 2002 RID="*" ID="*" Supported by grant No. R04–2001–000–00117–0 from the Korea Science & Engineering Foundation. RID="**" ID="**" Partially supported by the Research Management Committee of the University of Newcastle.     

Recursion Relations for Unitary Integrals,Combinatorics and the Toeplitz Lattice

 In a discussion in spring 2001, Alexei Borodin showed us recursion relations for the Toeplitz determinants going with the symbols e ^{t(z + z−1)} and \!. Borodin obtained these relations using Riemann-Hilbert methods; see the recent work of Borodin B and Baik Baik. The nature of Borodin's recursion relations pointed towards the Toeplitz lattice and its Virasoro algebra, introduced by us in AvM1. In this paper, we take the Toeplitz lattice and Virasoro algebra approach for a fairly large class of symbols, leading to a systematic way of generating recursion relations. The latter are very naturally expressed in terms of the L-matrices appearing in the Toeplitz lattice equations. As a surprise, we find, compared to Borodin's, a different set of relations, except for the 3-step relations associated with the symbol e ^{t(z + z−1)} . The Painlevé analysis of the Toeplitz lattice enables us to show the ``singularity confinement' for these recursion relations. Received: 30 January 2002 / Accepted: 6 January 2003 Published online: 19 May 2003 RID="⋆" ID="⋆" The support of a National Science Foundation grant DMS-01-00782 is gratefully acknowledged. RID="⋆⋆" ID="⋆⋆" The support of a National Science Foundation grant DMS-01-00782, a Nato, a FNRS and a Francqui Foundation grant is gratefully acknowledged. Communicated by L. Takhtajan     

Gerbes over Orbifolds and Twisted K-Theory

In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H³(X) via Chern-Weil theory. For an arbitrary gerbe , a twisting K_orb(X) of the orbifold K-theory of X is constructed, and shown to generalize previous twisting by Rosenberg [28], Witten [35], Atiyah-Segal [2] and Bowknegt et. al. [4] in the smooth case and by Adem-Ruan [1] for discrete torsion on an orbifold.The first author was partially supported by the National Science Foundation and Conacyt-México     

Projective Module Description of the q-Monopole

The Dirac q-monopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The Chern–Connes pairing of cyclic cohomology and K-theory is computed for the winding number −1. The non-triviality of this pairing is used to conclude that the quantum principal Hopf fibration is non-cleft. Among general results, we provide a left-right symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode. We also provide for arbitrary strong connections on algebraic quantum principal bundles (Hopf–Galois extensions) their associated covariant derivatives on projective modules. Received: Received: 4 September 1998 / Accepted: 16 October 1998     

Glauber Dynamics of the Random Energy Model

 We investigate the long-time behavior of the Glauber dynamics for the random energy model below the critical temperature. We give very precise estimates on the motion of the process to and between the states of extremal energies. We show that when disregarding time, the consecutive steps of the process on these states are governed by a Markov chain that jumps uniformly on all possible states. The mean times of these jumps are also computed very precisely and are seen to be asymptotically independent of the terminal point. A first indicator of aging is the observation that the mean time of arrival in the set of states that have waiting times of order T is itself of order T. The estimates proven in this paper will furnish crucial input for a follow-up paper where aging is analysed in full detail. Received: 9 October 2001 / Accepted: 17 October 2002 Published online: 28 February 2003 RID="*" ID="*" Work partially supported by the Swiss National Science Foundation under contract 21-65267.01 RID="⋆⋆" ID="⋆⋆" On leave from CPT-CNRS, Luminy, Case 907, 13288 Marseille Cedex 9, France. E-mail:veronique.gayrard@epfl.ch Communicated by M. Aizenman     

Conformal Subnets and Intermediate Subfactors

 Given an irreducible local conformal net 𝒜 of von Neumann algebras on S ¹ and a finite-index conformal subnet ℬ⊂𝒜, we show that 𝒜 is completely rational iff ℬ is completely rational. In particular this extends a result of F. Xu for the orbifold construction. By applying previous results of Xu, many coset models turn out to be completely rational and the structure results in [27] hold. Our proofs are based on an analysis of the net inclusion ℬ⊂𝒜; among other things we show that, for a fixed interval I, every von Neumann algebra  intermediate between ℬ(I) and 𝒜(I) comes from an intermediate conformal net ℒ between ℬ and 𝒜 with ℒ(I)=. We make use of a theorem of Watatani (type II case) and Teruya and Watatani (type III case) on the finiteness of the set ℑ(𝒩,ℳ) of intermediate subfactors in an irreducible inclusion of factors 𝒩⊂ℳ with finite Jones index [ℳ:𝒩]. We provide a unified proof of this result that gives in particular an explicit bound for the cardinality of ℑ(𝒩,ℳ) which depends only on [ℳ:𝒩]. Received: 21 December 2001 / Accepted: 28 February 2002 Published online: 14 March 2003 RID="⋆" ID="⋆" Supported in part by MIUR and INDAM-GNAMPA. Communicated by K. Fredenhagen     

The Characteristic Classes of Morita Equivalent Star Products on Symplectic Manifolds

In this paper we give a complete characterization of Morita equivalent star products on symplectic manifolds in terms of their characteristic classes: two star products ⋆ and ⋆' on (M,ω) are Morita equivalent if and only if there exists a symplectomorphism ψ\colon M M such that the relative class t(⋆, ψ^⋆(⋆')) is 2 π i-integral. For star products on cotangent bundles, we show that this integrality condition is related to Dirac's quantization condition for magnetic charges. Received: 19 July 2001 / Accepted: 23 January 2002     

A convergent star product for linear Poisson structures

An explicit star product ⋆ _α ^Γ on the dual of a general Lie algebra equipped with the linear Poisson bracket is constructed. An equivalence operator between this star product and the Kontsevich star product in [K₁] is given and diverse properties of the star product ⋆ _α ^Γ are studied. It is also proved that the star product ⋆ _α ^Γ provides a convergent deformation quantization in the sense of Rieffel [R₁].     

Large Time Behavior for Vortex Evolution in the Half-Plane

 In this article we study the long-time behavior of incompressible ideal flow in a half plane from the point of view of vortex scattering. Our main result is that certain asymptotic states for half-plane vortex dynamics decompose naturally into a nonlinear superposition of soliton-like states. Our approach is to combine techniques developed in the study of vortex confinement with weak convergence tools in order to study the asymptotic behavior of a self-similar rescaling of a solution of the incompressible 2D Euler equations on a half plane with compactly supported, nonnegative initial vorticity. Received: 28 June 2002 / Accepted: 6 January 2003 Published online: 5 May 2003 RID="⋆" ID="⋆" Research supported in part by CNPq grant 300.962/91-6 RID="⋆⋆" ID="⋆⋆" Research supported in part by CNPq grant 300.158/93-9 Communicated by P. Constantin