Categories of Holomorphic Vector Bundles on Noncommutative Two-Tori

 In this paper we study the category of standard holomorphic vector bundles on a noncommutative two-torus. We construct a functor from the derived category of such bundles to the derived category of coherent sheaves on an elliptic curve and prove that it induces an equivalence with the subcategory of stable objects. By the homological mirror symmetry for elliptic curves this implies an equivalence between the derived category of holomorphic bundles on a noncommutative two-torus and the Fukaya category of the corresponding symplectic (commutative) torus. Received: 24 November 2002 / Accepted: 25 November 2002 Published online: 28 February 2003 RID="⋆" ID="⋆" The work of both authors was partially supported by NSF grants. Communicated by A. Connes     

Noncommutative Line Bundle and Morita Equivalence

Global properties of Abelian noncommutative gauge theories based on -products which are deformation quantizations of arbitrary Poisson structures are studied. The consistency condition for finite noncommutative gauge transformations and its explicit solution in the Abelian case are given. It is shown that the local existence of invertible covariantizing maps (which are closely related to the Seiberg–Witten map) leads naturally to the notion of a noncommutative line bundle with noncommutative transition functions. We introduce the space of sections of such a line bundle and explicitly show that it is a projective module. The local covariantizing maps define a new star product which is shown to be Morita equivalent to .     

Contracted Representation of Yang's Spacetime Algebra and Buniy-Hsu-Zee's Discrete Spacetime

Motivated by the recent proposition by Buniy, Hsu, and Zee with respect to discrete spacetime and finite spatial degrees of freedom of our physical world with short- and long-distance scales, l _P and L, we reconsider the Lorentz-covariant Yang's quantized spacetime algebra (YSTA), which is intrinsically equipped with two such kinds of scale parameters, λ and R. In accordance with their proposition, we find the so-called contracted representation of YSTA with finite spatial degrees of freedom associated with the ratio R/λ, which gives a possibility of the divergence-free noncommutative field theory on YSTA. The canonical commutation relations familiar in the ordinary quantum mechanics appear as the cooperative Inonu-Wigner's contraction limit of YSTA, λ → 0 and R → ∓.     

Noncommutativity and Lorentz violation in relativistic heavy ion collisions

We show that relativistic heavy ion collisions at LHC energies could be used as an experimental probe to detect fundamental properties of spacetime long speculated about. Our results rely on the recent proposal that magnetic fields of intensity much larger than that of magnetars should be produced at the beginning of the collisions and this could have an important impact on the experimental manifestation of a noncommutative spacetime. Indeed, in the noncommutative generalization of electrodynamics the interplay between a nonzero noncommutative parameter and an external magnetic field leads us to predict the production of lepton pairs of low invariant mass by free photons (an event forbidden by Lorentz invariant electrodynamics) in relativistic heavy ion collisions at present and future available energies. This unique channel can be clearly considered as a signature of noncommutativity. On the other hand, the search for such decays is worth anyway because their absence would ameliorate of three orders of magnitude the current bound on the noncommutative parameter.     

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     

The Skyrme model with a U*(1) gauged Wess‐Zumino term in a noncommutative spacetime

The Skyrme model is generalized for a noncommutative spacetime with the Weyl‐operators of SU(2) matrices and the corresponding star‐product. The unitary condition and the topological current can be extended to star‐exponential matrices. The Wess‐Zumino term which breaks unphysical symmetries of the Skyrme action is gauged with the U_*(1) group to allow for electromagnetic processes in a noncommutative spacetime. Apart from corrections to the anomalous decay γ→π⁰π⁺π^– in commuting spacetime, the additional anomalous process γ→π⁰π⁰π⁰ is found in the U_*(1) gauged Wess‐Zumino action for a noncommutative spacetime.     

Quantum geons and noncommutative spacetimes

Physical considerations strongly indicate that spacetime at Planck scales is noncommutative. A popular model for such a spacetime is the Moyal plane. The Poincaré group algebra acts on it with a Drinfel’d-twisted coproduct, however the latter is not appropriate for more complicated spacetimes such as those containing Friedman-Sorkin (topological) geons. They have rich diffeomorphisms and mapping class groups, so that the statistics groups for N identical geons is strikingly different from the permutation group S _N . We generalise the Drinfel’d twist to (essentially all) generic groups including finite and discrete ones, and use it to deform the commutative spacetime algebras of geons to noncommutative algebras. The latter support twisted actions of diffeomorphisms of geon spacetimes and their associated twisted statistics. The notion of covariant quantum fields for geons is formulated and their twisted versions are constructed from their untwisted counterparts. Non-associative spacetime algebras arise naturally in our analysis. Physical consequences, such as the violation of Pauli’s principle, seem to be one of the outcomes of such nonassociativity. The richness of the statistics groups of identical geons comes from the nontrivial fundamental groups of their spatial slices. As discussed long ago, extended objects like rings and D-branes also have similar rich fundamental groups. This work is recalled and its relevance to the present quantum geon context is pointed out.     

Conceptual Unification of Gravity and Quanta

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra on the groupoid Γ=E×G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of , is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein’s equation. The algebra , when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra ℳ of random operators representing the quantum sector of the model. The Tomita–Takesaki theorem allows us to define the dynamics of random operators which depends on the state φ. The same state defines the noncommutative probability measure (in the sense of Voiculescu’s free probability theory). Moreover, the state φ satisfies the Kubo–Martin–Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra , one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not “feel” singularities.     

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     

QFT on homothetic Killing twist deformed curved spacetimes

We study the quantum field theory (QFT) of a free, real, massless and curvature coupled scalar field on self-similar symmetric spacetimes, which are deformed by an abelian Drinfel’d twist constructed from a Killing and a homothetic Killing vector field. In contrast to deformations solely by Killing vector fields, such as the Moyal-Weyl Minkowski spacetime, the equation of motion and Green’s operators are deformed. We show that there is a *-algebra isomorphism between the QFT on the deformed and the formal power series extension of the QFT on the undeformed spacetime. We study the convergent implementation of our deformations for toy-models. For these models it is found that there is a *-isomorphism between the deformed Weyl algebra and a reduced undeformed Weyl algebra, where certain strongly localized observables are excluded. Thus, our models realize the intuitive physical picture that noncommutative geometry prevents arbitrary localization in spacetime.     

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     

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     

Resonance Interaction Energy in κ-Minkowski Spacetime

If gravity is quantized, one of the consequences may be that the spacetime coordinates are quantized and become noncommutative. The κ-Minkowski spacetime is such kind of noncommutative spacetime. In this paper, the resonance interaction energy of a two-atom system coupled with a fluctuating vacuum scalar field in the κ-Minkowski spacetime is studied. It is found that the resonance interaction energy is dependent on the interatomic separation, the transition wavelength of the atoms, and the spacetime non-commutativity. When the interatomic separation is small compared with a characteristic length determined by the spacetime non-commutativity parameter and the transition wavelength, the resonance interaction energy is that in the Minkowski spacetime plus a correction due to the spacetime non-commutativity. When the interatomic separation is comparable to or larger than the characteristic length, the resonance interaction energy cannot be organized in the form of a Minkowski term plus a correction, which indicates that the long-range behavior of the vacuum in the κ-Minkowski spacetime is fundamentally different from that in the Minkowski spacetime.     

An arena for model building in the Cohen—Glashow very special relativity

The Cohen—Glashow Very Special Relativity (VSR) algebra is defined as the part of the Lorentz algebra which upon addition of CP or T invariance enhances to the full Lorentz group, plus the space—time translations. We show that noncommutative space—time, in particular noncommutative Moyal plane, with light- like noncommutativity provides a robust mathematical setting for quantum field theories which are VSR invariant and hence set the stage for building VSR invariant particle physics models. In our setting the VSR invariant theories are specified with a single deformation parameter, the noncommutativity scale ╕_NC. Preliminary analysis with the available data leads to ╕_NC ≳ 1–10 TeV.     

Three-dimensional noncommutative bosonization

We consider the extension of the (2+1)-dimensional$(2+1)\text{-dimensional}$ bosonization process in noncommutative (NC) spacetime. We show that the large mass limit of the effective action obtained by integrating out the fermionic fields in NC spacetime leads to the NC Chern–Simons action. The present result is valid to all orders in the noncommutative parameter θ. We also discuss how the NC Yang–Mills action is induced in the next to leading order.     

Noncommutative gravity, a ‘no strings attached’ quantum-classical duality, and the cosmological constant puzzle

There ought to exist a reformulation of quantum mechanics which does not refer to an external classical spacetime manifold. Such a reformulation can be achieved using the language of noncommutative differential geometry. A consequence which follows is that the ‘weakly quantum, strongly gravitational’ dynamics of a relativistic particle whose mass is much greater than Planck mass is dual to the ‘strongly quantum, weakly gravitational’ dynamics of another particle whose mass is much less than Planck mass. The masses of the two particles are inversely related to each other, and the product of their masses is equal to the square of Planck mass. This duality explains the observed value of the cosmological constant, and also why this value is nonzero but extremely small in Planck units. Second Award in the 2008 Essay Competition of the Gravity Research Foundation.     

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     

Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes

In this article we study the quantization of a free real scalar field on a class of noncommutative manifolds, obtained via formal deformation quantization using triangular Drinfel’d twists. We construct deformed quadratic action functionals and compute the corresponding equation of motion operators. The Green’s operators and the fundamental solution of the deformed equation of motion are obtained in terms of formal power series. It is shown that, using the deformed fundamental solution, we can define deformed *-algebras of field observables, which in general depend on the spacetime deformation parameter. This dependence is absent in the special case of Killing deformations, which include in particular the Moyal-Weyl deformation of the Minkowski spacetime.     

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     

Well-Posedness for the Linearized Motion of a Compressible Liquid with Free Surface Boundary

 We study the motion of a compressible perfect liquid body in vacuum. This can be through of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove linearized stability in Sobolev space assuming a ``physical condition', related to the fact that the pressure of a fluid has to be positive. Received: 23 September 2002 / Accepted: 2 December 2002 Published online: 14 April 2003 RID="⋆" ID="⋆" The author was supported in part by the National Science Foundation. Communicated by P. Constantin