Publications
samedi 29 novembre 2008 (mise à jour jeudi 1er avril 2010)
Some of my publications are downloadable in pdf, in the format in which they are published. Other are only accessible through their preprint version on arXiv.
Submitted and non published papers
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Compact $\kappa$-deformation and spectral triples
B. Iochum, T. Masson, T. Schücker, and A. Sitarzarxiv 1004.4190
Abstract
Abstract
Using $C^*$-algebras of groups, we construct discrete versions of $\kappa$-Minkowski space related to a certain compactness of the time coordinate. These models lead naturally to Baumslag--Solitar groups and dynamical systems. We discuss the construction and existence questions of finitely-summable spectral triples on them. -
Generalization of connections on Lie algebroids and derivation-based non-commutative geometry
S. Lazzarini and T. Massonarxiv 1003.6106
Abstract
Abstract
In this paper we study some generalized notions of connections on transitive Lie algebroids from an algebraic point of view. Differential calculi are introduced to manage connections $1$-forms and curvature $2$-forms. Two examples are study in details: the Atiyah Lie algebroid of a principal fiber bundle and the space of derivations of the endomorphisms algebra of a $SL(n)$-vector bundle. Using these two examples we show that the notion of generalized connections studied here is strongly related to the notion of connections on the derivation-based non-commutative geometry of this algebra of endomorphisms. As such, relative to ordinary connections, generalized connections on an Atiyah Lie algebroid is the same kind of generalization as (derivation-based) non-commutative connections.
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A Remark on the Spontaneous Symmetry Breaking Mechanism in the Standard Model
T. Masson and J.-C. Walletarxiv 1001.1176
Abstract
Abstract
In this paper we consider the Spontaneous Symmetry Breaking Mechanism (SSBM) in the Standard Model of particles in the unitary gauge. We show that the computation usually presented of this mechanism can be conveniently performed in a slightly different manner. As an outcome, the computation we present can change the interpretation of the SSBM in the Standard Model, in that it decouples the $SU(2)$-gauge symmetry in the final Lagrangian instead of breaking it.
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Noncommutative $\varepsilon$-graded connections
A. de Goursac, T. Masson, and J.-C. Walletarxiv 0811.3567(2009)
Abstract
Abstract
We introduce the new notion of $\varepsilon$-graded associative algebras which takes its root into the notion of commutation factors introduced in the context of $\varepsilon$-graded Lie algebras. We define and study the associated notion of $\varepsilon$-derivation-based differential calculus, which generalizes the derivation-based calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with the examples of $\varepsilon$-graded matrix algebras. Finally, we apply this formalism to a graded version of the Moyal algebra, for which we recover the recently constructed candidate for a renormalizable gauge action on Moyal space supplemented by terms built from a scalar field and a 2-covariant symmetric tensor field.
Refereed papers
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Examples of derivation-based differential calculi related to noncommutative gauge theories
T. MassonInternational Journal of Geometric Methods in Modern Physics 5 8 1315-1336 (2008)
Abstract
Abstract
Some derivation-based differential calculi which have been used to construct models of noncommutative gauge theories are presented and commented. Some comparisons between them are made.
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Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus
E. Cagnache, T. Masson, and J.-C. WalletJournal of Noncommutative Geometry (2010)
Abstract
Abstract
Derivations of a (noncommutative) algebra can be used to construct various consistent differential calculi, the so-called derivation-based differential calculi. We apply this framework to the noncommutative Moyal algebras for which all the derivations are inner and analyse in detail the case where the derivation algebras generating the differential calculus are related to area preserving diffeomorphisms. The ordinary derivations corresponding to spatial dimensions are supplemented by additional derivations necessarely related to additional covariant coordinates. It is shown that these latter have a natural interpretation as Higgs fields when involved in gauge invariant actions built from the noncommutative curvature. The UV/IR mixing problem for (some of) the resulting Yang-Mills-Higgs models is discussed. A comparition to other noncommutative geometries already considered in the litterature is given.
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Noncommutative generalization of $SU(n)$-principal fiber bundles: a review
T. MassonJournal of Physics: Conference Series 103 012003 (2008)
Abstract
Abstract
This is an extended version of a communication made at the international conference ``Noncommutative Geometry and Physics'' held at Orsay in april 2007. In this proceeding, we make a review of some noncommutative constructions connected to the ordinary fiber bundle theory. The noncommutative algebra is the endomorphism algebra of a $SU(n)$-vector bundle, and its differential calculus is based on its Lie algebra of derivations. It is shown that this noncommutative geometry contains some of the most important constructions introduced and used in the theory of connections on vector bundles, in particular, what is needed to introduce gauge models in physics, and it also contains naturally the essential aspects of the Higgs fields and its associated mechanics of mass generation. It permits one also to extend some previous constructions, as for instance symmetric reduction of (here noncommutative) connections. From a mathematical point of view, these geometrico-algebraic considerations highlight some new point on view, in particular we introduce a new construction of the Chern characteristic classes.
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Anyonic excitations in fast rotating Bose gases
A. Lakhoua, M. Lassaut, T. Masson, and J.-C. WalletPhysical Review A 73 023614 (2006)
Abstract
Abstract
The role of anyonic excitations in fast rotating harmonically trapped Bose gases in a fractional Quantum Hall state is examined. Standard Chern-Simons anyons as well as ``non standard'' anyons obtained from a statistical interaction having Maxwell-Chern-Simons dynamics and suitable non minimal coupling to matter are considered. Their respective ability to stabilize attractive Bose gases under fast rotation in the thermodynamical limit is studied. Stability can be obtained for standard anyons while for non standard anyons, stability requires that the range of the corresponding statistical interaction does not exceed the typical wavelenght of the atoms.
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Born-Infeld inspired bosonic action in a noncommutative geometry
R. Kerner, T. Masson, and E. SériéPhysical Review D 70 067701 (2004)
Abstract
Abstract
The Born-Infeld lagrangian for non-abelian gauge theory is adapted to the case of the generalized gauge fields arising in non-commutative matrix geometry. Basic properties of static and time dependent solutions of the scalar sector of this model are investigated.
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Invariant noncommutative connections
T. Masson and E. SériéJournal of Mathematical Physics 46 123503 (2005)
Abstract
Abstract
In this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle. It has been proven previously that this noncommutative algebra generalizes in a natural way the ordinary geometry of connections. We use explicitely some geometric constructions usually introduced to classify ordinary invariant connections, and we expand them using algebraic objects coming from the noncommutative setting. The main result is that the classification can be performed using a ``reduced'' algebra, an associated differential calculus and a module over this algebra.
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Vortex in Maxwell-Chern-Simons models coupled with external backgrounds
F. Chandelier, Y. Georgelin, M. Lassaut, T. Masson, and J.-C. WalletPhysical Review D 70 6 65016 (2004)
Abstract
Abstract
We consider Maxwell-Chern-Simons models involving different non-minimal coupling terms to a non relativistic massive scalar and further coupled to an external uniform background charge. We study how these models can be constrained to support static radially symmetric vortex configurations saturating the lower bound for the energy. Models involving Zeeman-type coupling support such vortices provided the potential has a ``symmetry breaking'' form and a relation between parameters holds. In models where minimal coupling is supplemented by magnetic and electric field dependant coupling terms, non trivial vortex configurations minimizing the energy occur only when a non linear potential is introduced. The corresponding vortices are studied numerically.
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Quantum Hall conductivity in a Landau type model with a realistic geometry II
F. Chandelier, Y. Georgelin, T. Masson, and J.-C. WalletAnnals of Physics 314 476 (2004)
Abstract
Abstract
We use a mathematical framework that we introduced in a previous paper to study geometrical and quantum mechanical aspects of a Hall system with finite size and general boundary conditions. Geometrical structures control possibly the integral or fractionnal quantization of the Hall conductivity depending on the value of $NB/2\pi$ ($N$ is the number of charge carriers and $B$ is the magnetic field). When $NB/2\pi$ is irrationnal, we show that monovalued wave functions can be constructed only on the graph of a free group with two generators. When $NB/2\pi$ is rationnal, the relevant space becomes a puncturated Riemann surface. We finally discuss our results from a phenomenological viewpoint.
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Non-Abelian generalization of Born-Infeld theory inspired by non-commutative geometry
R. Kerner, T. Masson, and E. SériéPhysical Review D 68 125003 (2003)
Abstract
Abstract
We present a new non-abelian generalization of the Born-Infeld Lagrangian. It is based on the observation that the basic quantity defining it is the generalized volume element, computed as the determinant of a linear combination of metric and Maxwell tensors. We propose to extend the notion of determinant to the tensor product of space-time and a matrix representation of the gauge group. We compute such a Lagrangian explicitly in the case of the $SU(2)$ gauge group and then explore the properties of static, spherically symmetric solutions in this model. We have found a one-parameter family of finite energy solutions. In the last section, the main properties of these solutions are displayed and discussed.
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Quantum Hall conductivity in a Landau type model with a realistic geometry
F. Chandelier, Y. Georgelin, T. Masson, and J.-C. WalletAnnals of Physics 305 60-78 (2003)
Abstract
Abstract
In this paper, we revisit some quantum mechanical aspects related to the Quantum Hall Effect. We consider a Landau type model, paying a special attention to the experimental and geometrical features of Quantum Hall experiments. The resulting formalism is then used to compute explicitely the Hall conductivity from a Kubo formula.
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Global quantum Hall phase diagram from visibility diagrams
F. Chandelier, Y. Georgelin, T. Masson, and J.-C. WalletPhysics Letters A 301 5-6 451-461 (2002)
Abstract
Abstract
We propose a construction of a global phase diagram for the quantum Hall effect. This global phase diagram is based on our previous constructions of visibility diagrams in the context of the Quantum Hall Effect. The topology of the phase diagram we obtain is in good agreement with experimental observations (when the spin effect can be neglected). This phase diagram does not show floating.
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Self-duality in Maxwell-Chern-Simon type effective theories
F. Chandelier, Y. Georgelin, T. Masson, and J.-C. WalletModern Physics Letters B 16 14 497-510 (2002)
Abstract
Abstract
We consider a class of $(2+1)$-dimensional nonlocal effective models with a Maxwell-Chern--Simons part for which the Maxwell term involves a suitable nonlocality that permits one to take into account some $(3+1$)-dimensional features of ``real'' planar systems. We show that this class of models exhibits a hidden duality symmetry stemming from the Maxwell-Chern-Simons part of the action. We discuss and illustrate this result in the framework of a $(2+1)$-dimensional effective model describing (massive) vortices and charges with realistic interactions.
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Visibility diagrams and experimental stripe structures in the quantum Hall effect
Y. Georgelin, T. Masson, and J.-C. WalletJournal of Physics A: Math. Gen. 33 8649-8662 (2000)
Abstract
Abstract
e analyze various properties of the visibility diagrams that can be used in the context of modular symmetries and confront them to some recent experimental developments in the Quantum Hall Effect. We show that a suitable physical interpretation of the visibility diagrams which permits one to describe successfully the observed architecture of the Quantum Hall states gives rise naturally to a stripe structure reproducing some of the experimental features that have been observed in the study of the quantum fluctuations of the Hall conductance. Furthermore, we exhibit new properties of the visibility diagrams stemming from the structure of subgroups of the full modular group.
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$\Gamma(2)$ modular symmetry, renormalization group flow and the quantum Hall effect
Y. Georgelin, T. Masson, and J.-C. WalletJournal of Physics A: Math. Gen. 33 39-55 (2000)
Abstract
Abstract
We construct a family of holomorphic $\beta$-functions whose RG flow preserves the $\Gamma(2)$ modular symmetry and reproduces the observed stability of the Hall plateaus. The semi-circle law relating the longitudinal and Hall conductivities that has been observed experimentally is obtained from the integration of the RG equations for any permitted transition which can be identified from the selection rules encoded in the flow diagram. The generic scale dependance of the conductivities is found to agree qualitatively with the present experimental data. The existence of a crossing point occuring in the crossover of the permitted transitions is discussed.
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On the noncommutative geometry of the endomorphism algebra of a vector bundle
T. MassonJournal of Geometry and Physics 31 142 (1999)
Abstract
Abstract
In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of noncommutative differential forms.
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$SU(n)$-connections and noncommutative differential geometry
M. Dubois-Violette and T. MassonJournal of Geometry and Physics 25 1,2 104 (1998)
Abstract
Abstract
We study the noncommutative differential geometry of the algebra of endomorphisms of any $SU(n)$-vector bundle. We show that ordinary connections on such $SU(n)$-vector bundle can be interpreted in a natural way as a noncommutative $1$-form on this algebra for the differential calculus based on derivations. We interpret the Lie algebra of derivations of the algebra of endomorphisms as a Lie algebroid. Then we look at noncommutative connections as generalizations of these usual connections.
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Modular groups, visibility diagram and quantum Hall effect
Y. Georgelin, T. Masson, and J.-C. WalletJournal of Physics A: Math. Gen. 30 5065-5075 (1997)
Abstract
Abstract
We consider the action of the modular group $\Gamma (2)$ on the set of positive rational fractions. From this, we derive a model for a classification of fractional (as well as integer) Hall states which can be visualized on two ``visibility'' diagrams, the first one being associated with even denominator fractions whereas the second one is linked to odd denominator fractions. We use this model to predict, among some interesting physical quantities, the relative ratios of the width of the different transversal resistivity plateaus. A numerical simulation of the tranversal resistivity plot based on this last prediction fits well with the present experimental data.
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On curvature in noncommutative geometry
M. Dubois-Violette, J. Madore, T. Masson, and J. MouradJournal of Mathematical Physics 37 8 4089 (1996)
Abstract
Abstract
A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.
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On the first order operators in bimodules
M. Dubois-Violette and T. MassonLetters in Mathematical Physics 37 467-474 (1996)
Abstract
Abstract
We analyse the structure of the first order operators in bimodules introduced by A. Connes. We apply this analysis to the theory of connections on bimodules generalizing thereby several proposals.
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Submanifolds and quotient manifolds in noncommutative geometry
T. MassonJournal of Mathematical Physics 37 5 2484-2497 (1996)
Abstract
Abstract
We define and study noncommutative generalizations of submanifolds and quotient manifolds, for the derivation-based differential calculus introduced by M.~Dubois-Violette and P.~Michor. We give examples to illustrate these definitions.
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On the noncommutative riemannian geometry of $GL_q(n)$
Y. Georgelin, J. Madore, T. Masson, and J. MouradJournal of Mathematical Physics 38 3263-3277 (1997)
Abstract
Abstract
A recently proposed definition of a linear connection in non-commutative geometry, based on a generalized permutation, is used to construct linear connections on $GL_q(n)$. Restrictions on the generalized permutation arising from the stability of linear connections under involution are discussed. Candidates for generalized permutation on $GL_q(n)$ are found. It is shown that, for a given generalized permutation, there exists one and only one associated linear connection. Properties of the linear connection are discussed, in particular its bicovariance, torsion and commutative limit.
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Linear connections on the two parameter quantum plane
Y. Georgelin, T. Masson, and J.-C. WalletReview of Mathematical Physics 8 8 1055-1060 (1996)
Abstract
Abstract
We apply a recently proposed definition of a linear connection in non commutative geometry based on the natural bimodule structure of the algebra of differential forms to the case of the two-parameter quantum plane. We find that there exists a non trivial family of linear connections only when the two parameters obeys a specific relation.
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Linear connections on matrix geometries
J. Madore, T. Masson, and J. MouradJournal of Classical and Quantum Gravity 12 1429 (1995)
Abstract
Abstract
A general definition of a linear connection in noncommutative geometry has been recently proposed. Two examples are given of linear connections in noncommutative geometries which are based on matrix algebras. They both possess a unique metric connection.
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Linear connections on the quantum plane
M. Dubois-Violette, J. Madore, T. Masson, and J. MouradLetters in Mathematical Physics 35 351 (1995)
Abstract
Abstract
A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric.
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Basic cohomology of associative algebras
M. Dubois-Violette and T. MassonJournal of Pure and Applied Algebra 114 39 (1996)
Abstract
Abstract
We define a new cohomology for associative algebras which we compute for algebras with units.
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A note about the comparison of the WCA and the Self-consistent WCA perturbation methods
L. Kazandjian and T. MassonJournal of Chemical Physics 99 2275 (1993)
Proceedings
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An informal introduction to the ideas and concepts of noncommutative geometry
T. MassonIntegrables Systems and Quantum Fields Theories, Physique-Mathématique, Travaux en cours, F. Hélein and J. Kouneiher (Ed.), Hermann, 2008Lecture given at the 6th Peyresq meeting 'Integrable systems and quantum field theory'
Abstract
Abstract
This informal introduction is an extended version of a three hours lecture given at the 6th Peyresq meeting ``Integrable systems and quantum field theory''. In this lecture, we make an overview of some of the mathematical results which motivated the development of what is called noncommutative geometry. The first of these results is the theorem by Gelfand and Neumark about commutative $C^\ast$-algebras; then come some aspects of the $K$-theories, first for topological spaces, then for $C^\ast$-algebras and finally the purely algebraic version. Cyclic homology is introduced, keeping in mind its relation to differential structures. The last result is the construction of the Chern character, which shows how these developments are related to each other.
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Linear connections in noncommutative geometry
M. Dubois-Violette, J. Madore, T. Masson, and J. MouradNew trends in Quantum Field Theory, Heron Press, Sofia, 1995
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On finite differential calculi
D. Kastler, J. Madore, and T. MassonGeometry and Nature, Conference on New Trends in Geometrical and Topological Methods, Contemporary Mathematics, H. Nencka and J.-P. Bourguignon (Ed.), 1997, vol. 203, p. 135
Scientific monographs
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Introduction aux (Co)Homologies
T. MassonHermann, Physique-Mathématique, Travaux en cours, 2008
Abstract
Abstract
Depuis son émergence dans la seconde moitié du XIXe siècle, les théories homologiques et cohomologiques ont dynamisé et considérablement enrichi de nombreux domaines des mathématiques. Elles sont parvenues à fertiliser et à éclairer certains travaux en physique théorique (cohomologie BRST des théories de jauge, théories des champs, théories conformes, homologie de Morse et supersymétrie, systèmes dynamiques$\dots$) et à inspirer de nombreux développement mathématiques récents. L'objectif de ce livre est, tout à la fois, d'introduire les méthodes à la base des théories (co)homologiques et de créer une certaine intuition de ces notions en abordant de nombreux exemples concrets et variés. Cette partie conceptuelle est complétée par plus d'une cinquantaine d'exercices et de problèmes complètement corrigés, couvrant un large spectre d'applications. Le contenu de ce livre peut être d'usage courant en fin de second cycle universitaire (quatrième et cinquième année L.M.D.). Il s'adresse aussi bien à des lecteurs engagés dans des études de mathématiques ou de physique théorique, qu'à des chercheurs confirmés voulant s'initier et en savoir plus sur ce domaine.
Thesis and Habilitation
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Quelques aspects de la géométrie non commutative en liaison avec la géométrie différentielle
T. MassonMémoire d'habilitation à diriger des recherches, présenté le 17 février 2009Université Paris XI
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Géométrie non commutative et applications à la théorie des champs
T. MassonThèse soutenue le 13 décembre 1995.Université Paris XI, 1995
Editor
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Proceedings of the International Workshop on Differential Geometry, Noncommutative Geometry, Homology and Fundamental Interactions in Honor of Michel Dubois-Violette
P. Dubois-Violette, T. Masson, and J.-C. Wallet (Ed.)International Journal of Geometric Methods in Modern Physics, 2008, vol. 5, n. 8
Pedagogical manuscripts
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Géométrie différentielle, groupes et algèbres de Lie, fibrés et connexions
T. MassonPolycopié de 195 pages d'introduction aux mathématiques de la géométrie différentielle(1998)
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