Publications

Submitted and non published papers

• Gauge Fixing in QFT and the Dressing Field Method
  M. Guillaud, S. Lazzarini, and T. Masson
  arxiv 2406.19937
  abstract
  In this paper, we revisit the Dressing Field Method (DFM) in the context of Quantum (Gauge) Field Theories (QFT). In order to adapt this method to the functional path integral formalism of QFT, we depart from the usual differential geometry approach used so far to study the DFM which also allows to tackle the infinite dimension of the field spaces. Our main result is that gauge fixing is an instance of the application of the DFM. The Faddeev-Popov gauge fixing procedure and the so-called unitary gauge are revisited in light of this result.
  arXiv

Refereed papers

• Lifting Bratteli Diagrams between Krajewski Diagrams: Spectral Triples, Spectral Actions, and $AF$ algebras
  T. Masson and G. Nieuviarts
  Journal of Geometry and Physics 187 104784 (2023)
  abstract
  In this paper, we present a framework to construct sequences of spectral triples on top of an inductive sequence defining an $AF$-algebra. One aim of this paper is to lift arrows of a Bratteli diagram to arrows between Krajewski diagrams. The spectral actions defining Non-commutative Gauge Field Theories associated to two spectral triples related by these arrows are compared (tensored by a commutative spectral triple to put us in the context of Almost Commutative manifolds). This paper is a follow up of a previous one in which this program was defined and physically illustrated in the framework of the derivation-based differential calculus, but the present paper focuses more on the mathematical structure without trying to study the physical implications.
  arXivpdfdoi
• Derivation-based Noncommutative Field Theories on $AF$ algebras
  T. Masson and G. Nieuviarts
  International Journal of Geometric Methods in Modern Physics 18 13 2150213 (2021)
  abstract
  In this paper, we start the investigation of a new natural approach to ``unifying'' noncommutative gauge field theories (NCGFT) based on approximately finite-dimensional ($AF$) $C^*$-algebras. The defining inductive sequence of an $AF$ $C^*$-algebra is lifted to enable the construction of a sequence of NCGFT of Yang-Mills-Higgs types. The present paper focus on derivation-based noncommutative field theories. A mathematical study of the ingredients involved in the construction of a NCGFT is given in the framework of $AF$ $C^*$-algebras: derivation-based differential calculus, modules, connections, metrics and Hodge $\star$-operators, Lagrangians… Some physical applications concerning mass spectra generated by Spontaneous Symmetry Breaking Mechanisms (SSBM) are proposed using numerical computations for specific situations.
  arXivpdfdoi
• Cartan Connections and Atiyah Lie Algebroids
  J. Attard, J. François, S. Lazzarini, and T. Masson
  Journal of Geometry and Physics 148 103541 (2020)
  abstract
  This work extends previous developments carried out by some of the authors on Ehresmann connections on Atiyah Lie algebroids. In this paper, we study Cartan connections in a framework relying on two Atiyah Lie algebroids based on a $H$-principal fiber bundle $\mathcal{P}$ and its associated $G$-principal fiber bundle $\mathcal{Q} := \mathcal{P} \times_H G$, where $H \subset G$ defines the model for a Cartan geometry. The first main result of this study is a commutative and exact diagram relating these two Atiyah Lie algebroids, which allows to completely characterize Cartan connections on $\mathcal{P}$. Furthermore, in the context of gravity and mixed anomalies, our construction answers a long standing mathematical question about the correct geometrico-algebraic setting in which to combine inner gauge transformations and infinitesimal diffeomorphisms.
  arXivpdfdoi
• Heat coefficient $a_4$ for nonminimal Laplace type operators
  B. Iochum and T. Masson
  Journal of Geometry and Physics 141 120--146 (2019)
  abstract
  Given a smooth hermitean vector bundle $V$ of fiber $\mathbb{C}^N$ over a compact Riemannian manifold and $\nabla$ a covariant derivative on $V$, let $P = -(\lvert g \rvert^{-1/2} \nabla_\mu \lvert g \rvert^{1/2} g^{\mu\nu} u \nabla_\nu + p^\mu \nabla_\mu +q)$ be a nonminimal Laplace type operator acting on smooth sections of $V$ where $u,\,p^\nu,\,q$ are $M_N(\mathbb{C})$-valued functions with $u$ positive and invertible. For any $a \in \Gamma(\text{End}(V))$, we consider the asymptotics $\text{Tr} \,a \,e^{-tP} \sim_{t \downarrow 0} \,\sum_{r=0}^\infty a_r(a, P)\,t^{(r-d)/2}$ where the coefficients $a_r(a, P)$ can be written as an integral of the functions $a_r(a, P)(x) = \text{tr}\,[a(x) \,\mathcal{R}_r(x)]$. This paper revisits the previous computation of $\mathcal{R}_2$ by the authors and is mainly devoted to a computation of $\mathcal{R}_4$. The results are presented with $u$-dependent operators which are universal (i.e. $P$-independent) and which act on tensor products of $u$, $p^\mu$, $q$ and their derivatives via (also universal) spectral functions which are fully described.
  arXivpdfdoi
• Heat asymptotics for nonminimal Laplace type operators and application to noncommutative tori
  B. Iochum and T. Masson
  Journal of Geometry and Physics 129 1--24 (2018)
  abstract
  Let $P$ be a Laplace type operator acting on a smooth hermitean vector bundle $V$ of fiber $\mathbb{C}^N$ over a compact Riemannian manifold given locally by $P = - [g^{\mu\nu} u(x)\partial_\mu\partial_\nu + v^\nu(x)\partial_\nu + w(x)]$ where $u,\,v^\nu,\,w$ are $M_N(\mathbb{C})$-valued functions with $u(x)$ positive and invertible. For any $a \in \Gamma(\text{End}(V))$, we consider the asymptotics $\text{Tr}(a e^{-tP}) \underset{t \downarrow 0^+}{\sim} \,\sum_{r=0}^\infty a_r(a, P)\,t^{(r-d)/2}$ where the coefficients $a_r(a, P)$ can be written locally as $a_r(a, P)(x) = \text{tr}[a(x) \mathcal{R}_r(x)]$. The computation of $\mathcal{R}_2$ is performed opening the opportunity to calculate the modular scalar curvature for noncommutative tori.
  arXivpdfdoi
• Heat trace for Laplace type operators with non-scalar symbols
  B. Iochum and T. Masson
  Journal of Geometry and Physics 116 90--118 (2017)
  abstract
  For an elliptic selfadjoint operator $P =-[u^{\mu\nu}\partial_\mu \partial_\nu +v^\nu \partial_\nu +w]$ acting on a fiber bundle over a Riemannian manifold, where $u^{\mu\nu},v^\mu,w$ are $N\times N$-matrices, we develop a method to compute the heat-trace coefficients $a_r$ which allows to get them by a pure computational machinery. It is exemplified in any even dimension by the value of $a_1$ written both in terms of $u^{\mu\nu}=g^{\mu\nu}u,v^\mu,w$ or diffeomorphic and gauge invariants. We also address the question: when is it possible to get explicit formulae for $a_r$?
  arXivpdfdoi
• Becchi-Rouet-Stora-Tyutin structure for the mixed Weyl-diffeomorphism residual symmetry
  J. François, S. Lazzarini, and T. Masson
  Journal of Mathematical Physics 57 033504 (2016)
  abstract
  We sum up known results on the inclusion of diffeomorphisms in a gauge theory so as to obtain the BRST algebra of a Einstein-Yang-Mills theory. We then show the compatibility of this operation with the (so-called) dressing field method which allows a systematic reduction of gauge symmetries. The robustness of the so obtained scheme is illustrated on the geometry of General Relativity and on the richer example of the second-order conformal structure.
  arXivpdfdoi
• Residual Weyl symmetry out of conformal geometry and its BRS structure
  J. François, S. Lazzarini, and T. Masson
  Journal of High Energy Physics 2015 9 (2015)
  abstract
  The conformal structure of second order in m-dimensions together with the so-called (normal) conformal Cartan connection, is considered as a framework for gauge theories. The dressing field scheme presented in a previous work amounts to a decoupling of both the inversion and the Lorentz symmetries such that the residual gauge symmetry is the Weyl symmetry. On the one hand, it provides straightforwardly the Riemannian parametrization of the normal conformal Cartan connection and its curvature. On the other hand, it also provides the finite transformation laws under the Weyl rescaling of the various geometric objects involved. Subsequently, the dressing field method is shown to fit the BRS differential algebra treatment of infinitesimal gauge symmetry. The dressed ghost field encoding the residual Weyl symmetry is presented. The related so-called algebraic connection supplies relevant combinations found in the literature in the algebraic study of the Weyl anomaly.
  arXivpdfdoi
• Nucleon spin decomposition and differential geometry
  J. François, S. Lazzarini, and T. Masson
  Physical Review D 91 4 045014 (2015)
  abstract
  In the last few years, the so-called Chen et al. approach of the nucleon spin decomposition has been widely discussed and elaborated on. In this letter we propose a genuine differential geometric understanding of this approach. We mainly highligth its relation to the ``dressing field method'' we advocated in [C. Fournel, J. François, S. Lazzarini, T. Masson, Int. J. Geom. Methods Mod. Phys. 11, 1450016 (2014)]. We are led to the conclusion that the claimed gauge-invariance of the Chen et al. decomposition is actually unreal.
  arXivpdfdoi
• Crossed product extensions of spectral triples
  B. Iochum and T. Masson
  Journal of Noncommutative Geometry 10 1 65--133 (2016)
  abstract
  Given a spectral triple $(A,H,D)$ and a $C^*$-dynamical system $(\mathbf{A}, G, \alpha)$ where $A$ is dense in $\mathbf{A}$ and $G$ is a locally compact group, we extend the triple to a triplet $(\mathcal{A},\mathcal{H},\mathcal{D})$ on the crossed product $G \ltimes_{\alpha, \text{red}} \mathbf{A}$ which can be promoted to a modular-type twisted spectral triple within a general procedure exemplified by two cases: the $C^*$-algebra of the affine group and the conformal group acting on a complete Riemannian spin manifold.
  arXivpdfdoi
• Gauge invariant composite fields out of connections, with examples
  C. Fournel, J. François, S. Lazzarini, and T. Masson
  International Journal of Geometric Methods in Modern Physics 11 1 1450016 (2014)
  abstract
  In this paper we put forward a systematic and unifying approach to construct gauge invariant composite fields out of connections. It relies on the existence in the theory of a group valued field with a prescribed gauge transformation. As an illustration, we detail some examples. Two of them are based on known results: the first one provides a reinterpretation of the symmetry breaking mechanism of the electroweak part of the Standard Model of particle physics; the second one is an application to Einstein's theory of gravity described as a gauge theory in terms of Cartan connections. The last example depicts a new situation: starting with a gauge field theory on Atiyah Lie algebroids, the gauge invariant composite fields describe massive vector fields. Some mathematical and physical discussions illustrate and highlight the relevance and the generality of this systematic approach.
  arXivpdfdoi
• $\kappa$-deformation, affine group and spectral triples
  B. Iochum, T. Masson, and A. Sitarz
  Banach Center Publ. 98 261--291 (2012)
  abstract
  A regular spectral triple is proposed for a two-dimensional $\kappa$-deformation. It is based on the naturally associated affine group $G$, a smooth subalgebra of $C^*(G)$, and an operator $\mathcal{D}$ defined by two derivations on this subalgebra. While $\mathcal{D}$ has metric dimension two, the spectral dimension of the triple is one. This bypasses an obstruction described in [Iochum, Masson, Schücker, and Sitarz, Compact $\kappa$-deformation and spectral triples, Reports on Mathematical Physics, 68(1):37--64, 2011], on existence of finitely-summable spectral triples for a compactified $\kappa$-deformation.
  arXivpdfdoi
• Formulation of gauge theories on transitive Lie algebroids
  C. Fournel, S. Lazzarini, and T. Masson
  Journal of Geometry and Physics 64 174--191 (2013)
  abstract
  In this paper we introduce and study some mathematical structures on top of transitive Lie algebroids in order to formulate gauge theories in terms of generalized connections and their curvature: metrics, Hodge star operator and integration along the algebraic part of the transitive Lie algebroid (its kernel). Explicit action functionals are given in terms of global objects and in terms of their local description as well. We investigate applications of these constructions to Atiyah Lie algebroids and to derivations on a vector bundle. The obtained gauge theories are discussed with respect to ordinary and to similar noncommutative gauge theories.
  arXivpdfdoi
• $\kappa$-deformation and spectral triples
  B. Iochum, T. Masson, T. Schücker, and A. Sitarz
  Acta Physica Polonica B Proceedings Supplement 4 3 305 (2011)
  abstract
  The aim of the paper is to answer the following question: does $\kappa$-deformation fit into the framework of noncommutative geometry in the sense of spectral triples? Using a compactification of time, we get a discrete version of $\kappa$-Minkowski deformation via $C^*$-algebras of groups. The dynamical system of the underlying groups (including some Baumslag--Solitar groups) is used in order to construct finitely summable spectral triples. This allows to bypass an obstruction to finite-summability appearing when using the common regular representation.
  arXivpdfdoi
• Compact $\kappa$-deformation and spectral triples
  B. Iochum, T. Masson, T. Schücker, and A. Sitarz
  Reports on Mathematical Physics 68 1 37--64 (2011)
  abstract
  We construct discrete versions of $\kappa$-Minkowski space related to a certain compactness of the time coordinate. We show that these models fit into the framework of noncommutative geometry in the sense of spectral triples. The dynamical system of the underlying discrete groups (which include some Baumslag--Solitar groups) is heavily used in order to construct finitely summable spectral triples. This allows to bypass an obstruction to finite-summability appearing when using the common regular representation. The dimension of these spectral triples is unrelated to the number of coordinates defining the $\kappa$-deformed Minkowski spaces.
  arXivpdfdoi
• Connections on Lie algebroids and on derivation-based non-commutative geometry
  S. Lazzarini and T. Masson
  Journal of Geometry and Physics 62 387--402 (2012)
  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.
  arXivpdfdoi
• Noncommutative $\varepsilon$-graded connections
  A. de Goursac, T. Masson, and J.-C. Wallet
  Journal of Noncommutative Geometry 6 2 343--387 (2012)
  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.
  arXivpdfdoi
• Examples of derivation-based differential calculi related to noncommutative gauge theories
  T. Masson
  International Journal of Geometric Methods in Modern Physics 5 8 1315-1336 (2008)
  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.
  arXivpdfdoi
• Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus
  E. Cagnache, T. Masson, and J.-C. Wallet
  Journal of Noncommutative Geometry 5 1 39--67 (2011)
  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.
  arXivpdfdoi
• Noncommutative generalization of $SU(n)$-principal fiber bundles: a review
  T. Masson
  Journal of Physics: Conference Series 103 012003 (2008)
  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.
  arXivpdfdoi
• Anyonic excitations in fast rotating Bose gases
  A. Lakhoua, M. Lassaut, T. Masson, and J.-C. Wallet
  Physical Review A 73 023614 (2006)
  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.
  arXivpdfdoi
• Born-Infeld inspired bosonic action in a noncommutative geometry
  R. Kerner, T. Masson, and E. Sérié
  Physical Review D 70 067701 (2004)
  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.
  arXivpdfdoi
• Invariant noncommutative connections
  T. Masson and E. Sérié
  Journal of Mathematical Physics 46 123503 (2005)
  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.
  arXivpdfdoi
• Vortex in Maxwell-Chern-Simons models coupled with external backgrounds
  F. Chandelier, Y. Georgelin, M. Lassaut, T. Masson, and J.-C. Wallet
  Physical Review D 70 6 65016 (2004)
  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.
  arXivpdfdoi
• Quantum Hall conductivity in a Landau type model with a realistic geometry II
  F. Chandelier, Y. Georgelin, T. Masson, and J.-C. Wallet
  Annals of Physics 314 476 (2004)
  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.
  arXivpdfdoi
• Non-Abelian generalization of Born-Infeld theory inspired by non-commutative geometry
  R. Kerner, T. Masson, and E. Sérié
  Physical Review D 68 125003 (2003)
  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.
  arXivpdfdoi
• Quantum Hall conductivity in a Landau type model with a realistic geometry
  F. Chandelier, Y. Georgelin, T. Masson, and J.-C. Wallet
  Annals of Physics 305 60-78 (2003)
  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.
  arXivpdfdoi
• Global quantum Hall phase diagram from visibility diagrams
  F. Chandelier, Y. Georgelin, T. Masson, and J.-C. Wallet
  Physics Letters A 301 5-6 451-461 (2002)
  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.
  arXivpdfdoi
• Self-duality in Maxwell-Chern-Simon type effective theories
  F. Chandelier, Y. Georgelin, T. Masson, and J.-C. Wallet
  Modern Physics Letters B 16 14 497-510 (2002)
  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.
  arXivpdfdoi
• Visibility diagrams and experimental stripe structures in the quantum Hall effect
  Y. Georgelin, T. Masson, and J.-C. Wallet
  Journal of Physics A: Math. Gen. 33 8649-8662 (2000)
  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.
  arXivpdfdoi
• $\Gamma(2)$ modular symmetry, renormalization group flow and the quantum Hall effect
  Y. Georgelin, T. Masson, and J.-C. Wallet
  Journal of Physics A: Math. Gen. 33 39-55 (2000)
  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.
  arXivpdfdoi
• On the noncommutative geometry of the endomorphism algebra of a vector bundle
  T. Masson
  Journal of Geometry and Physics 31 142 (1999)
  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.
  arXivpdfdoi
• $SU(n)$-connections and noncommutative differential geometry
  M. Dubois-Violette and T. Masson
  Journal of Geometry and Physics 25 1,2 104 (1998)
  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.
  arXivpdfdoi
• Modular groups, visibility diagram and quantum Hall effect
  Y. Georgelin, T. Masson, and J.-C. Wallet
  Journal of Physics A: Math. Gen. 30 5065-5075 (1997)
  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.
  arXivpdfdoi
• On curvature in noncommutative geometry
  M. Dubois-Violette, J. Madore, T. Masson, and J. Mourad
  Journal of Mathematical Physics 37 8 4089 (1996)
  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.
  arXivpdfdoi
• On the first order operators in bimodules
  M. Dubois-Violette and T. Masson
  Letters in Mathematical Physics 37 467-474 (1996)
  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.
  arXivpdfdoi
• Submanifolds and quotient manifolds in noncommutative geometry
  T. Masson
  Journal of Mathematical Physics 37 5 2484-2497 (1996)
  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.
  arXivpdfdoi
• On the noncommutative riemannian geometry of $GL_q(n)$
  Y. Georgelin, J. Madore, T. Masson, and J. Mourad
  Journal of Mathematical Physics 38 3263-3277 (1997)
  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.
  arXivpdfdoi
• Linear connections on the two parameter quantum plane
  Y. Georgelin, T. Masson, and J.-C. Wallet
  Review of Mathematical Physics 8 8 1055-1060 (1996)
  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.
  arXivpdfdoi
• Linear connections on matrix geometries
  J. Madore, T. Masson, and J. Mourad
  Classical and Quantum Gravity 12 1429 (1995)
  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.
  arXivpdfdoi
• Linear connections on the quantum plane
  M. Dubois-Violette, J. Madore, T. Masson, and J. Mourad
  Letters in Mathematical Physics 35 351 (1995)
  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.
  arXivpdfdoi
• Basic cohomology of associative algebras
  M. Dubois-Violette and T. Masson
  Journal of Pure and Applied Algebra 114 39 (1996)
  abstract
  We define a new cohomology for associative algebras which we compute for algebras with units.
  arXivpdfdoi
• A note about the comparison of the WCA and the Self-consistent WCA perturbation methods
  L. Kazandjian and T. Masson
  Journal of Chemical Physics 99 2275 (1993)
  abstract
  The self‐consistent WCA method is generally regarded as more accurate than the WCA method at high densities, which may seem surprising in view of the orders of approximation expected. It is shown in this note that, in fact, the relative ranking of the two methods depends on the representation chosen for the indirect correlation function of the reference hard‐sphere fluid.
  pdfdoi

Proceedings

• An informal introduction to the ideas and concepts of noncommutative geometry
  T. Masson
  Integrables Systems and Quantum Fields Theories, Physique-Mathématique, Travaux en cours, F. Hélein and J. Kouneiher (Ed.), Hermann, 2008
  Lecture given at the 6th Peyresq meeting 'Integrable systems and quantum field theory'
  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.
  arXiv
• Linear connections in noncommutative geometry
  M. Dubois-Violette, J. Madore, T. Masson, and J. Mourad
  New trends in Quantum Field Theory, proceedings of the 2nd Bulgarian Workshop, Razlog, Bulgaria, A. Ganchev, R. Kerner, and I. T. Todorov (Ed.), Heron Press, Sofia, 1995
• On finite differential calculi
  D. Kastler, J. Madore, and T. Masson
  Geometry 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
  pdf

Scientific monographs

• Introduction aux (Co)Homologies
  T. Masson
  Hermann, Physique-Mathématique, Travaux en cours, 2008
  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.

Chapters in Books

• La physique du XIX$^e$ siècle face au concept de paradigme
  T. Masson
  , 2018
  À paraître dans un ouvrage collectif aux Presses Universitaires d'Aix-Marseille, collection Inter-normes
  abstract
  Nous interrogeons la pertinence du concept de paradigme de Kuhn, ainsi que de son schéma des révolutions scientifiques, en les confrontant à l'histoire des fluides impondérables au XIX\textsuperscript{e} siècle au sein de la thermodynamique et de l'électromagnétisme.
  pdf
• The dressing field method of gauge symmetry reduction, a review with examples
  J. Attard, J. François, S. Lazzarini, and T. Masson
  Foundations of Mathematics and Physics one Century After Hilbert, New Perspectives, J. Kouneiher (Ed.), Springer, 2018
  abstract
  Gauge symmetries are a cornerstone of modern physics but they come with technical difficulties when it comes to quantization, to accurately describe particles phenomenology or to extract observables in general. These shortcomings must be met by essentially finding a way to effectively reduce gauge symmetries. We propose a review of a way to do so which we call the dressing field method. We show how the BRST algebra satisfied by gauge fields, encoding their gauge transformations, is modified. We outline noticeable applications of the method, such as the electroweak sector of the Standard Model and the local twistors of Penrose.
  arXiv
• Gauge field theories: various mathematical approaches
  J. François, S. Lazzarini, and T. Masson
  Mathematical Structures of the Universe, M. Eckstein, M. Heller, and S. J. Szybka (Ed.), Copernicus Center Press, 2014, p. 177--225
  abstract
  This paper presents relevant modern mathematical formulations for (classical) gauge field theories, namely, ordinary differential geometry, noncommutative geometry, and transitive Lie algebroids. They provide rigorous frameworks to describe Yang-Mills-Higgs theories or gravitation theories, and each of them improves the paradigm of gauge field theories. A brief comparison between them is carried out, essentially due to the various notions of connection. However they reveal a compelling common mathematical pattern on which the paper concludes.
  arXivpdf
• Gauge theories in noncommutative geometry
  T. Masson
  FFP11 Symposium Proceedings, AIP, 2012, p. 73--98
  abstract
  In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommutative geometry are covered: the one based on derivations and the one based on spectral triples. Examples of noncommutative gauge field theories are given to illustrate the constructions and to display some of the common features.
  arXiv

Thesis and Habilitation

• Quelques aspects de la géométrie non commutative en liaison avec la géométrie différentielle
  T. Masson
  Habilitation à diriger des recherches
  Mémoire présenté le 17 février 2009
  Université Paris XI
  pdf
• Géométrie non commutative et applications à la théorie des champs
  T. Masson
  Thèse de doctorat
  Thèse soutenue le 13 décembre 1995.
  Université Paris XI, 1995
  pdf

Editor

• ``Fundamental Frontiers of Physics 14''
  E. Kajfasz, T. Masson, and R. Triay (Ed.)
  ``Fundamental Frontiers of Physics 14'', Proceedings of Science, 2014
• 11th annual international meeting ``Fundamental Frontiers of Physics''
  C. Barbachoux, J. Kouneier, T. Masson, and D. Vey (Ed.)
  11th annual international meeting ``Fundamental Frontiers of Physics'', AIP, 2012
• 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

• Géométrie différentielle, groupes et algèbres de Lie, fibrés et connexions
  T. Masson
  Polycopié de 195 pages d'introduction aux mathématiques de la géométrie différentielle
  (1998)
  pdf

Other publications

• La physique quantique, 100 ans de questions (II)
  T. Masson
  Les Cahiers Rationalistes 570 (2004)
  pdf
• La physique quantique, 100 ans de questions (I)
  T. Masson
  Les Cahiers Rationalistes 569 (2004)
  pdf

Discontinued papers

Papers that will not be published for different reasons.

• Local description of generalized forms on transitive Lie algebroids and applications
  C. Fournel, S. Lazzarini, and T. Masson
  arxiv 1109.4282v1
  (2011)
  abstract
  In this paper we study the local description of spaces of forms on transitive Lie algebroids. We use this local description to introduce global structures like metrics, $\ast$-Hodge operation and integration along the algebraic part of the transitive Lie algebroid (its kernel). We construct a Čech-de Rham bicomplex with a Leray-Serre spectral sequence. We apply the general theory to Atiyah Lie algebroids and to derivations on a vector bundle.
  arXiv
• A Remark on the Spontaneous Symmetry Breaking Mechanism in the Standard Model
  T. Masson and J.-C. Wallet
  arxiv 1001.1176
  (2010)
  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.
  arXiv
• Fractional Quantum Hall States in Fast Rotating Bose Gases
  A. Lakhoua, T. Masson, and J.-C. Wallet
  cond-mat/0612479
  (2006)
  abstract
  We use a Chern Simons Landau-Ginzburg (CSLG) framework related to hierarchies of composite bosons to describe 2D harmonically trapped fast rotating Bose gases in Fractional Quantum Hall Effect (FQHE) states. The predicted values for $\nu$ (ratio of particle to vortex numbers) are $\nu = \frac{p}{q}$ ($p$, $q$ are any integers) with even product $pq$, including numerically favored values previously found and predicting a richer set of values. We show that those values can be understood from a bosonic analog of the law of the corresponding states relevant to the electronic FQHE. A tentative global phase diagram for the bosonic system for $\nu <1$ is also proposed.
  arXiv