Cette thèse est consacrée à l’étude de trois questions différentes concernant les groupoïdes de Lie et leurs applications. Le premier chapitre présente quelques préliminaires sur les groupoïdes de Lie. Dans le chapitre 2, on exprime la déformation de Witten à l’aide d’une déformation au cone normal et la théorie de C∗-modules ce qui nous permet de retrouver les inégalités de Morse. Notre méthode se généralise au cas des feuilletages. Dans le chapitre 3, on donne une construction simple du groupoïde de déformation construit par Choi-Pönge et Van Erp-Yuncken. Rappelons que celui-ci décrit le calcule pseudo-différentiel inhomogène grâce au travail de Debord-Skandalis et Van Erp- Yuncken. Notre construction montre que le groupoïde de déformation est en fait une déformation au cone normal classique itérée. Dans le chapitre 4, suivant le travail de Antonini, Azzali et Skandalis, on construit un élément en KK-théorie équivariante qui permet d’exprimer directement les invariants de Chern-Simons en K-théorie. Dans l’appendice on donne quelques rappels sur la KK-théorie équivariante et la KK-théorie réelle introduite par Antonini, Azzali et Skandalis. / This thesis is devoted to the study of three different questions concerning Lie groupoids and their applications. The first chapter presents some preliminaries on Lie groupoids. In Chapter 2, Witten’s deformation is expressed using deformation to the normal cone construction and the theory of C∗-modules, which allows us to reprove the Morse inequalities. Our method is generalised to the case of foliations. In Chapter 3, we give a simple construction of the deformation groupoid built by Choi-Pönge and Van Erp-Yuncken. Recall that this groupoid describes the inhomogeneous pseudo-differential calculus thanks to the work of Debord-Skandalis and Van Erp-Yuncken. Our construction shows that the deformation groupoid is actually an iterated classical deformation to the normal cone. In Chapter 4, following the work of Antonini, Azzali and Skandalis, we construct an element in equivariant KK-theory that allows us to express the Chern-Simons invariants directly in K-theory. In the appendix we give some reminders about the equivariant KK-theory and the real KK-theory introduced by Antonini, Azzali and Skandalis.

Texte synthétique de présentation des théories quantiques des champs dites "hyperboliques" , définies par l'auteur en collaboration avec R. Benedetti. Leur place en topologie quantique, et leurs relations avec la conjecture du volume et les invariants de Chern-Simons, sont développés.

Dans cette thèse, nous traitons de l'interaction entre deux états de<br />bord dans le régime de l'effet Hall fractionnaire. Nous nous sommes<br />appuyés sur les réalisations expérimentales récentes de structures<br />(les jonctions quantiques étendues) qui favorisent largement ces<br />interactions.<br /><br />Nous avons d'abord introduit l'effet Hall quantique en mettant<br />l'accent sur la physique des états de bords.<br /><br />Nous avons ensuite étudié le transport tunnel à travers une jonction<br />quantique étendue, en considérant les bords comme des liquides de<br />Luttinger chiraux. Nous exposons différents régimes de courant en<br />fonction de la longueur de la jonction, la force des interactions,<br />le facteur de remplissage et la température. Nous calculons<br />également le bruit associé à ce courant.<br /><br />Nous avons généralisé ces résultats à une jonction en coin, dans<br />laquelle les canaux de bord peuvent être contre ou copropageant.<br />Dans le cas contre-propageant, il apparaît une transition de phase<br />commensurable-incommensurable en fonction des interactions et d'un<br />paramêtre supplémentaire. Dans le cas co-propageant, nous calculons<br />le courant tunnel en perturbation dans un modèle de sine-Gordon<br />chiral.<br /><br />Enfin, nous déterminons les profils de densité des bords et les<br />excitations. Nous mettons en évidence l'apparition d'une instabilité<br />dans ces jonctions quand l'interaction entre les bords est trop<br />forte. La théorie de Chern-Simons montre qu'il se produit alors une<br />reconstruction des bords par les interactions.

Nous présentons dans ce travail deux exemples de modèles topologiques faisant appel à la cohomologie : - dans le premier exemple nous montrons comment obtenir des invariants topologiques, tels que ceux de Donaldson, de Mumford, de Mathaï-Quillen ou de gravité topologique, en utilisant la cohomologie équivariante. Nous présentons une méthode universelle permettant d'obtenir de tels invariants topologiques en se basant sur une approche de type BRST. Nous rappelons qu'il existe différents " schémas " caractérisant une théorie équivariante et nous montrons comment le schéma de Kalkman permet une construction optimisée des invariants. - dans le second exemple nous étudions les théories abéliennes de Chern-Simons. Nous montrons comment une approche basée sur la cohomologie de Deligne-Beilinson permet de traiter ces théories sur des variétés fermées de dimension trois. Nous montrons comment la structure de ces espaces de cohomologie induit canoniquement la quantification de la constante de couplage et des charges, tout en fournissant les informations nécessaires et suffisantes pour obtenir via l'intégration fonctionnelle les invariants de liens usuellement obtenus à partir de procédures de chirurgie sur la sphère. Cette méthode admet un prolongement naturel qui permet de traiter plus généralement les variétés de dimension 4n+3.

This thesis consists of two parts, which can be read separately. In the first part we study aspects of topological field theories. We show how to topologically twist three-dimensional N=2 supersymmetric Chern-Simons theory using a contact structure on the underlying manifold. This gives us a formulation of Chern-Simons theory together with a set of auxiliary fields and an odd symmetry. For Seifert manifolds, we show how to use this odd symmetry to localize the path integral of Chern-Simons theory. The formulation of three-dimensional Chern-Simons theory using a contact structure admits natural generalizations to higher dimensions. We introduce and study these theories. The focus is on the five-dimensional theory, which can be understood as a topologically twisted version of N=1 supersymmetric Yang-Mills theory. When formulated on contact manifolds that are circle fibrations over a symplectic manifold, it localizes to contact instantons. For the theory on the five-sphere, we show that the perturbative part of the partition function is given by a matrix model. In the second part of the thesis, we study supersymmetric sigma models in the Hamiltonian formalism, both in a classical and in a quantum mechanical setup. We argue that the so called Chiral de Rham complex, which is a sheaf of vertex algebras, is a natural framework to understand quantum aspects of supersymmetric sigma models in the Hamiltonian formalism. We show how a class of currents which generate symmetry algebras for the classical sigma model can be defined within the Chiral de Rham complex framework, and for a six-dimensional Calabi-Yau manifold we calculate the equal-time commutators between the currents and show that they generate the Odake algebra.

This dissertation splits into two distinct halves. The first half is devoted to the study of the holography of higher spin gauge theory in AdS$_3$. We present a conjecture that the holographic dual of $W_N$ minimal model in a 't Hooft-like large $N$ limit is an unusual ``semi-local" higher spin gauge theory on AdS$_3\times $S$^1$. At each point on the S$^1$ lives a copy of three-dimensional Vasiliev theory, that contains an infinite tower of higher spin gauge fields coupled to a single massive complex scalar propagating in AdS$_3$. The Vasiliev theories at different points on the S$^1$ are correlated only through the AdS$_3$ boundary conditions on the massive scalars. All but one single tower of higher spin symmetries are broken by the boundary conditions. This conjecture is checked by comparing tree-level two- and three-point functions, and also one-loop partition functions on both side of the duality. The second half focuses on the holography of higher spin gauge theory in AdS$_4$. We demonstrate that a supersymmetric and parity violating version of Vasiliev's higher spin gauge theory in AdS$_4$ admits boundary conditions that preserve ${\cal N}=0,1,2,3,4$ or $6$ supersymmetries. In particular, we argue that the Vasiliev theory with $U(M)$ Chan-Paton and ${\cal N}=6$ boundary condition is holographically dual to the 2+1 dimensional $U(N)_k\times U(M)_{-k}$ ABJ theory in the limit of large $N,k$ and finite $M$. In this system all bulk higher spin fields transform in the adjoint of the $U(M)$ gauge group, whose bulk t'Hooft coupling is $\frac{M}{N}$. Our picture suggests that the supersymmetric Vasiliev theory can be obtained as a limit of type IIA string theory in AdS$_4\times \mathbb{CP}^3$, and that the non-Abelian Vasiliev theory at strong bulk 't Hooft coupling smoothly turn into a string field theory. The fundamental string is a singlet bound state of Vasiliev's higher spin particles held together by $U(M)$ gauge interactions. / Physics

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<p>The fractional quantum Hall effect (FQHE), discovered in 1982 in a two-dimensional electron system, has generated a wealth of successful theory and new concepts in condensed matter physics, but is still not fully understood. The possibility of having nonabelian quasiparticle statistics has recently attracted attention on purely theoretical grounds but also because of its potential applications in topologically protected quantum computing.</p><p>This thesis focuses on the quasiparticles using three different approaches. The first is an effective Chern-Simons theory description, where the noncommutativity imposed on the classical space variables captures the incompressibility. We propose a construction of the quasielectron and illustrate how many-body quantum effects are emulated by a classical noncommutative theory.</p><p>The second approach involves a study of quantum Hall states on a torus where one of the periods is taken to be almost zero. Characteristic quantum Hall properties survive in this limit in which they become very simple to understand. We illustrate this by giving a simple counting argument for degeneracy 2<i>n</i>^-1, pertinent to nonabelian statistics, in the presence of 2<i>n</i> quasiholes in the Moore-Read state and generalise this result to 2<i>n</i>-<i>k</i> quasiholes and <i>k </i>quasielectrons.</p><p>In the third approach, we study the topological nature of the degeneracy 2<i>n</i>^-1 by using a recently proposed analogy between the Moore-Read state and the two-dimensional spin-polarized p-wave BCS state. We study a version of this problem where one can use techniques developed in the context of high-<i>T</i>c superconductors to turn the vortex background into an effective gauge field in a Dirac equation. Topological arguments in the form of index theory gives the degeneracy 2<i>n</i>^-1 for 2<i>n</i> vortices.</p>

We study three types of theories, using supersymmetry and ideas from string theory as tools to gain understanding of systems of more general interest. Firstly, we introduce non-relativistic Chern-Simons-matter field theories in three dimensions and study their anyonic spectrum in a conformal phase. These theories have supersymmetric completions, which in the non-relativistic case suffices to protect certain would-be BPS quantities from corrections. This allows us to compute one-loop exact anomalous dimensions of various bound states of non-Abelian anyons, analyse some interesting unitarity bound violations, and test some recently proposed bosonization dualities. Secondly, we turn on a chemical potential and break conformal invariance, putting the theory into the regime of the Fractional Quantum Hall Effect (FQHE). This is illustrated in detail: the theory supports would-be BPS vortices which model the electrons of the FQHE, and they form bag-like states with the appropriate filling fractions, Hall conductivities, and anyonic excitations. This formalism makes possible some novel explicit computations: an analytic calculation of the anyonic phases experienced by Abelian quasiholes; analytic relationships to the boundary Wess-Zumino-Witten model; and derivations of a wide class of QHE wavefunctions from a bulk field theory. We also further test the three-dimensional bosonization dualities in this new setting. Along the way, we accumulate new descriptions of the QHE. Finally, we turn away from flat space and investigate a problem in (3+1)-dimensional quantum gravity. We find that even as an effective theory, the theory has enough structure to suggest the inclusion of certain gravitational instantons in the path integral. An explicit computation in a minimally supersymmetric case illustrates the principles at work, and highlights the role of a hitherto unidentified scale in quantum gravity. It also is an interesting result in itself: a non-perturbative quantum instability of a flat supersymmetric Kaluza-Klein compactification.

The fractional quantum Hall effect (FQHE), discovered in 1982 in a two-dimensional electron system, has generated a wealth of successful theory and new concepts in condensed matter physics, but is still not fully understood. The possibility of having nonabelian quasiparticle statistics has recently attracted attention on purely theoretical grounds but also because of its potential applications in topologically protected quantum computing. This thesis focuses on the quasiparticles using three different approaches. The first is an effective Chern-Simons theory description, where the noncommutativity imposed on the classical space variables captures the incompressibility. We propose a construction of the quasielectron and illustrate how many-body quantum effects are emulated by a classical noncommutative theory. The second approach involves a study of quantum Hall states on a torus where one of the periods is taken to be almost zero. Characteristic quantum Hall properties survive in this limit in which they become very simple to understand. We illustrate this by giving a simple counting argument for degeneracy 2n-1, pertinent to nonabelian statistics, in the presence of 2n quasiholes in the Moore-Read state and generalise this result to 2n-k quasiholes and k quasielectrons. In the third approach, we study the topological nature of the degeneracy 2n-1 by using a recently proposed analogy between the Moore-Read state and the two-dimensional spin-polarized p-wave BCS state. We study a version of this problem where one can use techniques developed in the context of high-Tc superconductors to turn the vortex background into an effective gauge field in a Dirac equation. Topological arguments in the form of index theory gives the degeneracy 2n-1 for 2n vortices.