Дисертації з теми "Gauge theory on noncommutative spaces"
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1
Bizi, Nadir. "Semi-riemannian noncommutative geometry, gauge theory, and the standard model of particle physics." Thesis, Sorbonne université, 2018. http://www.theses.fr/2018SORUS413/document.
Повний текст джерелаАнотація:
Dans cette thèse, nous nous intéressons à la géométrie non-commutative - aux triplets spectraux en particulier - comme moyen d'unifier gravitation et modèle standard de la physique des particules. Des triplets spectraux permettant une telle unification on déjà été construits dans le cas des variétés riemanniennes. Il s'agit donc ici de généraliser au cas des variétés semi-riemanniennes, et d'appliquer ensuite au cas lorentzien, qui est d'une importance particulière en physique. C'est ce que nous faisons dans la première partie de la thèse, ou le passage du cas riemannien au cas semi-riemannien nous oblige à nous intéresser à des espaces vectoriels de signatures indéfinies (et non définies positives), dits espaces de Krein. Ceci est une conséquence de notre étude des algèbres de Clifford indéfinies et des structures Spin sur variétés semi-riemanniennes. Nous généralisons ensuite les triplets spectraux en triplets dits indéfinis en conséquence de cela. Dans la deuxième partie de la thèse, nous appliquons le formalisme des formes différentielles non-commutatives à nos triplets indéfinis pour formuler des théories de jauge non-commutatives sur espace-temps lorentzien. Nous montrons ensuite comment obtenir le modèle standardThe subject of this thesis is noncommutative geometry - more specifically spectral triples - and how it can be used to unify General Relativity with the Standard Model of particle physics. This unification has already been achieved with spectral triples for Riemannian manifolds. The main concern of this thesis is to generalize this construction to semi-Riemannian manifolds generally, and Lorentzian manifolds in particular. The first half of this thesis will thus be dedicated to the transition from Riemannian to semi-Riemannian manifolds. This entails a study of Clifford algebras for indefinite vector spaces and Spin structures on semi-Riemannian manifolds. An important consequence of this is the introduction of complex vector spaces of indefinite signature. These are the so-called Krein spaces, which will enable us to generalize spectral triples to indefinite spectral triples. In the second half of this thesis, we will apply the formalism of noncommutative differential forms to indefinite spectral triples to construct noncommutative gauge theories on Lorentzian spacetimes. We will then demonstrate how to recover the Standard Model
2
Meyer, Frank. "Gauge-Field Theories and Gravity on Noncommutative Spaces." Diss., lmu, 2006. http://nbn-resolving.de/urn:nbn:de:bvb:19-53357.
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Behr, Wolfgang. "Noncommutative Gauge Theory beyond the Canonical Case." Diss., lmu, 2005. http://nbn-resolving.de/urn:nbn:de:bvb:19-44251.
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Möller, Lutz. "Noncommutative gauge theory and the kappa-deformed spacetime." Diss., lmu, 2004. http://nbn-resolving.de/urn:nbn:de:bvb:19-23608.
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Mielke, Thomas Martin. "Mapping spaces, configuration spaces and gauge theory." Thesis, University of Warwick, 1995. http://wrap.warwick.ac.uk/109201/.
Повний текст джерелаАнотація:
The present thesis considers the space of connections modulo based gauge equivalence on a principal SU(2) bundle over a closed simply-connected smooth four-dimensional manifold M. Up to homotopy equivalence, this is the space of basepoint-preserving maps from M to BSU('2), the classifying space of SU(2). It depends only on the homotopy type of M which is characterized by the intersection form. The Z/pZ-homology of the mapping space for p a prime not equal to 3 is computed and given in terms of the data associated to the intersection form. For the prime 3, partial results are obtained. The main method is to consider a fibration associated to a CW decomposition of M and to show that the corresponding Eilenberg- Moore spectral sequence collapses. These results generalize from manifolds to spaces homotopy equivalent to a bouquet of 2-spheres with a single 4-cell attached. For the possible homotopy types the space of connections modulo gauge equivalence ran attain, a classification is obtained in the following sense. The homotopy type of this space is uniquely determined by the rank, type and signature modulo eight of the intersection form. On the other hand, the homotopy type determines the rank, type and signature modulo four of the intersection form. Both results together give a complete classification for the case of spin manifolds. The homotopy types of the spaces of connections modulo gauge equivalence over two spin manifolds agree if and only if the intersection forms are of the same rank. These results use a classification of unimodular bilinear forms over the ring Z/4Z. In a final part, a map is constructed from the labelled configuration spaces of points in the manifold to the mapping space. This map is shown to be asymptotically surjective in homology with Z/2Z-coefficients. For homology with general coefficients, classes are constructed which are not approximated by this map.
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Grosse, Harald, Thomas Krajewski, Raimar Wulkenhaar, and grosse@doppler thp univie ac at. "Renormalization of Noncommutative Yang-Mills Theories: A Simple Example." ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi914.ps.
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Sykora, Andreas. "The application of star-products to noncommutative geometry and gauge theory." Diss., lmu, 2004. http://nbn-resolving.de/urn:nbn:de:bvb:19-28937.
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Volkholz, Jan. "Nonperturbative studies of quantum field theories on noncommutative spaces." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät I, 2007. http://dx.doi.org/10.18452/15712.
Повний текст джерелаАнотація:
Diese Arbeit befasst sich mit Quantenfeldtheorien auf nicht-kommutativen Räumen. Solche Modelle treten im Zusammenhang mit der Stringtheorie und mit der Quantengravitation auf. Ihre nicht-störungstheoretische Behandlung ist üblicherweise schwierig. Hier untersuchen wir jedoch drei nicht-kommutative Quantenfeldtheorien nicht-perturbativ, indem wir die Wirkungsfunktionale in eine äquivalente Matrixformulierung übersetzen. In der Matrixdarstellung kann die jeweilige Theorie dann numerisch behandelt werden. Als erstes betrachten wir ein regularisiertes skalares Modell auf der nicht-kommutativen Ebene und untersuchen den Kontinuumslimes bei festgehaltener Nicht-Kommutativität. Dies wird auch als Doppelskalierungslimes bezeichnet. Insbesondere untersuchen wir das Verhalten der gestreiften Phase. Wir finden keinerlei Hinweise auf die Existenz dieser Phase im Doppelskalierungslimes. Im Anschluss daran betrachten wir eine vier-dimensionale U(1) Eichtheorie. Hierbei sind zwei der räumlichen Richtungen nicht-kommutativ. Wir untersuchen sowohl die Phasenstruktur als auch den Doppelskalierungslimes. Es stellt sich heraus, dass neben den Phasen starker und schwacher Kopplung eine weitere Phase existiert, die gebrochene Phase. Dann bestätigen wir die Existenz eines endlichen Doppelskalierungslimes, und damit die Renormierbarkeit der Theorie. Weiterhin untersuchen wir die Dispersionsrelation des Photons. In der Phase mit schwacher Kopplung stimmen unsere Ergebnisse mit störungstheoretischen Berechnungen überein, die eine Infrarot-Instabilität vorhersagen. Andererseits finden wir in der gebrochenen Phase die Dispersionsrelation, die einem masselosen Teilchen entspricht. Als dritte Theorie betrachten wir ein einfaches, in seiner Kontinuumsform supersymmetrisches Modell, welches auf der "Fuzzy Sphere" formuliert wird. Hier wechselwirken neutrale skalare Bosonen mit Majorana-Fermionen. Wir untersuchen die Phasenstruktur dieses Modells, wobei wir drei unterschiedliche Phasen finden.This work deals with three quantum field theories on spaces with noncommuting position operators. Noncommutative models occur in the study of string theories and quantum gravity. They usually elude treatment beyond the perturbative level. Due to the technique of dimensional reduction, however, we are able to investigate these theories nonperturbatively. This entails translating the action functionals into a matrix language, which is suitable for numerical simulations. First we explore a scalar model on a noncommutative plane. We investigate the continuum limit at fixed noncommutativity, which is known as the double scaling limit. Here we focus especially on the fate of the striped phase, a phase peculiar to the noncommutative version of the regularized scalar model. We find no evidence for its existence in the double scaling limit. Next we examine the U(1) gauge theory on a four-dimensional spacetime, where two spatial directions are noncommutative. We examine the phase structure and find a new phase with a spontaneously broken translation symmetry. In addition we demonstrate the existence of a finite double scaling limit which confirms the renormalizability of the theory. Furthermore we investigate the dispersion relation of the photon. In the weak coupling phase our results are consistent with an infrared instability predicted by perturbation theory. If the translational symmetry is broken, however, we find a dispersion relation corresponding to a massless particle. Finally, we investigate a supersymmetric theory on the fuzzy sphere, which features scalar neutral bosons and Majorana fermions. The supersymmetry is exact in the limit of infinitely large matrices. We investigate the phase structure of the model and find three distinct phases. Summarizing, we study noncommutative field theories beyond perturbation theory. Moreover, we simulate a supersymmetric theory on the fuzzy sphere, which might provide an alternative to attempted lattice formulations.
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Daemi, Aliakbar. "Symmetric Spaces and Knot Invariants from Gauge Theory." Thesis, Harvard University, 2014. http://dissertations.umi.com/gsas.harvard:11563.
Повний текст джерелаАнотація:
In this thesis, we set up a framework to define knot invariants for each choice of a symmetric space. In order to address this task, we start by defining appropriate notions of singular bundles and singular connections for a given symmetric space. We can associate a moduli space to any singular bundle defined over a compact 4-manifold with possibly non-empty boundary. We study these moduli spaces and show that they enjoy nice properties. For example, in the case of the symmetric space SU(n)/SO(n) the moduli space can be perturbed to an orientable manifold. Although this manifold is not necessarily compact, we introduce a comapctification of it. We then use this moduli space for singular bundles defined over 4-manifolds of the form YxR to define knot invariants. In another direction we mimic the construction of Donaldson invariants to define polynomial invariants for closed 4-manifolds equipped with smooth action of Z/2Z.Mathematics
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Sardis, Ioannis E. E. "Gluing maps, moduli spaces of connections and Donaldson invariants." Thesis, University of Warwick, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.263327.
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Ivankov, Nikolay [Verfasser]. "Unbounded Bivariant K-theory and an Approach to Noncommutative Fréchet Spaces / Nicolay Ivankov." Bonn : Universitäts- und Landesbibliothek Bonn, 2011. http://d-nb.info/1016150652/34.
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SURIANO, LUCA. "A Quantum distance for noncommutative measure spaces and an application to quantum field theory." Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 2010. http://hdl.handle.net/2108/1326.
Повний текст джерелаАнотація:
Nella prima parte della Tesi, presentiamo una versione "puntata" della topologia di Gromov-Hausdorff quantistica introdotta da Rieffel per spazi metrici quantistici compatti (cioè, spazi con unità d'ordine e una seminorma Lipschitz che metrizza la topologia *-debole sullo spazio dei funzionali positivi normalizzati). In particolare, proporremo una nozione di cono tangente quantistico di uno spazio metrico quantistico, come analogo noncommutativo del cono tangente di Gromov in un punto di uno spazio metrico ordinario, basata su una opportuna procedura di riscalamento della seminorma Lipschitz definita su uno spazio metrico quantistico. Tale costruzione estende effettivamente la corrispondente costruzione valida per spazi metrici ordinari. Infine, a titolo di esempio, descriveremo il cono tangente quantistico del toro noncommutativo bidimensionale.
Nella seconda parte, invece, introduciamo una particolare distanza quantistica sull'insieme delle algebre di von Neumann Lip-normate (cioè, dotate di una ulteriore norma che metrizza la topologia debole sui sottoinsiemi limitati nella norma C*). Come avviene per le distanze di tipo Gromov-Hausdorff, anche questa distanza G.-H. duale è una pseudo-distanza, e diviene una vera distanza solo sulle classi di equivalenza isometrica (rispetto alla norma Lip) delle algebre di von Neumann Lip-normate. Inoltre, dimostreremo un criterio di precompatteza per famiglie di algebre di vN Lip-normate (fortemente) uniformemente limitate, utilizzando la nozione di ultraprodotto (ristretto) di algebre di vN Lip-normate. Infine, nell'ambito del'approccio algebrico alla teoria quantistica dei campi, applicheremo tale costruzione allo studio del limite di scala (cioè, quando si fanno tendere a un punto le regioni dello spaziotempo su cui sono definiti gli osservabili della teoria) di una rete locale di algebre di vN (le algebre degli osservabili), confrontando l'approccio tramite ultraprodotti (e con la convergenza nella distanza quantistica) con la costruzione delle algebre "limite di scala" di Buchholz e Verch, mostrando che nel caso del campo libero bosonico le due procedure forniscono lo stesso risultato.In the first part of this dissertation, we study a pointed version of Rieffel's quantum Gromov-Hausdorff topology for compact quantum metric spaces (i.e, order-unit spaces with a Lipschitz-like seminorm inducing a distance on the space of positive normalized linear functionals which metrizes the w*-topology). In particular, in analogy with Gromov's notion of metric tangent cone at a point of an (abstract) proper metric space, we propose a similar construction for (compact) quantum metric spaces, based on a suitable procedure of rescaling the Lipschitz seminorm on a given quantum metric space. As a result, we get a quantum analogue of the Gromov tangent cone, which extends the classical (say, commutative) construction. As a case study, we apply this procedure to the two-dimensional noncommutative torus, and we obtain what we call a noncommutative solenoid. In the second part, we introduce a quantum distance on the set of dual Lip-von Neumann algebras (i.e., vN algebras with a dual Lip-norm which metrizes the w*-topology on bounded subset). As for the other G.-H. distances (classical or quantum), this dual quantum Gromov-Hausdorff (pseudo-)distance turns out to be a true distance on the (Lip-)isometry classes of Lip-vN algebras. We give also a precompactness criterion, relating the limit of a (strongly) uniform sequence of Lip-vN algebras to the (restricted) ultraproduct, over an ultrafilter, of the same sequence. As an application, we apply this construction to the study of the Buchholz-Verch scaling limit theory of a local net of (algebras of) observables in the algebraic quantum field theory framework, showing that the two approaches lead to the same result for the (real scalar) free field model.
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Nikoofard, Vahid. "Some considerations about field theories in commutative and noncommutative spaces." Universidade Federal de Juiz de Fora (UFJF), 2015. https://repositorio.ufjf.br/jspui/handle/ufjf/3057.
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CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Esta tese é composta por assuntos distintos entre si de teorias quânticas de campos onde alguns deles são descritos em espaços não-comutativos (NC). Em primeiro lugar, analisamos a dinâmica de uma partícula livre sobre uma 2-esfera e através da dinâmica das suas equações de movimento, obtivemos as perturbações NCs neste espaço de fase. Este modelo sugere uma origem para o Zitterbewegung do elétron. Depois disso, consideramos umaversãoNCdasegundaleideNewtonparaestemodelo, quefoiobtidocomestecenário geométrico aplicado a este modelo. Em seguida, discutimos um formalismo alternativo relacionado à não-comutatividade chamado DFR onde o parâmetro NC é considerado uma coordenada e demonstramos exatamente que ela tem obrigatoriamente um momento conjugado neste espaço de fase DFR, diferentemente do que alguns autores da atual literatura sobre DFR afirmam. No próximo assunto, usando o formalismo de solda que, em poucaspalavras,colocapartículascomquiralidadesopostasnomesmomultipleto,soldamos algumas versões NCs de modelos bem conhecidos como modelos de Schwinger quirais e modelos (anti) auto duais no espaço-tempo de Minkowski estendido. Em outro assunto estudado aqui, também construímos a versão NC do modelo de Jackiw-Pi com um grupo de calibre arbitrário e usamos o mapeamento bem conhecido de Seiberg-Witten para obter este modelo NC em termos de variáveis comutativos. Finalmente, utilizamos o formalismo de campos e anticampos (ou método BV) para construir a ação de Batalin-Vilkovisky (BV) do modelo Jackiw-Pi estendido e após o prEntendiocedimento de fixação de calibre chegamos a uma ação completa, pronta para quantização.
This thesis is composed of distinct aspects of quantum field theories where some of them are described in noncommutative (NC) spaces. Firstly, we have analyzed the dynamics of a free particle over a 2-sphere and through the dynamics of the equations of motion we have derived its NC perturbations in the phase-space. This model suggests an origin for Zitterbewegung feature of the electron. After that we have considered the NC version of Newton’s second law for this model, which was obtained with the geometricalscenarioappliedtothismodel. Thenwehavediscussedtheso-calledDoplicher– Fredenhagen–Roberts (DFR) alternative formalism concerning noncommutativity where the NC parameter has a coordinate role and we showed exactly that it has a conjugated momentum in the DFR phase-space, differently of what some authors of the current DFR-literature claims. In the next issue, using the soldering formalism which, in few words, put opposite chiral particles in the same multiplet, we have soldered some NC versions of well known models like the chiral Schwinger model and (anti)self dual models in the extended Minkowski spacetime. Changing the subject, we have constructed the NC spacetime version of Jackiw-Pi model with an arbitrary gauge group and we used the well known Seiberg-Witten map to obtain the NC model expressed in terms of commutative variables. Finally, we have used the field-antifield (or BV method) formalism to construct the Batalin-Vilkovisky (BV) action of the extended Jackiw-Pi model and after the gauge fixing procedure we have arrived at a quantized-ready action for this model.
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Sundin, Per. "Perturbative quantization of superstring theory in Anti de-Sitter spaces." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät I, 2011. http://dx.doi.org/10.18452/16320.
Повний текст джерелаАнотація:
Um das mikroskopische Verhalten der Gravitation zu beschreiben, ist es nötig, Quantenfeldtheorie und allgemeine Relativitätstheorie in einer vereinheitlichten Sprache zu formulieren. Eine Möglichkeit dieses Problem anzugehen ist es, die Punktteilchen der Quantenfeldtheorie durch fadenförmige Strings zu ersetzen. Allerdings erfordert die mathematische Konsistenz, dass sich die String in höherdimensionalen Raum-Zeiten bewegen; dies macht es jedoch sehr schwer, physikalische Konsequenzen zu extrahieren. Eine mögliche Lösung dieses Problems ist die Verwendung von String-Dualitäten, welche die Stringtheorie mittels holographischer Beschreibungen mit Eichtheorien auf dem Rand der Raum-Zeit verbinden. Die Dualitäten sind begründete Vermutungen, die die String- und Eichtheorie bei unterschiedlichen Werten der Kopplung gleichsetzen. Nicht zuletzt deshalb ist eine direkte Überprüfung der Dualitäten schwierig durchführbar. Hier hilft jedoch die sehr bemerkenswerte Tatsache, dass eine verborgene Eigenschaft der Vermutungen Integrabilität zu sein scheint, welche eine Extrapolation zwischen starker und schwacher Kopplung ermöglicht. Desweiteren kann das gesamte Spektrum, in gewissen vereinfachenden Grenzfällen, durch einen kompakten Satz von Bethe-Gleichungen ausgedrückt werden. Die Bethe-Gleichungen, welche aus Eichtheorierechnungen hergeleitet und geraten werden, bieten ein exzellentes Hilfsmittel, die vermuteten Dualitäten zu prüfen. Durch das Vergleichen der Vorhersagen der Gleichungen und expliziten Berechnungen in der Stringtheorie erhält man starke Argumente für die Gültigkeit der Vermutung und der angenommenen Integrabilität.In this thesis we study superstring theory on AdS$_5\, \times\,$S$^5$, AdS$_3\,\times\,$S$^3$ and $\adsfour$. A shared feature of each theory is that their corresponding symmetry algebras allows for a decomposition under a $\mathbb{Z}_4$ grading. The grading can be realized through an automorphism which allows for a convenient construction of the string Lagrangians directly in terms of graded components. We adopt a uniform light-cone gauge and expand in a near plane wave limit, or equivalently, an expansion in transverse string coordinates. With a main focus on the two critical string theories, we perform a perturbative quantization up to quartic order in the number of fields. Each string theory is, through holographic descriptions, conjectured to be dual to lower dimensional gauge theories. The conjectures imply that the conformal dimensions of single trace operators in gauge theory should be equal to the energy of string states. What is more, through the use of integrable methods, one can write down a set of Bethe equations whose solutions encode the full spectral problem. One main theme of this thesis is to match the predictions of these equations, written in a language suitable for the light-cone gauge we employ, against explicit string theory calculations. We do this for a large class of string states and the perfect agreement we find lends strong support for the validity of the conjectures.
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Johnston, Ann. "Markov Bases for Noncommutative Harmonic Analysis of Partially Ranked Data." Scholarship @ Claremont, 2011. http://scholarship.claremont.edu/hmc_theses/4.
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Given the result $v_0$ of a survey and a nested collection of summary statistics that could be used to describe that result, it is natural to ask which of these summary statistics best describe $v_0$. In 1998 Diaconis and Sturmfels presented an approach for determining the conditional significance of a higher order statistic, after sampling a space conditioned on the value of a lower order statistic. Their approach involves the computation of a Markov basis, followed by the use of a Markov process with stationary hypergeometric distribution to generate a sample.This technique for data analysis has become an accepted tool of algebraic statistics, particularly for the study of fully ranked data. In this thesis, we explore the extension of this technique for data analysis to the study of partially ranked data, focusing on data from surveys in which participants are asked to identify their top $k$ choices of $n$ items. Before we move on to our own data analysis, though, we present a thorough discussion of the Diaconis–Sturmfels algorithm and its use in data analysis. In this discussion, we attempt to collect together all of the background on Markov bases, Markov proceses, Gröbner bases, implicitization theory, and elimination theory, that is necessary for a full understanding of this approach to data analysis.
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Becker, Christian. "On the Riemannian geometry of Seiberg-Witten moduli spaces." Phd thesis, [S.l. : s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=975744771.
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Sundin, Per [Verfasser], Sergey [Akademischer Betreuer] Frolov, Stefan [Akademischer Betreuer] Theisen, and Peter [Akademischer Betreuer] Uwer. "Perturbative quantization of superstring theory in Anti de-Sitter spaces : integrability in gauge / string dualities / Per Sundin. Gutachter: Sergey Frolov ; Stefan Theisen ; Peter Uwer." Berlin : Humboldt Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät I, 2011. http://d-nb.info/1015081290/34.
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Kailasvuori, Janik. "Quasiparticles in the Quantum Hall Effect." Doctoral thesis, Stockholm : Department of Physics, Stockholm University, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-1250.
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De, Notariis Kevin. "Light hyperweak new gauge bosons from kinetic mixing in string models." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/19491/.
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String theory is at the moment our best candidate for a unified quantum theory of gravity, aiming to reconcile all the known (and unknown) interactions with gravity as well as provide insights for currently mysterious phenomena that the Standard Model and the modern Cosmology are not able to explain. In fact, it is believed that most of the problems associated to the Standard Model can indeed be resolved in string theory. Supersymmetry is supposed to be an elegant solution to the Hierarchy problem (even though more and more stringent bounds in this direction are being placed by the fact that we have been unable to experimentally find supersymmetry yet), while all the axions that compactifications bring into play can be used to resolve the strong CP problem as well as provide good candidates for Dark Matter. Inflationary models can also be constructed in string theory, providing, then, the most diffused solution to the Horizon problem. This work, in particular, is formulated in type IIB string theory compactified on an orientifolded Calabi-Yau three-fold in LARGE Volume Scenario (LVS) and focuses on the stabilisation of all the moduli in play compatible with the construction of a hidden gauge sector whose gauge boson kinetically mixes to the visible sector U(1), acquiring a mass via a completely stringy process resulting in the St{\"u}ckelberg mechanism. The "compatibility" regards the fact that certain experimental bounds should be respected combined with recent data extrapolated by Coherent Elastic Neutrino-Nucleus Scattering (CE$\nu$NS) events at the Spallation Neutron Source at Oak Ridge National Laboratory. We are going to see that in this context we will be able to fix all the moduli as well as present a brane and fluxes set-up reproducing the correct mass and coupling of the hidden gauge boson. We also get a TeV scale supersymmetry, since the gravitino in this model will be of order O(TeV), with an uplifted vacuum to reproduce a de Sitter universe as well.
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Cazassus, Guillem. "Homologie instanton-symplectique : somme connexe, chirurgie de Dehn, et applications induites par cobordismes." Thesis, Toulouse 3, 2016. http://www.theses.fr/2016TOU30043/document.
Повний текст джерелаАнотація:
L'homologie instanton-symplectique est un invariant associé à une variété de dimension trois close orientée, qui a été dé?ni par Manolescu et Woodward, et qui correspond conjecturalement à une version symplectique d'une homologie des instantons de Floer. Dans cette thèse nous étudions le comportement de cet invariant sous l'effet d'une somme connexe, d'une chirurgie de Dehn, et d'un cobordisme de dimension quatre. Nous établissons une formule de Künneth pour la somme connexe : si Y et Y' désignent deux variétés closes orientées de dimension trois, l'homologie instanton-symplectique associée à leur somme connexe est isomorphe à la somme directe du produit tensoriel de leurs groupes d'homologie instantonsymplectique respectifs, et de leur produit de torsion (après décalage des degrés). Nous définissons des versions tordues de cette homologie, et prouvons un analogue de la suite exacte de Floer, reliant les groupes associés à une triade de chirurgie. Cette suite exacte nous permet de calculer le rang des groupes associés à des familles de variétés, notamment les revêtements doubles ramifiés d'entrelacs quasi-alternés, des chirurgies entières de grande pente le long de certains noeuds, ainsi que certaines variétés obtenues par plombage de fibrés en disques au-dessus de sphères. Nous définissons enfin des invariants pour des cobordismes de dimension 4 prenant la forme d'applications entre groupes d'homologie instantonsymplectique des bords, et prouvons que deux des morphismes intervenant dans la suite exacte de chirurgie s'interprètent comme de telles applications, associées aux cobordismes d'attachement d'anses. Nous donnons également un critère d'annulation pour de telles applications associées à des éclatementsSymplectic instanton homology is an invariant for closed oriented three-manifolds, defined by Manolescu and Woodward, which conjecturally corresponds to a symplectic version of a variant of Floer's instanton homology. In this thesis we study the behaviour of this invariant under connected sum, Dehn surgery, and four-dimensional cobordisms. We prove a Künneth-type formula for the connected sum: let Y and Y' be two closed oriented three-manifolds, we show that the symplectic instanton homology of their connected sum is isomorphic to the direct sum of the tensor product of their symplectic instanton homology, and a shift of their torsion product. We define twisted versions of this homology, and then prove an analog of the Floer exact sequence, relating the invariants of a Dehn surgery triad. We use this exact sequence to compute the rank of the groups associated to branched double covers of quasi-alternating links, some plumbings of disc bundles over spheres, and some integral Dehn surgeries along certain knots. We then define invariants for four dimensional cobordisms as maps between the symplectic instanton homology of the two boundaries. We show that among the three morphisms in the surgery exact sequence, two are such maps, associated to the handle-attachment cobordisms. We also give a vanishing criteria for such maps associated to blow-ups
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Chandra, Nitin. "Topics In Noncommutative Gauge Theories And Deformed Relativistic Theories." Thesis, 2012. http://etd.iisc.ernet.in/handle/2005/2468.
Повний текст джерелаАнотація:
There is a growing consensus among physicists that the classical notion of spacetime has to be drastically revised in order to nd a consistent formulation of quantum mechanics and gravity. One such nontrivial attempt comprises of replacing functions of continuous spacetime coordinates with functions over noncommutative algebra. Dynamics on such noncommutative spacetimes (noncommutative theories) are of great interest for a variety of reasons among the physicists. Additionally arguments combining quantum uncertain-ties with classical gravity provide an alternative motivation for their study, and it is hoped that these theories can provide a self-consistent deformation of ordinary quantum field theories at small distances, yielding non-locality, or create a framework for finite truncation of quantum field theories while preserving symmetries.
In this thesis we study the gauge theories on noncommutative Moyal space. We nd new static solitons and instantons in terms of the so-called generalized Bose operators (GBO). GBOs are constructed to describe reducible representation of the oscillator algebra. They create/annihilate k-quanta, k being a positive integer. We start with giving an alternative description to the already found static magnetic flux tube solutions of the noncommutative gauge theories in terms of GBOs. The Nielsen-Olesen vortex solutions found in terms of these operators also reduce to the ones known in the literature. On the other hand, we nd a class of new instanton solutions which are unitarily inequivalent to the ones found from ADHM construction on noncommutative space. The charge of the instanton has a description in terms of the index representing the reducibility of the Fock space representation, i.e., k. After studying the static soliton solutions in noncommutative Minkowski space and the instanton solutions in noncommutative Euclidean space we go on to study the implications of the time-space noncommutativity in Minkowski space. To understand it properly we study the time-dependent transitions of a forced harmonic oscillator in noncommutative 1+1 dimensional spacetime. We also provide an interpretation of our results in the context of non-linear quantum optics. We then shift to the so-called DSR theories which are related to a different kind of noncommutative ( -Minkowski) space. DSR (Doubly/Deformed Special Relativity) aims to search for an alternate relativistic theory which keeps a length/energy scale (the Planck scale) and a velocity scale (the speed of light scale) invariant. We study thermodynamics of an ideal gas in such a scenario.
In first chapter we introduce the subjects of the noncommutative quantum theories and the DSR. Chapter 2 starts with describing the GBOs. They correspond to reducible representations of the harmonic oscillator algebra. We demonstrate their relevance in the construction of topologically non-trivial solutions in noncommutative gauge theories, focusing our attention to flux tubes, vortices, and instantons. Our method provides a simple new relation between the topological charge and the number of times the basic irreducible representation occurs in the reducible representation underlying the GBO. When used in conjunction with the noncommutative ADHM construction, we nd that these new instantons are in general not unitarily equivalent to the ones currently known in literature.
Chapter 3 studies the time dependent transitions of quantum forced harmonic oscillator (QFHO) in noncommutative R1;1 perturbatively to linear order in the noncommutativity . We show that the Poisson distribution gets modified, and that the vacuum state evolves into a \squeezed" state rather than a coherent state. The time evolutions of un-certainties in position and momentum in vacuum are also studied and imply interesting consequences for modelling nonlinear phenomena in quantum optics.
In chapter 4 we study thermodynamics of an ideal gas in Doubly Special Relativity. We obtain a series solution for the partition function and derive thermodynamic quantities. We observe that DSR thermodynamics is non-perturbative in the SR and massless limits. A stiffer equation of state is found. We conclude our results in the last chapter.
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Meyer, Frank [Verfasser]. "Gauge-field theories and gravity on noncommutative spaces / vorgelegt von Frank Meyer." 2006. http://d-nb.info/980449014/34.
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Behr, Wolfgang [Verfasser]. "Noncommutative gauge theory beyond the canonical case / vorgelegt von Wolfgang Behr." 2005. http://d-nb.info/97757055X/34.
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Marcillaud, de Goursac Axel [Verfasser]. "Noncommutative geometry, gauge theory and renormalization / vorgeelgt von Axel Marcillaud de Goursac." 2009. http://d-nb.info/1002113490/34.
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Sykora, Andreas [Verfasser]. "The application of *-products to noncommutative geometry and gauge theory / Andreas Sykora." 2004. http://d-nb.info/974061336/34.
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Möller, Lutz [Verfasser]. "Noncommutative gauge theory and κ-deformed [kappa-deformed] spacetime / vorgelegt von Lutz Möller". 2004. http://d-nb.info/972021833/34.
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Alekseev, Vadim. "Noncommutative manifolds and Seiberg-Witten-equations." Doctoral thesis, 2011. http://hdl.handle.net/11858/00-1735-0000-0006-B3ED-D.
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