Дисертації з теми "Weyle invariance"

In this thesis work I analyze higher spin field theories from a first quantized perspective, finding in particular new equations describing complex higher spin fields on Kaehler manifolds. They are studied by means of worldline path integrals and canonical quantization, in the framework of supersymmetric spinning particle theories, in order to investigate their quantum properties both in flat and curved backgrounds. For instance, by quantizing a spinning particle with one complex extended supersymmetry, I describe quantum massless (p,0)-forms and find a worldline representation for their effective action on a Kaehler background, as well as exact duality relations. Interesting results are found also in the definition of the functional integral for the so called O(N) spinning particles, that will allow to study real higher spins on curved spaces. In the second part, I study Weyl invariant field theories by using a particular mathematical framework known as tractor calculus, that enable to maintain at each step manifest Weyl covariance.

In this thesis work I analyze higher spin field theories from a first quantized perspective, finding in particular new equations describing complex higher spin fields on Kaehler manifolds. They are studied by means of worldline path integrals and canonical quantization, in the framework of supersymmetric spinning particle theories, in order to investigate their quantum properties both in flat and curved backgrounds. For instance, by quantizing a spinning particle with one complex extended supersymmetry, I describe quantum massless (p,0)-forms and find a worldline representation for their effective action on a Kaehler background, as well as exact duality relations. Interesting results are found also in the definition of the functional integral for the so called O(N) spinning particles, that will allow to study real higher spins on curved spaces. In the second part, I study Weyl invariant field theories by using a particular mathematical framework known as tractor calculus, that enable to maintain at each step manifest Weyl covariance.

Cette thèse comprend deux parties indépendantes. Dans la première partie, nous développons une approche fondée sur la théorie des formes de Jacobi dont l'indice est un réseau pour classifier les formes modulaires réflexives sur des réseaux de niveau arbitraire. Les formes modulaires réflexives ont des applications en géométrie algébrique, en algèbre de Lie et en arithmétique. La classification des formes modulaires réflexives est un problème ouvert et a été étudiée par Borcherds, Gritsenko, Nikulin, Scheithauer et Ma depuis 1998. Dans cette partie, nous établissons de nouvelles conditions nécessaires à l'existence d'une forme modulaire réflexive. Nous prouvons la non-existence de formes modulaires réflexives et de formes modulaires 2-réflexives sur des réseaux de grand rang. Nous donnons également une classification complète des formes modulaires 2-réflexives sur des réseaux contenant deux plans hyperboliques.La deuxième partie est consacrée à l’étude des formes de Jacobi de $W(E_8)$-invariantes. Ce type de formes de Jacobi a une signification dans les variétés de Frobenius, la théorie de Gromov-Witten et la théorie des cordes. En 1992, Wirthm\"{u}ller a prouvé que l’espace des formes de Jacobi pour tout système de racines irréductible excepté $E_8$ est une algèbre polynomiale. Très peu de choses sont connues dans le cas de $E_8$. Dans cette partie, nous montrons que l'anneau bigradué des formes de Jacobi $W(E_8)$-invariantes n'est pas une algèbre polynomiale et prouvons que chacune de ces formes de Jacobi peut être exprimée uniquement sous la forme d'un polynôme en neuf formes de Jacobi algébriquement indépendantes introduites par Sakai avec des coefficients méromorphes $\SL_2(\ZZ)$-modulaires. Ce dernier résultat implique que, à indice fixé, l’espace des formes de Jacobi $W(E_8)$-invariantes est un module libre sur l’anneau des formes $\SL_2(\ZZ)$-modulaires et que le nombre de générateurs peut être calculé via une série génératrice. Nous déterminons et construisons tous les générateurs pour des indices petits. Ces résultats étendent un théorème de type de Chevalley au cas du réseau $E_8$
This thesis consists of two independent parts. In the first part we develop an approach based on the theory of Jacobi forms of lattice index to classify reflective modular forms on lattices of arbitrary level. Reflective modular forms have applications in algebraic geometry, Lie algebra and arithmetic. The classification of reflective modular forms is an open problem and has been investigated by Borcherds, Gritsenko, Nikulin, Scheithauer and Ma since 1998. In this part, we establish new necessary conditions for the existence of a reflective modular form. We prove non-existence of reflective modular forms and 2-reflective modular forms on lattices of large rank. We also give a complete classification of 2-reflective modular forms on lattices containing two hyperbolic planes. The second part is devoted to the study of Weyl invariant $E_8$ Jacobi forms. This type of Jacobi forms has significance in Frobenius manifolds, Gromov--Witten theory and string theory. In 1992, Wirthm\"{u}ller proved that the space of Jacobi forms for any irreducible root system not of type $E_8$ is a polynomial algebra. But very little has been known about the case of $E_8$. In this paper we show that the bigraded ring of Weyl invariant $E_8$ Jacobi forms is not a polynomial algebra and prove that every such Jacobi form can be expressed uniquely as a polynomial in nine algebraically independent Jacobi forms introduced by Sakai with coefficients which are meromorphic $\SL_2(\ZZ)$ modular forms. The latter result implies that the space of Weyl invariant $E_8$ Jacobi forms of fixed index is a free module over the ring of $\SL_2(\ZZ)$ modular forms and that the number of generators can be calculated by a generating series. We determine and construct all generators of small index. These results give a proper extension of the Chevalley type theorem to the case of $E_8$

In this thesis we present a novel scheme for calculating the bosonic string partition function on certain curved backgrounds related to Anti-de Sitter [AdS] space. We take the concept of a large expansion from nonlinear sigma models in particle physics and apply it to the bosonic string theory sigma model, where the analogous large dimensionless parameter is the dimension of the target space, D. We then perform a perturbative expansion in negative powers of D, rather than in positive powers of α/ι(^2)(the conventional expansion parameter).As a specific example of a curved geometry of interest, we focus on an example of the metric proposed by Polyakov [1] to describe the dynamics of the Wilson loop of pure SU(N) Yang-Mills theory, namely AdS space. Using heat kernel methods, we find that within the large-D scheme one can obtain different conditions for Weyl invariance than those found in [2]. This is because our scheme is valid for backgrounds where a is no longer small. In particular, we find that it is possible to have a dilaton that depends on the holographic coordinate only, provided one allows mixing of the ghost and matter sectors of the worldsheet theory. This field preserves Poincare invariance in the gauge theory, unlike the conventional dilaton. We also compute a simple string amplitude by constructing certain vertex operators for a scalar field in AdS, and discuss the consequences for the string spectrum.

Nous entendons confronter pratique des mathématiques et réflexions sur les mathématiques dans l'œuvre de Weyl. Nous étudierons : (a) ses monographies en analyse complexe, en relativité générale et en mécanique quantique, (b) les articles en lien avec ces ouvrages, (c) certains de ses cours, (d) sa correspondance avec divers scientifiques, principalement A. Einstein, E. Cartan, J. von Neumann. Nous voulons savoir si les théories mathématiques qu'il investit conditionnent ses positions sur les fondements des mathématiques. Inversement, nous montrerons que les philosophies auxquelles il se réfère – essentiellement le criticisme kantien, l'idéalisme fichtéen et la phénoménologie de Husserl – conditionnent ses recherches. Tout d'abord, nous reviendrons sur Die Idee der Riemannschen Fläche (première éd. 1913). Nous montrerons qu'il opte alors pour un formalisme mitigé. Il se revendique de deux traditions incarnées par Klein et par Hilbert. Ensuite, nous étudierons les éditions successives de Raum, Zeit, Materie (1918-1923). Nous aborderons le projet d'une géométrie purement infinitésimale qui permet à Weyl de proposer une théorie unifiée des champs, cette dernière étant réfutée par Einstein, Pauli, Reichenbach, Hilbert and Eddington. Nous décrirons aussi la construction et la résolution de son « problème de l'espace » (1921-1923). Nous indiquerons comment la référence aux philosophies de Fichte et de Husserl permet d'éclairer ces deux projets. Enfin, nous commenterons l'article de Weyl sur les groupes de Lie (1925-1926) ainsi que son ouvrage Gruppentheorie und Quantenmechanik (1928, 1931). Son article sur les groupes de Lie manifeste la voie moyenne entre formalisme et intuitionnisme qu'il adopte en 1924. Son ouvrage en mécanique quantique incarne quant à lui un « tournant empirique » dans son épistémologie qu'il conviendra de comparer \`a l'« empirisme logique »
Our purpose consists in comparing Weyl's mathematical practice with his philosophical reflections on mathematics. We will study (a) his monographs on complex analysis, general relativity and quantum mechanics, (b) the articles which are linked to these books, (c) some of his lecture courses, (d) his correspondence with different scientists, mainly A. Einstein, E. Cartan, J. von Neumann. We will show that his mathematical research has a strong influence on the different stands he successively takes regarding the foundations of mathematics. Conversely, we will show that the philosophical systems he refers to (mainly kantian criticism, fichtean idealism and husserlian phenomenology) have a real impact on his investigations in mathematics. We will first analyse Die Idee der Riemannschen Fläche (first edition 1913). In this book, Weyl seems to take up a formalist point of view, but this is partly true. In fact, he is influenced by two traditions respectively embodied by Hilbert and Klein. Then, we will study the successive editions of Raum, Zeit, Materie (1918-1923). We will describe Weyl's project of a “purely infinitesimal geometry”. Thanks to this geometrical framework, he builds a unified fields theory, which will be disproved by Einstein, Pauli, Reichenbach, Hilbert and Eddington. During this short period, Weyl also constructs and solves the so-called space problem (1921-1923). Weyl's references to Fichte and Husserl have a significant impact on these two projects. Finally, we will comment Weyl's main article on Lie groups (1925-1926) and his monograph on quantum mechanics, i.e. Gruppentheorie und Quantenmechanik (1rst ed. 1928, 2nd ed. 1931). Weyl's article on Lie groups is in accordance with his compromise between intuitionism and formalism (1924). On the other hand, Weyl's book on quantum mechanics encapsulates an “empirical turn” in his epistemology, which will be compared with the so-called empirical logicism

Thesis (Ph. D.)--University of Oregon, 2007.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 107-108). Also available for download via the World Wide Web; free to University of Oregon users.

In this thesis we study the trace anomaly of a Weyl fermion in an abelian gauge background. We first introduce the topic of anomaly in Quantum Fields Theory and provide case studies of a global and a gauge anomaly. Then, we review the lagrangians of the Weyl fermion and Dirac fermion, the models that are the focus of our chiral and trace anomaly computations. Since we evaluate the anomalies using Pauli-Villars (PV) regularization, we present different PV masses and discuss the classical symmetries they break. We identify the differential operators that enter our regularization schemes and we review the method that we use to evaluate anomalies: we read them from the path integral à la Fujikawa and compute them with heat kernel formulas. Then, we evaluate the chiral and trace anomaly of the models we are interested in. The chiral anomaly is well studied in the literature and we reproduce the standard result. The trace anomaly is our original result and, although the presence of the chiral anomaly implies a breakdown of gauge invariance, we find that the trace anomaly can be cast in a gauge invariant form. The issue is analogous to the one recently discussed in the literature about a conjectured contribution of an odd-parity term to the trace anomaly of a Weyl fermion in curved backgrounds. With an abelian gauge background, this odd-parity term would be a Chern-Pontryagin density, that does not appear in our final results.

Nel presente elaborato studiamo un modello fermionico libero ed invariante di scala con derivate di ordine elevato. In particolare, controlliamo che la simmetria di scala sia estendibile all'intero gruppo conforme. Essendoci derivate di ordine più alto il modello non è unitario, ma costituisce un nuovo esempio di teoria conforme libera. Nelle prime sezioni riguardiamo la teoria generale del bosone libero, partendo dapprima con modelli semplici con derivate di ordine basso, per poi estenderci a dimensioni arbitrarie e derivate più alte. In questo modo illustriamo la tecnica che ci permette di ottenere un modello conforme da un modello invariante di scala, attraverso l'accoppiamento con la gravità e richiedendo l'ulteriore invarianza di Weyl. Se questo è possibile, il modello originale ammette certamente l'intera simmetria conforme, che emerge come generata dai vettori di Killing conformi. Nel modello scalare l'accoppiamento con la gravità necessita di nuovi termini nell'azione, indispensabili anche la teoria sia appunto invariante di Weyl. La costruzione di questi nuovi termini viene ripetuta per un particolare modello fermionico, con azione contenente l'operatore di Dirac al cubo, per il quale dimostriamo l'invarianza conforme. Tale modello descrive equazioni del moto con derivate al terzo ordine. Dal momento che l'invarianza di Weyl garantisce anche l'invarianza conforme, ci si aspetta che il tensore energia-impulso corrispondente sia a traccia nulla. Per ogni modello introdotto controlliamo sistematicamente che tale condizione sia verifiata, ed in particolar modo per il caso della teoria fermionica con operator di Dirac cubico, che rappresenta il contributo originale di questa tesi.

Soit G un groupe algébrique réductif connexe de centre connexe défini sur un corps fini de caractéristique p>0. On munit cette structure d'un endomorphisme de Frobenius F et l'on note G^F l'ensemble des points de G fixes pour l'action de F : G^F est un groupe fini. On suppose que la caractéristique p est bonne pour G.

On définit alors une application Phi_G de l'ensemble des classes de conjugaison spéciales de G^* dans l'ensemble des classes unipotentes de G. Cette application décrit le support unipotent des différentes classes de faisceaux-caractères définis sur G.

Parallèlement à cela, via la correspondance de Springer, on définit différents invariants, dont les d-invariants, pour les caractères d'un groupe de Weyl W. Nous avons étudié le lien entre l'induction de caractères spéciaux de certains sous groupes de W et les d-invariants. A l'aide de ceci, on démontre que Phi_G, restreinte à certaines classes spéciales particulières de G^* est surjective. On a montré que la stabilité vis-à-vis du Frobenius pouvait être introduite dans ce résultat.

On en déduit deux résultats. Le premier est un lien étroit entre les restrictions aux éléments unipotents de faisceaux-caractères de certaines classes et différents systèmes locaux irréductibles et G-équivariants sur les classes unipotentes de G.

Le second est une preuve d'une conjecture de Kawanaka sur les caractères de Gelfand-Graev généralisés de G : ils forment une base du Z-module des caractères virtuels de G^F à support unipotent.

Classical partial transport barriers govern both classical and quantum dynamics of generic Hamiltonian systems. Chaotic eigenstates of quantum systems are known to localize on either side of a partial barrier if the flux connecting the two sides is not resolved by means of Heisenberg's uncertainty. Surprisingly, in open systems, in which orbits can escape, chaotic resonance states exhibit such a localization even if the flux across the partial barrier is quantum mechanically resolved. We explain this using the concept of conditionally invariant measures by introducing a new quantum mechanically relevant class of such fractal measures. We numerically find quantum-to-classical correspondence for localization transitions depending on the openness of the system and on the decay rate of resonance states. Moreover, we show that the number of long-lived chaotic resonance states that localize on one particular side of the partial barrier is described by an individual fractal Weyl law. For a generic phase space, this implies a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of phase space.

Nodal semimetals host either degenerate points (Dirac/Weyl points) or lines whose band topology in Brillouin zone can be classified either as trivial (normal nodal semimetals) or non trivial (topological nodal semimetals). This thesis investigates the electronic structure of two different categories of topological nodal semimetals probed by angleresolved photoemission spectroscopy (ARPES): The first material is Indium Bismuth (InBi). InBi is a semimetal with simple tetragonal structure with P4/nmm space group. This space group is predicted to host protected nodal lines along the perpendicular momentum direction at the high symmetry lines of the Brillouin zone boundary even under strong spin-orbit coupling (SOC) situation. As a semimetal with two heavy elements, InBi is a suitable candidate to test the prediction. The investigation by ARPES demonstrates not only that InBi hosts the nodal line in the presence of strong SOC, it also shows the signature of type-II Dirac crossing along the perpendicular momentum direction from the center of Brillouin zone. However, as the nodal line observed is trivial in nature, there is no exotic drumhead surface states observed in this material. This finding demonstrates that Dirac crossings can be protected in a non-symmorphic space group. The second material is NbIrTe₄ which is a semimetal that breaks inversion symmetry predicted to host only four Weyl points. This simplest configuration is confirmed by the measurement from the top and bottom surface of NbIrTe₄ showing only a pair of Fermi arcs each. Furthermore, it is found that the Fermi arc connectivity on the bottom surface experiences re-wiring as it evolves from Weyl points energy to the ARPES Fermi energy level. This change is attributed to the hybridisation between the surface and the bulk states as their projection lie within the vicinity of each other. The finding in this work demonstrates that although Fermi arcs are guaranteed in Weyl semimetals, their shape and connectivity are not protected and may be altered accordingly.

Topological states of matter are a novel family of phases that elude the conventional Landau paradigm of phase transitions. Topological phases are characterized by global topological invariants which are typically reflected in the quantization of physical observables. Moreover, their characteristic bulk-boundary correspondence often gives rise to robust surface modes with exceptional features, such as dissipationless charge transport or non-Abelian statistics. In this way, the study of topological states of matter not only broadens our knowledge of matter but could potentially lead to a whole new range of technologies and applications. In this light, it is of great interest to find novel topological phases and to study their unique properties. In this work, novel manifestations of topological states of matter are studied as they arise when materials are subject to additional symmetries. It is demonstrated how symmetries can profoundly enrich the topology of a system. More specifically, it is shown how symmetries lead to additional nontrivial states in systems which are already topological, drive trivial systems into a topological phase, lead to the quantization of formerly non-quantized observables, and give rise to novel manifestations of topological surface states. In doing so, this work concentrates on weakly interacting systems that can theoretically be described in a single-particle picture. In particular, insulating and semi-metallic topological phases in one, two, and three dimensions are investigated theoretically using single-particle techniques.

博士
國立交通大學
物理研究所
105
We construct a model with Weyl-invariance and massive gravity, study its eld equations, and nd some analytic solutions in the Bianchi type I space. In this model, massive gravity provides an eective cosmological constant, while Weyl-invariance provides a cold dark matter. After we add a quartic potential of the Weyl gauge vector into the Lagrangian density, the Weyl-invariant matter will be extreme hot initially although cold today, which acts like a mixture of cold matter and hot matter. Furthermore, a free in aton is added to control an anisotropy in equations of motion so that our solution can contain an isotropic background.

Classical partial transport barriers govern both classical and quantum dynamics of generic Hamiltonian systems. Chaotic eigenstates of quantum systems are known to localize on either side of a partial barrier if the flux connecting the two sides is not resolved by means of Heisenberg's uncertainty. Surprisingly, in open systems, in which orbits can escape, chaotic resonance states exhibit such a localization even if the flux across the partial barrier is quantum mechanically resolved. We explain this using the concept of conditionally invariant measures by introducing a new quantum mechanically relevant class of such fractal measures. We numerically find quantum-to-classical correspondence for localization transitions depending on the openness of the system and on the decay rate of resonance states. Moreover, we show that the number of long-lived chaotic resonance states that localize on one particular side of the partial barrier is described by an individual fractal Weyl law. For a generic phase space, this implies a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of phase space.

Topological states of matter are a novel family of phases that elude the conventional Landau paradigm of phase transitions. Topological phases are characterized by global topological invariants which are typically reflected in the quantization of physical observables. Moreover, their characteristic bulk-boundary correspondence often gives rise to robust surface modes with exceptional features, such as dissipationless charge transport or non-Abelian statistics. In this way, the study of topological states of matter not only broadens our knowledge of matter but could potentially lead to a whole new range of technologies and applications. In this light, it is of great interest to find novel topological phases and to study their unique properties. In this work, novel manifestations of topological states of matter are studied as they arise when materials are subject to additional symmetries. It is demonstrated how symmetries can profoundly enrich the topology of a system. More specifically, it is shown how symmetries lead to additional nontrivial states in systems which are already topological, drive trivial systems into a topological phase, lead to the quantization of formerly non-quantized observables, and give rise to novel manifestations of topological surface states. In doing so, this work concentrates on weakly interacting systems that can theoretically be described in a single-particle picture. In particular, insulating and semi-metallic topological phases in one, two, and three dimensions are investigated theoretically using single-particle techniques.