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Index Theorems and SupersymmetryEriksson, Andreas January 2014 (has links)
The Atiyah-Singer index theorem, the Euler number, and the Hirzebruch signature are derived via the supersymmetric path integral. Concisely, the supersymmetric path integral is a combination of a bosonic and a femionic path integral. The action in the supersymmetric path integral includes here bosonic, fermionic- and isospin fields (backgroundfields), where the cross terms in the Lagrangian are nicely eliminated due to scaling of the fields and using techniques from spontaneous breaking of supersymmetry (that give rise to a mechanism, analogous to the Higgs-mechanism, but here regarding the so called superparticles instead). Thus, the supersymmetric path integral is a product of three pathintegrals over the three given fields, respectively, that can be evaluated exactly by means of Gaussian integrals. The closely related Witten index is a measure of the failure of spontaneous breaking of supersymmetry. In addition, the basic concepts of supersymmetry breaking are reviewed.
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Limites adiabatiques, fibrations holomorphes plates et théorème de R.R.G. / Adiabatic limits, holomorphic flat fibrations and R.R.G. theoremZhang, Yeping 21 September 2016 (has links)
Cette thèse est faite de deux parties. La première partie est un article rédigé conjointementavec Martin Puchol et Jialin Zhu. La deuxième partie est une série de résultats obtenus par moi-même liés au théorème de Riemann-Roch-Grothendieck pour les fibrés vectoriels plats. Dans la première partie, nous donnons une preuve analytique d'un résultat décrivant le comportement de la torsion analytique en théorie de de Rham lorsque la variété considérée est séparée en deux par une hypersurface. Plus précisément, nous donnons une formule liant la torsion analytique de la variété entière aux torsions analytiques associées aux variétés à bord avec des conditions limites relative ou absolue le long de l'hypersurface. Dans la deuxième partie de cette thèse, nous raffinons les résultats de Bismut-Lott pour les images directes des fibrés vectoriels plats au cas où le fibré vectoriel plat en question est lui-même la cohomologie holomorphe d'un fibré vectoriel le long d'une fibration plate à fibres complexes. Dans ce contexte, nous donnons une formule de Riemann-Roch-Grothendieck dans laquelle la classe de Todd du fibré tangent relatif apparaît explicitement. En remplaçant les classes de cohomologie par des formes explicites qui les représentent en théorie de Chern-Weil, nous généralisons ainsi des constructions de Bismut-Lott. / This thesis consists of two parts. The first part is an article written jointly with Martin Puchol and Jialin Zhu, the second part is a series of results obtained by myself in connection with the Riemann-Roch-Grothendieck theorem for flat vector bundles. In the first part, we give an analytic approach to the behavior of classical Ray-Singer analytic torsion in de Rham theory when a manifold is separated along a hypersurface. More precisely, we give a formula relating the analytic torsion of the full manifold, and the analytic torsion associated with relative or absolute boundary conditions along the hypersurface. In the second part of this thesis, we refine the results of Bismut-Lott on direct images of flat vector bundles to the case where the considered flat vector bundle is itself the fiberwise holomorphic cohomology of a vector bundle along a flat fibration by complex manifolds. In this context, we give a formula of Riemann-Roch-Grothendieck in which the Todd class of the relative holomorphic tangent bundle appears explicitly. By replacing cohomology classes by explicit differential forms in Chern-Weil theory, we extend the constructions of Bismut-Lott in this context.
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An L²‐index formula for monopoles with Dirac-type singularities / Dirac型特異点付きモノポールのL²‐指数定理Yoshino, Masaki 23 March 2020 (has links)
京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第22234号 / 理博第4548号 / 新制||理||1653(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 望月 拓郎, 教授 大槻 知忠, 教授 加藤 毅 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM
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Analogues of eta invariants for even dimensional manifoldsXie, Zhizhang 27 July 2011 (has links)
No description available.
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Teorema do índice em superfícies curvas de grafeno e fases de BerryLopes, Mirleide Dantas 02 December 2010 (has links)
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Previous issue date: 2010-12-02 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The graphene consists of a two-dimensional hexagonal structure formed only by
carbon atoms. It is a peculiar molecule, because in low energy its Hamiltonian
can be described by the Dirac operator and this gives it some unusual characteristics.
In this work the index theorem will be applied to graphene. This allows to
estimates the number of zero modes of geometric variants of graphene by means
of topological features of these molecules. Finally, it is observed that the index of
the Hamiltonian of this system can be described in terms of Berry phases. And so,
it is investigated the possibility of doing holonomic quantum computation using
the topology of such molecules. / O grafeno consiste em uma estrutura bidimensional hexagonal constituída apenas
por átomos de carbono. Trata-se de uma molécula bastante peculiar, pois em
baixas energias o seu hamiltoniano pode ser descrito pelo operador de Dirac e
isso lhe confere características incomuns. Neste trabalho o teorema do índice será
aplicado ao grafeno. Teorema que permite estimar o número de modos zero das
variantes geométricas do grafeno por meio das características topológicas destas
moléculas. Por fim, observa-se que o índice do hamiltoniano deste sistema pode ser
descrito em termos das fases de Berry. E dessa forma, investiga-se a possibilidade
de fazer computação quântica holonômica, a partir da topologia de tais moléculas.
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Secondary large-scale index theory and positive scalar curvatureZeidler, Rudolf 24 August 2016 (has links)
No description available.
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D-bar and Dirac Type Operators on Classical and Quantum DomainsMcBride, Matthew Scott 29 August 2012 (has links)
Indiana University-Purdue University Indianapolis (IUPUI) / I study d-bar and Dirac operators on classical and quantum domains subject to the APS boundary conditions, APS like boundary conditions, and other types of global boundary conditions. Moreover, the inverse or inverse modulo compact operators to these operators are computed. These inverses/parametrices are also shown to be bounded and are also shown to be compact, if possible. Also the index of some of the d-bar operators are computed when it doesn't have trivial index. Finally a certain type of limit statement can be said between the classical and quantum d-bar operators on specialized complex domains.
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The Index Bundle for Gap-Continuous Families, Morse-Type Index Theorems and Bifurcation / Das Indexbündel für Graphenstetige Familien, Morseartige Indexsätze und BifurkationWaterstraat, Nils 31 October 2011 (has links)
No description available.
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Supersymmetric Quantum Mechanics, Index Theorems and Equivariant CohomologyNguyen, Hans January 2018 (has links)
In this thesis, we investigate supersymmetric quantum mechanics (SUSYQM) and its relation to index theorems and equivariant cohomology. We define some basic constructions on super vector spaces in order to set the language for the rest of the thesis. The path integral in quantum mechanics is reviewed together with some related calculational methods and we give a path integral expression for the Witten index. Thereafter, we discuss the structure of SUSYQM in general. One shows that the Witten index can be taken to be the difference in dimension of the bosonic and fermionic zero energy eigenspaces. In the subsequent section, we derive index theorems. The models investigated are the supersymmetric non-linear sigma models with one or two supercharges. The former produces the index theorem for the spin-complex and the latter the Chern-Gauss-Bonnet Theorem. We then generalise to the case when a group action (by a compact connected Lie group) is included and want to consider the orbit space as the underlying space, in which case equivariant cohomology is introduced. In particular, the Weil and Cartan models are investigated and SUSYQM Lagrangians are derived using the obtained differentials. The goal was to relate this to gauge quantum mechanics, which was unfortunately not successful. However, what was shown was that the Euler characteristics of a closed oriented manifold and its homotopy quotient by U(1)n coincide.
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