京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第22582号 / 情博第719号 / 新制||情||123(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 矢ヶ崎 一幸, 教授 中村 佳正, 教授 梅野 健 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

The N= 4 Super Yang-Mills theory in four dimensions admits deformations and the exactly marginal deformations of its SU(3) R-symmetry sub-sector are known as Leigh-Strassler. Leigh-Strassler deformations break the N= 4 supersymmetry down to N= 1 while preserving conformal symmetry. With exactly marginal deformations only the F-terms are deformed thus Leigh-Strassler deformations only affect the superpotential in the Lagrangian. In this thesis we study the symmetry of the marginally deformed N= 4 SYM and demonstrate that its algebraic structure can be understood in terms of quasi-Hopf algebras. Quasi-Hopf algebras have a notion of twisting due to Drinfeld which makes them a natural mathematical language with which to treat deformations. Furthermore the deformation of the N= 4 SYM superpotential is automated by the definition of a suitable star product. / Thesis (PhD)--University of Pretoria, 2020. / NiTheP / Physics / PhD / Unrestricted

A dynamical system originally invented by Boltzmann has had recent developments. The system consists of a particle in a gravitational potential with an added centrifugal force, which is subject to reflection against a wall that separates the system from the gravitational center. The recent developments are with regards to the integrability of the system in the special case of vanishing centrifugal term. The purpose of this essay is to explicate these developments. / Ett dynamiskt system, ursprungligen uppfunnet av Boltzmann, har nyligen sett utvecklingar. Systemet består av en partikel i en gravitationspotential med en tillagd centrifugalkraft, som reflekterar vid kontakt med en vägg som skiljer partikeln och gravitationscentrumet. De nya utvecklingarna är inom systemets integrabilitet i det specialfall att centrifugalkraften är borttagen. Syftet med denna uppsats är att explicera dessa framtaganden.

Superconformal field theories (SCFTs) are an important class of quantum field theories. These SCFTs have been a significant component in exploring and comprehending the fundamental framework of quantum field theories. In the vast realm of quantum field theories, integrability plays a crucial role, providing powerful analytic tools that allow us to solve certain physical quantities exactly. In this thesis, we focus on the representation theory of the algebraic structure in six-dimensional (6D) SCFTs and investigate the intricate interplay between 6D SCFTs and integrability. To begin, we delve into the fundamental concepts of representation theory, establishing a solid foundation for our subsequent analysis. The discussion then will move on to all possible generators in the SCFTs, explaining how they are realized in terms of bosonic and fermionic oscillators. Finally, we investigate spin chains and their application in 6D SCFTs. We demonstrate that symmetry arguments derived from representation theory are not sufficient to establish the integrability of the spin chains in 6D SCFTs. This conclusion does not imply the absence of integrable systems within 6D SCFTs; rather, it suggest there are other potential methods available e.g., correlation functions, to explore the appearance of integrable systems in 6D SCFTs. / Superkonforma fältteorier (SCFTs) är en viktig klass av kvantfältteorier. Dessa SCFTs utgör en viktig komponent för att utforska och förstå det fundamentala ramverket för kvantfältteorin. Inom det stora riket av kvantfältteori spelar integrabilitet en avgörande roll, vilket tillhandahåller kraftfulla analytiska verktyg som gör att vi kan lösa vissa fysiska storheter exakt. I denna avhandling fokuserar vi på representationsteorin av den algebraiska strukturen i sexdimensionella (6D) SCFTs och undersöker det intrikat samspelet mellan 6D SCFTs och integrabilitet. Till att börja med kommer vi att fördjupa oss i de grundläggande begreppen inom representationsteori och skapa en gedigen grund för vår efterföljande analys.  Diskussionen kommer sedan att gå vidare till alla möjliga generatorer i SCFTs, och förklarar hur de realiseras i termer av bosoniska och fermioniska oscillatorer. Slutligen kommer spinnkedjor och dess tillämpningar i 6D SCFTs att undersökas. Vi kommer visa att symmetriargument som härleds från representationsteori inte är  tillräckliga för att fastställa integrerbarhet av spinnkedjor i 6D SCFTs. Denna slutsats innebär inte att integrerbara system inte existerar inom 6D SCFTs, utan föreslår att det finns andra potentiella metoder, till exempel korrelationsfunktioner, för att utforska existensen av integrerbara system i 6D SCFTs.

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GaugThe first half of the thesis concerns Abelian vortices and Yang-Mills theory. It is proved that the 5 types of vortices recently proposed by Manton are actually symmetry reductions of (anti-)self-dual Yang-Mills equations with suitable gauge groups and symmetry groups acting as isometries in a 4-manifold. As a consequence, the twistor integrability results of such vortices can be derived. It is presented a natural definition of their kinetic energy and thus the metric of the moduli space was calculated by the Samols' localisation method. Then, a modified version of the Abelian–Higgs model is proposed in such a way that spontaneous symmetry breaking and the Bogomolny argument still hold. The Painlevé test, when applied to its soliton equations, reveals a complete list of its integrable cases. The corresponding solutions are given in terms of third Painlevé transcendents and can be interpreted as original vortices on surfaces with conical singularity. The last two chapters present the following results in projective differential geometry and Hamiltonians of hydrodynamic-type systems. It is shown that the projective structures defined by the Painlevé equations are not metrisable unless either the corresponding equations admit first integrals quadratic in first derivatives or they define projectively flat structures. The corresponding first integrals can be derived from Killing vectors associated to the metrics that solve the metrisability problem. Secondly, it is given a complete set of necessary and sufficient conditions for an arbitrary affine connection in 2D to admit, locally, 0, 1, 2 or 3 Killing forms. These conditions are tensorial and simpler than the ones in previous literature. By defining suitable affine connections, it is shown that the problem of existence of Killing forms is equivalent to the conditions of the existence of Hamiltonian structures for hydrodynamic-type systems of two components.

India’s geographical location renders it with ample solar-energy potential ranging from 4-7 kWh/m2 daily and 2,300–3,200 sunshine hours annually. The diverse nature of human settlements (scattered low-rise to dense high-rise) in India is one of the unexplored avenues of harnessing solar energy through electricity generation using photovoltaic (PV) technology. Solar energy is a promising alternative that carries adequate potential to support the growing energy demands of India’s burgeoning population. A previous study estimates, by the year 2070, with 425 million households (of which utilizing only 20 %), about 90 TWh of electrical energy can be generated utilizing solar energy. PV is viable for onsite distributed (decentralized) power generation offering advantages of size and scale variability, modularity, relatively low maintenance and integration into buildings (no additional demand land). The application of solar PV technology as the building envelope viz., walls, façade, fenestration, roof and skylights is termed Building Integrated Photovoltaic (BIPV). Apart from generating electricity, PV has to also function as a building envelope, which makes BIPV systems unique. Even with a gradual rise in the number of BIPV installations across the world over the years, a common consensus on their evaluation has not yet been developed. Unlike PV in a ground mounted system, its application in buildings as an envelope has huge implications on both PV and building performance. The functions of PV as a building material translates well beyond electricity generation alone and would also have to look into various aspects like the thermal comfort, weather proofing, structural rigidity, natural lighting, thermal insulation, shading, noise protection safety and aesthetics. To integrate PV into a residential building successfully serving the purpose (given the low energy densities of PV and initial cost), would also mean considering factors like the buildings electricity requirement and economic viability. As many studies have revealed, 40% of electricity consumed in a building is utilized for maintaining indoor thermal comfort. Tropical regions, such as India, are generally characterized by high temperatures and humidity attributed to good sunlight, therefore, the externality considered for this study has been the impact of BIPV on the thermal comfort. Passive designs need to regulate the buildings solar exposure by integrating a combination of appropriate thermal massing, material selection, space orientation and natural ventilation. On the other hand, PV design primarily aims to maximize solar to generate maximum energy. The design requirements for climate-responsive building design may thus infringe upon those required for optimal PV performance. Regulating indoor thermal comfort in tropical regions poses a particular challenge under such conditions, as the indoor temperature is likely to be sensitive to external temperature variations. In addition, given current performance efficiencies for various PVs, high initial cost and space requirement, it is also crucial to ascertain PV’s ability to efficiently support buildings energy requirement. Thus, BIPV would require addressing, concurrently, design requirements for energy-efficient building performance, effective PV integration, and societal feasibility. A real time roof integrated BIPV system (5.25 kW) installed at the Center for Sustainable Technologies at the Indian Institute of Science, Bangalore has been studied for its PV and building thermal performance. The study aims at understanding a BIPV system (based on crystalline silicon) from the technical (climate-responsiveness and PV performance), social (energy requirement and energy efficiency) and economical (costs and benefits) grounds and identifies relevant factors to quantify performance of any BIPV system. A methodology for BIPV evaluation has been proposed (Integrability Methodology), especially for urban localities, which can also be adopted for various PV configurations, building typologies and climatic zones. In the process, a novel parameter (thermal comfort energy) to evaluate the thermal performance of naturally ventilated buildings combining climate-responsiveness and thermal comfort aspects has also been developed. An Integrability Index has also been devised, integrating various building performance factors, to evaluate and compare the performance of BIPV structures. The methodology has been applied to the 5.25 kW BIPV system and the index has been computed to be 0.17 (on a scale of 0 – 1). An insulated BIPV system (building applied photovoltaic system) has been found to be favorable for the climate of Bangalore than BIPV. BIPV systems have also been compared across three different climates (Bangalore, Shillong and Delhi) and given the consideration of the same system for comparison, the system in Delhi is predicted to have a higher Integrability than the other two systems. The current research work is a maiden effort, that aims at developing and testing a framework to evaluate BIPV systems comprising technical, social and economic factors.

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.

In this thesis, we study the FX option pricing problem and put forward a 4-factor hybrid stochastic-local volatility model. The model, which describes the dynamics of an exchange rate, its volatility and the domestic and foreign short rates, allows for a perfect calibration to European options and has a good hedging performance. Due to the high-dimensionality of the problem, we propose a Monte Carlo simulation scheme that combines the full truncation Euler scheme for the stochastic volatility component and the stochastic short rates with the log-Euler scheme for the exchange rate. We analyze exponential integrability properties of Euler discretizations for the square-root process driving the stochastic volatility and the short rates, properties which play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a large class of stochastic differential equations in finance, including the ones studied in this thesis. Hence, we prove the strong convergence of the exchange rate approximations and the convergence of Monte Carlo estimators for a number of vanilla and exotic options. Then, we calibrate the model to market data and discuss its fitness for pricing FX options. Next, due to the relatively slow convergence of the Monte Carlo method in the number of simulations, we examine a variance reduction technique obtained by mixing Monte Carlo and finite difference methods via conditioning. We consider a purely stochastic version of the model and price vanilla and exotic options by simulating the paths of the volatility and the short rates, and then evaluating the "inner" Black-Scholes-type expectation by means of a partial differential equation. We prove the convergence of numerical approximations and carry out a theoretical variance reduction analysis. Finally, we illustrate the efficiency of the method through a detailed quantitative assessment.

In this thesis we shall examine the role of measurability in the theory of Lebesgue Integration. This shall be done in the context of the real line where we define the notion of an integral of a bouuded real-valued function over a set of bounded outer measure without a prior assumption of measurability concerning the function and the domain of integration