Gauge/gravity duality is an important tool for learning about strongly coupled gauge theories. This thesis explores a set of examples of this duality in which the field theories have SU(2|4) supersymmetry and discrete sets of vacuum solutions. Specifically, we use the duality to propose Lagrangian definitions of type IIA Little String Theory on S⁵ as double-scaling limits of the Plane-Wave Matrix Model, maximally supersymmetric Yang-Mills theory on R x S² and N=4 supersymmetric Yang-Mills theory on R×S³/Zk. We find the supergravity solutions dual to generic vacua of the Plane-Wave Matrix Model and maximally supersymmetric Yang-Mills theory on R×S². We use the supergravity duals to calculate new instanton amplitudes for the Plane-Wave Matrix Model at strong coupling. Finally, we study a natural coarse-graining of the vacua, and find that the associated geometries are singular. We define an entropy functional that vanishes for regular geometries, is non-zero for singular geometries, and is maximized by the thermal state. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

In this thesis we study anti–self–duality equations in four and eight dimensions on manifolds of special Riemannian holonomy, among these hyper–Kähler, Quaternion–Kähler and Spin(7)–manifolds. We first consider the octonionic anti–self–duality equations on manifolds with holonomy Spin(7). We construct explicit solutions to their symmetry reductions, the non–abelian Seiberg–Witten equations, with gauge group SU(2). These solutions are singular for flat and Eguchi–Hanson backgrounds, however we find a solution on a co–homogeneity one hyper–Kähler metric with a domain wall, and the solution is regular away from the wall. We then turn to Quaternion–Kähler four–manifolds, which are locally determined by one scalar function subject to Przanowski’s equation. Using twistorial methods we construct a Lax Pair for Przanowski’s equation, confirming its integrability. The Lee form of a compatible local complex structure gives rise to a conformally invariant differential operator, special cases of the associated generalised Laplace operator are the conformal Laplacian and the linearised Przanowski operator. Using recursion relations we construct a contour integral formula for perturbations of Przanowski’s function. Finally, we construct an algorithm to retrieve Przanowski’s function from twistor data. At last, we investigate the relationship between anti–self–dual Einstein metrics with non–null symmetry in neutral signature and pseudo–, para– and null–Kähler metrics. We classify real–analytic anti–self–dual null–Kähler metrics with a Killing vector that are conformally Einstein. This allows us to formulate a neutral signature version of Tod’s result, showing that around non-singular points all real–analytic anti–self–dual Einstein metrics with symmetry are conformally pseudo– or para–Kähler.

Let K be a class of finite graphs and F = {F1, F2, ..., Fm} be a set of finite graphs. Then, K is said to have finite-duality if there exists a graph U in K such that for any graph G in K, G is homomorphic to U if and only if Fi is not homomorphic to G, for all i = 1, 2, ..., m. Nešetřil asked in [J. Nešetřil, Homonolo Combinatorics Workshop, Nova Louka, Czech Rep., (2003)] if non-trivial examples can be found. In this note, we answer this positively by showing classes containing arbitrary long anti-chains and yet having the finite-duality property.

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With the increase in the number of functions in the modern age, it has become quite a challenge for an architect to satisfy the requirements and still have an appropriate architectural expression for the materials and the underlying structure. Such challenges may be found both in nature and man made machines, giving rise to elements with multiple functions. This thesis attempts to explore such precedents offered in both architecture and other areas, and apply the ideas and principles in the design of a primary school. / Master of Architecture

Geometry. Pure Form. What are the limits of these concepts in architecture? To what extent can they be realized through constructive means? To the architect, these concepts are often the originating forces driving their work, but their nature is intangible, and can be best understood through reason. There exists then, a dichotomy that the architect is left to resolve: that which is solely of an intelligible nature and that which can exist within the physical limitations of our reality. While architectonic limitations are that of the physical, Architecture itself exists within both of these realms, the duality of the mind and of the body, and it is the charge of the architect to reconcile their inherent contradictions. The limitations of the mind and the body are incompatible at an absolute level, but there exists a degree of overlap within which architecture is found. Place is a catalyst that can trigger this dissonance. The intelligible exists in a placeless space, a space that was given a framework by René Descartes in his notion of extension, and exists as a free body. In contrast, the architectonic is contingent on placement and the forces of gravity. They are simultaneously contradictory and co-related. This thesis pursues the limits of this contradiction; its culmination more akin to an architectural follie than the original intent: a modest hiking shelter. / Master of Architecture / This thesis explores the duality and contradictions that arise when the realm of reason and the realm of that which is built coincide. The framework through which this exploration takes place is in the conception and design of a small structure in the mountains of Ticino, an Italian canton of southern Switzerland. It is a building whose purpose is pleasure, nothing more. The pursuit of ideal form in place is a catalyst for the series of contradictions that exist within not only this thesis, but the realm of architecture. Place and space. Mind and body. Intelligible and sensible.

In the context of (0, 2) gauged linear sigma models, we explore chains of perturbatively dual heterotic string compactifications. The notion of target space duality (TSD) originates in non-geometric phases and can be used to generate distinct GLSMs with shared geometric phases leading to apparently identical target space theories. To date, this duality has largely been studied at the level of counting states in the effective theories. We extend this analysis in several ways. First, we consider the correspondence including the effective potential and loci of enhanced symmetry in dual theories. By engineering vector bundles with non-trivial constraints arising from slope-stability (i.e. D-terms) and holomorphy (i.e. F-terms) the detailed structure of the vacuum space of the dual theories can be explored. Our results give new evidence that GLSM target space duality may provide important hints towards a more complete understanding of (0,2) string dualities. In addition, we consider TSD theories on elliptically fibered Calabi-Yau manifolds. In this context, each half of the "dual" heterotic theories must in turn have an F-theory dual. Moreover, the apparent relationship between two heterotic compactifications seen in (0,2) heterotic target space dual pairs should, in principle, induce some putative correspondence between the dual F-theory geometries. It has previously been conjectured in the literature that (0,2) target space duality might manifest in F-theory as multiple K3- fibrations of the same elliptically fibered Calabi-Yau manifold. In this work we investigate this conjecture in the context of both six-dimensional and four-dimensional effective theories and demonstrate that in general, (0,2) target space duality cannot be explained by such a simple phenomenon alone. Finally, we consider Landau-Ginzburg (LG) phases of TSD theories and explore their massless spectrum. In particular, we investigate TSD pairs involving geometric singularities. We study resolutions of these singularities and their relationship to the duality. / Doctor of Philosophy / In string theory, the space-time has “hidden” dimensions beyond the three spatial and one time-like dimensions macroscopically seen in our universe. We want to study how the geometries of this “internal space” can affect observable physics, and which geometries are compatible with our universe. Target space duality is a relationship that connects two or more geometries together. In target space duality, gauged linear sigma models (related to string theories) share a common locus (called a Landau-Ginzburg phase) in their parameter space, but are distinct theories. To date, this duality has largely been studied at the level of counting states in the effective theories. In this dissertation, target space duality is studied in more depth. First we extend the analysis to the effective potential and loci of enhanced symmetry. By engineering examples with non-trivial constraints, the detailed structure of the vacuum space of the dual theories can be explored. Our results give new evidence that target space duality may provide important hints towards a more complete understanding of string dualities. We also investigate the conjecture that target space duality might manifest in F-theory, a higher dimensional string theory, as multiple fibrations of the same manifold. We demonstrate that in general, target space duality cannot be explained by such a simple phenomenon alone. In our cases, we provide evidence that non-geometric data in F-theory must play at least some role in the induced F-theory correspondence, while leaving the full determination of the putative new F-theory duality to future work. Finally we explore the complete massless spectrum of the Landau-Ginzburg (LG) phase. Specifically, we calculate the full LG spectra for both sides, and compare the theory with the geometric phases. We find examples in which half of the target space dual geometry is singular. We have probed some approaches to resolving the singularity.

This thesis deals with string theory at finite temperature. String theory has attracted considerable attention in recent years because of its ability to unify the fundamental forces and particles in nature and provide a quantized description of gravity. However, many aspects of this theory remain mysterious, including its behavior at high temperature. One guiding principle for finite temperature string theory is the observation that a quantum theory at finite temperature can be recast as a zero-temperature theory in which a Euclidean time dimension is compactified on a circle. This temperature/radius correspondence holds in quantum mechanics as well as quantum field theory, and is normally assumed to hold in string theory as well. However it was shown recently that this correspondence fails for a class of string theories, called heterotic strings. This motivates a search for an alternate way to restore this correspondence, as well as a reevaluation of the thermodynamic behaviour of other classes of string theories, namely Type~II and Type~I. We find that contrary to the established wisdom, all ten dimensional string theories have a similar behaviour at finite temperature. This also leads us to the conclusion that the Heterotic and Type~I theory behave in a dual way at finite temperature.

No description available.

Made available in DSpace on 2014-06-11T19:26:15Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-03-02Bitstream added on 2014-06-13T19:33:40Z : No. of bitstreams: 1 gazon_ab_me_sjrp.pdf: 514461 bytes, checksum: a28c7f2428994893238d1ea3bcd3a9b1 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Baseado na teoria de cohomologia de grupos, Andrade e Fanti definiram um invariante algébrico, denotado por E(G;S;M), onde G é um grupo, S é uma família de subgrupos de G de índice finito e Mé um Z 2G-módulo. O objetivo deste trabalho é definir um invariante dual a E(G;S;M), que denotaremos por E (G;S;M), utilizando a homologia de grupos em vez da cohomologia. Com este invariante, obtemos diversos resultados e aplicações, principalmente nas teorias de grupos e pares de dualidade e de decomposição de grupos. Estes resultados fornecem uma maneira alternativa de obter aplicações e propriedades nestas teorias. E, para desenvolver este trabalho, estudamos as teorias de (co)homologia absoluta e relativa de grupos, bem como suas interpretações topológicas, e a teoria de grupos e pares de dualidade / Based on the cohomology theory of groups, Andrade and Fanti defined an algebraic invariant, denoted by E(G;S;M), where G is a group, S is a family of subgroups of G with nite index and M is a Z 2G-module. The purpose of this work is to define a dual invariant of E(G;S;M), which we denote by E (G;S;M), using the homology of groups instead of cohomology. With this invariant, we obtain many results and applications, especially in the duality and splitting theories of groups. These results provide an alternative way to get applications and properties in these theories. And to develop this work, we studied the absolute and relative (co)homology theories of groups, as well as their topological interpretations, and the theories of duality groups and pairs