Thèses sur le sujet « Dualitie »

La dinamica a basse energie di teorie quantistiche di campo è spesso caratterizzata da diversi interessanti fenomeni non perturbativi. Di questi, nella presente tesi studieremo dualità infrarosse e enhancement di simmetrie globali in tre e quattro dimensioni. Ci concentreremo su teorie supersimmetriche, per le quali varie tecniche computazionali esatte sono disponibili. Particolarmente importante è la localizzazione supersimmetrica, la quale permette di calcolare funzioni di partizione su vari spazi compatti. Queste risultano essere invarianti sotto il flusso del gruppo di rinormalizzazione e per questo sono estremamente utili per investigare dualità ed enhancement di simmetria. Inoltre, con l'ausilio delle funzioni di partizione supersimmetriche è possibile studiare efficacemente flussi tra diverse dimensioni spaziotemporali. Equipaggiati di queste tecniche, in questa tesi investigheremo diversi punti di vista sul programma di trovare e organizzare dualità e enhancement di simmetria in 3d e 4d. Gli approcci che useremo coinvolgeranno diversi concetti che compaiono nello studio di teorie supersimmetriche. Il primo riguarda la possibilità di fluire tra diverse dimensioni attraverso compattificazioni dello spaziotempo. In questo modo è possibile trovare nuove dualità e enhancement di simmetria, sia riducendo dimensionalmente a dimensioni inferiori sia promuovendo a dimensioni superiori alcuni esempi già noti. Un'altra possibilità è quella di compattificare una teoria in un maggior numero di dimensioni, come per esempio una teoria 6d su una superficie di Riemann in modo da ottenere una teoria 4d, e di utilizzare questa costruzione geometrica per predirre e sistematizzare dualità e enhancement di simmetria. L'ultimo ingrediente che giocherà un ruolo importante nella nostra analisi è la corrispondenza gauge/CFT, la quale mette in relazione le funzioni di partizione di certe teorie supersimmetriche e le funzioni di correlazione di alcune CFT. Questo tipo di corrispondenze può essere sfruttato per trovare nuovi risultati in teorie di gauge partendo da risultati noti dal lato di CFT. In questo lavoro di tesi discuteremo diversi esempi di applicazione di queste idee. In primo luogo presenteremo una relazione tra dualità 3d N = 2 e identità per correlatori in teoria libera di CFT in 2d, e spiegheremo come essa può essere utilizzata per promuovere risultati noti in 2d a nuovi aspetti di teorie 3d. Il secondo tema affrontato sarà la compattificazione di una particolare SCFT 6d N = (1,0), nota come teoria E-string di rango N, su superfici di Riemann con flussi in modo da ottenere teorie 4d N = 1. Questa costruzione ci permetterà di predirre dualità e enhancement di simmetria partendo dalle note proprietà della teoria 6d e da considerazioni geometriche. Infine, discuteremo un nuovo tipo di dualità 4d N = 1 che rappresenta un antenato in una dimensione superiore della più nota mirror symmetry in 3d.
The low energy dynamics of quantum field theories can be characterized by several interesting non-perturbative phenomena. Among these, in this thesis we study infra-red dualities and global symmetry enhancements in three and four dimensions. We focus on supersymmetric theories, for which various exact computational tools are at our disposal. Of particular relevance is supersymmetric localization, which allows us to compute partition functions on various compact manifolds. These turn out to be invariant under the renormalization group flow and are thus powerful to probe dualities and symmetry enhancements. Moreover, one can also effectively study flows across dimensions through these supersymmetric partition functions. Equipped with these tools, in this thesis we investigate different perspectives on the program of finding and organizing dualities and symmetry enhancements in 3d and 4d. The approaches that we will employ combine several different concepts that appear in the study of supersymmetric theories. This first one is that we can flow across dimensions via spacetime compactifications. In this way we can find new dualities and symmetry enhancements, by either dimensionally reducing to lower dimensions or uplifting to higher dimensions some that are already known. Another possibility is to compactify a higher dimensional theory, like a 6d theory on a Riemann surface so to get a 4d theory, and use this geometric construction to predict and systematize dualities and symmetry enhancements. The last important ingredient that will play a role in our analysis consists of correspondences, in particular gauge/CFT correspondences relating partition functions of certein supersymmetric theories to correlation functions of some CFTs. These kind of correspondences can also be exploited to find new results on the gauge theory side from known results on the CFT side. In this thesis we discuss various examples of applications of these ideas. We first present a relation between 3d N = 2 dualities and identities for 2d CFT free field correlators, and we explain how this can be used to uplift known results about 2d free fields to new aspects of 3d theories. The second topic is the compactification of a particular 6d N = (1, 0) SCFT, known as the rank-N E-string theory, on Riemann surfaces with fluxes so to get 4d N = 1 theories. This construction allows us to predict dualities and symmetry enhancements from known properties of the 6d SCFT and geometric considerations. Finally, we discuss a new type of duality for 4d N = 1 theories that represents a higher dimensional ancestor of the well-known 3d mirror symmetry.

Motivated by the AdS/CFT correspondence, new supersymmetric solutions to Type IIB and Type IIA supergravity are presented. These solutions contain $AdS_5$ or $AdS_4$ factors and are generated using T-duality symmetries of supergravity. The technique used to generate these solutions consists of performing a series of non-Abelian and Abelian T-dualities, sometimes with coordinate shifts in-between, to Freund-Rubin type seed backgrounds. An added bonus of the gauge fixing procedure inherent in non-Abelian T-Duality is the freedom to generate backgrounds with extra free parameters, some examples of which are presented. Aspects of the dual field theories of these new solutions are analyzed using holography techniques. The supersymmetry of these new backgrounds is also discussed. In addition to supergravity backgrounds with AdS, the study of generalized Calabi-Yau manifolds in the context of flux compactifications is briefly reviewed. The particular case of the resolved cone over $Y^{p,q}$ and its admission of generalized SU(3) structure solutions is examined. Contrary to geometries with $AdS$ factors, whose field theory duals are conformal field theories, these types of geometries can be phenomenologically interesting to study, as their gauge theory duals are minimally supersymmetric and confining, thus they could someday help aid our understanding of strongly-coupled QCD (Quantum Chromodynamics).

Cette étude consiste à examiner le thème du flâneur chez Baudelaire dans la perspective du dualisme non-dualiste du Son bouddhisme coréen. Comme le faisaient les maîtres du Son, il se méfie de la rationalité. Pour être libre de celle-ci, Baudelaire prend l'ivresse comme moyen, parce que l'ivresse rend possible toutes les réversibilités, y compris du « moi» et de l'Autrui. Cette expérience de l'ivresse correspond à celle de l'éveil Son bouddhique dans la mesure où elle constitue une sorte d'expérience de la vérité. Gardant cet éveil du vrai «moi» et du vrai «monde », notre homme décide de participer au monde ordinaire avec son imagination. À la fois gardant et perdant ses distances à l'égard de la foule, telle est l'activité du flâneur. Du fait que tout est réversible dans le monde baudelairien, la participation se termine par l'orientation vers l'observation, le solidaire vers le solitaire. De plus, les deux coexistent sans aucun conflit mais de façon dynamique: ce qui correspond au dualisme non-dualiste du Son bouddhisme.

The paradoxical nature of society leads to great dualities in the study of motion and space, creating conflicting relationships. The role of the architectural design is to facilitate these motions to become a stage for the spectacle in its immediate context. The dissertation presents an architectural proposal that addresses the notion of duality. By identifying the possibilities within a liminal context, an architectural narrative creates scenarios as a possible response. The author investigates the manifestation of multiple programmes, a leather workshop, bakery and bar within the South African urban context of Pretoria West. The architectural exploration aims to enrich public life in a liminal space on the periphery of the city. Challenges associated with the typology of duality are addressed through the integration with the immediate context to ensure its sustainability. The aim is to celebrate the edge condition and the spaces commonly overlooked.
Dissertation MArch(Prof)-University of Pretoria, 2012.
Architecture
MArch(Prof)
Unrestricted

L'une des théories les plus prometteuses qui vise à unifier la mécanique quantique et la relativité générale est actuellement la théorie des cordes. La recherche d'une formulation supersymétrique des cordes a conduit à cinq théories de supercordes cohérentes en dix dimensions qui ont ensuite été unifiées dans le cadre de la théorie M. L'aspect fondamental de cette unification est la découverte d'un réseau de dualités entre les cinq théories des supercordes et la supergravité à onze dimensions.Cette thèse aborde les dualités dans le contexte de la théorie F en huit dimensions. La théorie F est douze dimensionnelle et fournit une formulation non perturbative de la supergravité de type IIB avec 7-branes. La théorie des champs exceptionnels, quant à elle, fournit une formulation U-dual de la supergravité de type IIB. Nous nous concentrons donc sur les liens possibles entre ces deux formulations. La théorie F est également supposée être duale à la théorie des cordes hétérotique en 8 dimensions. La structure des groupes de jauge apparaît radicalement différemment dans ces deux formulations. Dans la théorie F, elle est interprétée comme un choix particulier de structure algébrique d'une surface K3 elliptique, tandis que dans le cadre de la corde hétérotique, elle est principalement déterminée par les lignes de Wilson. Bien qu'étudiée dans le contexte de la théorie des cordes de type IIB, l'identification entre les modules de la théorie F et de la théorie des cordes hétérotique n'est que peu connue.Dans la première partie de cette thèse, nous présentons les notions de base de la théorie des cordes, des compactifications, des branes et des dualités. Dans la seconde, nous montrons que la théorie des champs exceptionnels R+ X E₃(₃)en huit dimensions présente des aspects de la théorie F dans un cadre spécifique, et permet en particulier de décrire les monodromies des(p; q) 7-branes. Enfin, dans la troisième partie, nous étudions la dualité entre la compactification de la théorie F sur une surface K3 elliptique et la corde hétérotique sur un deux-tore. Nous présentons comment construire des surfaces K3 elliptiques via des polyèdres réflexifs qui peuvent être interprétés en termes de lignes de Wilson dans la théorie des cordes hétérotique duale
One of the most promising theories to unify quantum mechanics and general relativity is currently string theory. The search for a supersymmetric formulation of strings led to five consistent ten dimensional superstring theories which were later unified under the scope of M-theory. The fundamental aspect of this unification is the discovery of a web of dualities between the five superstring theories and eleven dimensional supergravity.This thesis addresses dualities in the context of F-theory in eight dimensions. F-theory is twelve dimensional and provides a non-perturbative formulation of type IIB supergravity with 7-branes. On the other hand exceptional field theory provides a U-dual formulation of type IIB supergravity and we therefore focus on the possible links between these two formulations. F-theory is also conjectured to be dual to the heterotic string in 8 dimensions. The gauge group appears radically differently in these two formulations. In F-theory it is interpreted as a particular algebraic structure of an elliptically fibered K3 surface, while on the heterotic string it is principally determined the Wilson lines. Although studied in the context of type IIB string theory, the explicit map between the moduli in F-theory and its heterotic dual are still quite unknown.In the first part of this thesis we present basic notions of string theory, compactifications, branes and dualities. In the second one, we show that R+ X E₃(₃) exceptional field theory in eight dimensions can incorporate aspects of F-theory in a specific setting, and in particular describes the monodromies of (p; q) 7-branes. Finally, in the third part we study the duality between F-theory compactified on an elliptic K3 surface and the heterotic string on a two-torus. We present how to construct elliptically fibered K3 surfaces via reflexive polytopes which can be understood in terms of Wilson lines in the dual heterotic string theory

In this work we prove dualities for Diers categories, Barr categories and small Barr-exact categories. The latter duality solves a problem of M. Makkai. Further, we prove a stronger version of the strong completeness theorem on $ kappa$-Barr-exact categories. Finally we prove that an accessibly embedded subcategory of a locally presentable category satisfies the solution-set condition if it is accessible. This improves work of J. Adamek and J. Rosicky on injectivity in locally presentable categories.

In (2), Brian Davey poses the question: "Which finite algebras admit a duality?" The answer to this question has eluded mathematicians for many years and a simple characterization of those algebras which are dualizable (in a reasonable way) seems an unlikely feat. I will consider the question "Which finite (operational) algebras admit a duality which may be expressed in a purely operational form?". This is a substantially less ambitious question, for there are many algebras which are dualizable but not operationally so. However, it does have a clear answer.

Non-abelian target space duality and its effect on low energy string backgrounds is investigated, with particular emphasis on Taub-NUT space. The dual solutions are analysed and the physical effects of the duality are discussed, along with the interrelationship between successive applications of target space duality. A full quantum mechanical analysis of non-abelian duality is then considered. Conditions for the transformation to be anomaly free are presented, along with the quantum corrections to the standard non-abelian duality formulae. A low energy analysis is also performed. The study of D-branes creates an interest in massive sigma models. The supersymmetric abelian target space duals of such models are constructed. From the dual models the correct form of the dual potentials may be inferred. Finally the effects of the duality on the solitons of the massive model are discussed. The general problem of branes in twelve dimensions is studied. A supersymmetric action principle is formulated, but in order to do this the notions of supersymmetric must generalised. The resulting (2 + 2)- brane lives in a new type of superspace with two timelike directions. It is then shown that the traditional spacetime strings and membranes may be obtained by dimensional reduction. Once one has adapted to the possibility that there are many different supersymmetry theories, it is natural to attempt a classification. Details of such a procedure are given, and the corresponding superspaces constructed. In addition, complete brane scans are given for branes in any spacetime dimension and signature, and the algebras for which they are possible. The (2+2)-brane requires there to be two time directions. The concrete properties which any supersymmetric theory with two times must have are thus discussed. The BPS states of the two-time algebras are investigated in an analysis similar to that used in supergravity. Consistency conditions are discussed and the intersection properties of the branes detailed.

We introduce and review some of the building blocks to what is known as the AdS/CFT correspondence. We then present, using the framework of the correspondence, two applications: one to QCD (Chapters 2 and 3) and one to superconductivity (Chapter 4). The work presented in these chapters was carried out by the author and collaborators during the author's graduate studies.

Dans ce travail, nous avons traité du genre « roman autobiographique » illustré par trois œuvres algériennes d'expression française: Jacinthe noire de Taos Amrouche, Le Fils du pauvre de Mouloud Feraoun et de L'amour la fantasia de Assia Djebar.Nouvellement défini par la poétique moderne, le roman autobiographique se démarque avanttout avec sa stratégie d'ambiguïté générique. Nous avons repéré cela systématiquement dans nos œuvres, notamment du point de vue structurel, il s'agit des éléments textuels qui fonctionnent comme des opérateurs génériques, tels que le paratexte, l'intertexte, mais aussi les mécanismes d'énonciations sur les axes référentiel et fictionnel.Partant du point de vue du contexte, ce sont d'autres ambiguïtés qui viennent alimenter la nature du genre que nous avons souhaité traiter sous le concept de dualité que le contexte social et individuelle fait apparaître. Ce concept nous l'avons par la suite décliné en concept du double qui affecte essentiellement la figure de l'auteur ainsi que ses personnages.Dépassées cette figure du double, c'est l'unité du Moi et de l'identité personnelle qui se trouvent remises en question. Avant de statuer sur différentes théories qui avaient tentées de mettreen œuvre des pistes de réception qui puissent valider herméneutiquement le genre dans ses divergentes acceptations.Mots clés : Roman autobiographique, fiction, réalité, ambiguïté, dualité, double, identité du Moi, amalgame, réception
In this piece of work, we have discussed on the ‘autobiographical novel’ as a literary genre exemplified by three Algerian works written in French – namelyTaos Amrouche’s Jacinthe noire de, Mouloud Feraoun’s Le Fils du pauvre and Assia Djebar’s L'amour la fantasia.First and foremost, the autobiographical novel – recently defined by modern poetic – marks itself off from the other genres with its strategy of generic ambiguity. This has been systematically noticed in these works, especially from a structural point of view where text components play the part of generic operators such as paratext, intertext, but also enunciating mechanisms on fictional and referential axes.Starting from the contextual standpoint, other ambiguities nurture the nature of the genre we have wished to tackle through the concept of duality that is unveiled by the social and individual context. This very concept has then been developed into the concept of the double. It affects mostly the figure of the author as well as their characters.Moving beyond the figure of the double amounts to questioning the very unity of the Self and personal identity before rendering a final verdict on various theories that tried to implement reception that may hermeneutically validate the genre in all its diverging acceptations.Keywords: autobiographical novel, fiction, reality, ambiguity, duality, double, self identity, amalgam, reception

We investigate various relationships between categories of functors. The major examples are given by extending some duality to a larger structure, such as an adjunction or a recollement of abelian categories. We prove a theorem which provides a method of constructing recollements which uses 0-th derived functors. We will show that the hypotheses of this theorem are very commonly satisfied by giving many examples. In our most important example we show that the well-known Auslander-Gruson-Jensen equivalence extends to a recollement. We show that two recollements, both arising from different characterisations of purity, are strongly related to each other via a commutative diagram. This provides a structural explanation for the equivalence between two functorial characterisations of purity for modules. We show that the Auslander-Reiten formulas are a consequence of this commutative diagram. We define and characterise the contravariant functors which arise from a pp-pair. When working over an artin algebra, this provides a contravariant analogue of the well-known relationship between pp-pairs and covariant functors. We show that some of these results can be generalised to studying contravariant functors on locally finitely presented categories whose category of finitely presented objects is a dualising variety.

This thesis consists in a self-contained study of gauge/gravity dualities in the line of the Klebanov-Witten model. Here we explore first the known Maldacena duality that relates N=4 SYM theory in four dimensions to type IIB supergravity on AdS_5×S^5 in reasonable detail, after some necessary preliminaries on supersymmetric gauge theories, where we display in detail the supersymmetry algebra and representations for N 1 supersymmetry. There we also construct the so-called superfields that will be helpful to write invariant lagrangians for gauge theoriesmreadily, and then useful to construct the gauge theory side of the Klebanov-Witten model. In the original AdS/CFT correspondence and its phenomenologically interesting extensions, Dp-branes as solutions of supergravity and nonperturbative objects in string theory where gauge theory lives are crucial. So, to preserve the self-contained nature of this work, we include a brief review of superstring theory addressed to understand the need to include this higher-dimensional objects by T-duality and, at low-energy limit of the string theory, as solutions of the Einstein equations. The first climax of this work occurs when we use all we learned to establish the Maldacena conjecture, N=4 SU(Nc) SYM theory we study in the supersymmetry chapter, living on the four-dimensional worldvolume of a stack of Nc D3-branes in a flat-space, corresponds exactly to type IIB supergravity on AdS_5×S^5 .In order to prove it, we match symmetries and operators with states in both sides. But actually this corresponds to the weak form of the correspondence, because it is not possible to handle neither string theory or gauge theory at strong coupling. The focus and main motive to have to learn the first hundred of pages here will be to extend the dual gauge theory we studied in AdS/CFT towards more realistic gauge theories as duals of some supergravity theory. The Klebanov-Witten model, consists in replacing the five-sphere in the gravity background of type IIB for a more interesting Einstein manifold X5 , a coset space called T^1,1 .The resulting dual gauge theory is expected to be less supersymmetric, and it is indeed N = 1 superconformal field theory with matter content in the bifundamental representation of the gauge group SU(N)×SU(N), and a quartic superpotential that exhibits SU(2)×SU(2)×U(1) global symmetry, which is precisely the symmetry of the coset space in the gravity side. This is not the end of the story, the Klebanov-Witten model extended the Maldacena correspondence and found as a dual gauge theory a less supersymmetric but still conformal theory. Breaking of the conformal theory, proposed by Klebanov, Nekrasov and Tseytlin, is achieved by introducing fractional M D3-branes in addition to the N regular D3-branes. The resulting theory is an SU(N+M)×SU(N) gauge theory with N = 1 supersymmetry, no longer conformal and then a little more interesting as a part of the crusade to find a QCD-like theory. This is still not the end, the last model suffers from a singularity in the deep IR, rendering the gravitational description invalid in that regime. It was conjectured that the strong dynamics of the gauge theory should somehow resolve this problem. Klebanov, again, and Strassler showed that this conjecture was correct, and argue that the RG flow is in fact an infinite series of Seiberg duality transformations- a cascade - in which the number of colors repeatedly drops from N NM , so the gaugegroup changes from SU(N+M)×SU(N) to SU(NM) ×SU(N). This process can be repeated until the IR limit where the gauge group simply becomes SU(M). So, at the end we get a N=1 SU(M) gauge theory, a QCD-like theory. We say that the standard model itself may lie at the base of a duality cascade.
Essa tese consiste num estudo autocontido das dualidades calibre/gravidade na linha do modelo do Klebanov-Witten. Aqui nos exploramos primeiro de um jeito razoavelmente detalhado,a conhecida dualidade do Maldacena que relaciona a teoria N=4 SYM em quatro dimensões com as supercordas tipo IIB no espaço AdS_5×S^5, depois de alguns preliminares necessários sobre teorias supersimétricas de calibre, onde nós mostramos em detalhe à algebra supersimétrica e as representações para N 1 supersimetria. Nós também construímos os conhecidos supercampos que são úteis para escrever lagrangianas invariantes para teorias de calibre facilmente, e então serão úteis para construir a teoría de calibre do modelo de Klebanov-Witten. Na correspondência AdS/CFT original e as suas extensões fenomenologicamente interessantes, as Dp-branas, como soluções de supergravidade e objetos não perturbativos na teoria de cordas onde as teorias de calibre moram, são essenciais. Assim ,a fim de preservar a natureza autocontida desse trabalho, nós incluímos uma breve revisão sobre teoria de supercordas dirigida a entender a necessidade de incluir esses objetos extra-dimensionais usando dualidade-T e, no limite de baixa-energia da teoria de cordas, como soluções das equações de Einstein. O primeiro clímax desse trabalho ocorre quando nós usamos tudo o que aprendemos para estabelecer a conjectura do Maldacena, a teoria de calibre N=4 SYM que nós estudamos no capítulo de supersimetria, morando no volume de mundo quadridimensional de uma pilha de Nc D3-branas (sim, o subscrito c significa cor!) em espaço plano, corresponde exatamente à teoria de supergravidade tipo IIB no espaço AdS_5×S^5 . A fim de testar ela, nós identificamos simetrias e operadores com estados em ambos lados da dualidade. Mas na verdade isto corresponde à forma fraca da correspondência, porque não é possível lidar nem com a teoria de cordas nem com a teoria de calibre no limite de acoplamento forte. O foco e motivo principal de porque nós temos que aprender as primeiras cem páginas aqui, será estender a teoria de calibre dual que estudamos em AdS/CFT, para teorias de calibre mais realisticas como duais de alguma teoria de supergravidade. O modelo do Klebanov-Witten, consiste em substituir a esfera de cinco dimensões no fundo de supergravidade da teoria de supercordas tipo IIB por um espaço que é mais interessante X5, um espaço coset chamado T^1,1. Nós esperamos que a teoria de calibre dual que resulta é menos supersimetrica, e na verdade é N =1 superconforme com um conteúdo de matéria na representação bifundamental do grupo de calibre SU(N)×SU(N), e um superpotencial quártico que tem simetria global SU(2)×SU(2)×U(1), que é precisamente a simetria do espaço coset no lado da gravidade. Mas isso não é tudo, o modelo do Klebanov-Witten estendeu a correspondência do Maldacena e encontrou como teoria dual uma teoria menos supersimetrica mas ainda conforme. A quebra da simetria conforme, proposta pelo Klebanov, Nekrasov e Tseytlin, é obtida introduzindo M D3-branas fracionais além das N D3-branas regulares. A teoria resultante é uma teoria de calibre SU(N+M)×SU(N) com N = 1 supersimetria, não mais conforme e então um pouco mais interessante como parte da nossa cruzada para encontrar uma teoria tipo-QCD. Isso ainda não é o final, o modelo anterior sofre de uma singularidade no IR profundo, tornando inválido a descrição gravitacional. Foi conjeturado então que a dinâmica do acoplamento forte na teoria de gauge deveria de algum jeito resolver esse problema. Klebanov, de novo, e Strassler mostraram que essa conjetura foi correta, e argumentaram que o fluxo do GR é de fato uma serie infinita de transformações de dualidade de Seiberg - uma cascata - onde o numero de cores cai repetidamente de NNM, e o grupo de calibre muda de SU(N+M)×SU(N) a SU(NM)×SU(N). O processo pode ser repetido até o limite IV onde o grupo de calibre simplesmente torna-se SU(M). Então, no final nós obtemos uma N = 1 teoria de calibre SU (M ), ou seja uma teoria tipo-QCD. Então, nós dissemos que o modelo padrão mesmo pode se situar na base da cascata de dualidade.

In dieser Arbeit behandeln wir exakte, nicht-perturbative Ergebnisse, die mithilfe der superkonformen Index-Technik, in supersymmetrischen Eichtheorien mit vier Superladungen (d. h. N=1 Supersymmetrie in vier Dimensionen und N=2 in drei Dimensionen) gewonnen wurden. Wir benutzen die superkonforme Index-Technik um mehrere Dualitäts Vermutungen in supersymmetrischen Eichtheorien zu testen. Wir führen Tests der dreidimensionalen Spiegelsymmetrie und Seiberg ähnlicher Dualitäten durch. Das Ziel dieser Promotionsarbeit ist es moderne Fortschritte in nicht-perturbativen supersymmetrischen Eichtheorien und ihre Beziehung zu mathematischer Physik darzustellen. Im Speziellen diskutieren wir einige interessante Identitäten der Integrale, denen einfache und hypergeometrische Funktionen genügen und ihren Bezug zu supersymmetrischen Dualitäten in drei und vier Dimensionen. Methoden der exakten Berechnungen in supersymmertischen Eichtheorien sind auch auf integrierbare statistische Modelle anwendbar. Dies wird im letzten Kapitel der vorliegenden Arbeit behandelt.
In this thesis we discuss exact, non-perturbative results achieved using superconformal index technique in supersymmetric gauge theories with four supercharges (which is N = 1 supersymmetry in four dimensions and N = 2 supersymmetry in three). We use the superconformal index technique to test several duality conjectures for supersymmetric gauge theories. We perform tests of three-dimensional mirror symmetry and Seiberg-like dualities. The purpose of this thesis is to present recent progress in non-perturbative supersymmetric gauge theories in relation to mathematical physics. In particular, we discuss some interesting integral identities satisfied by basic and elliptic hypergeometric functions and their relation to supersymmetric dualities in three and four dimensions. Methods of exact computations in supersymmetric theories are also applicable to integrable statistical models, which we discuss in the last chapter of the thesis.

The main contribution of this thesis is the concept of Fenchel duality with a focus on its application in the field of machine learning problems and image restoration tasks. We formulate a general optimization problem for modeling support vector machine tasks and assign a Fenchel dual problem to it, prove weak and strong duality statements as well as necessary and sufficient optimality conditions for that primal-dual pair. In addition, several special instances of the general optimization problem are derived for different choices of loss functions for both the regression and the classifification task. The convenience of these approaches is demonstrated by numerically solving several problems. We formulate a general nonsmooth optimization problem and assign a Fenchel dual problem to it. It is shown that the optimal objective values of the primal and the dual one coincide and that the primal problem has an optimal solution under certain assumptions. The dual problem turns out to be nonsmooth in general and therefore a regularization is performed twice to obtain an approximate dual problem that can be solved efficiently via a fast gradient algorithm. We show how an approximate optimal and feasible primal solution can be constructed by means of some sequences of proximal points closely related to the dual iterates. Furthermore, we show that the solution will indeed converge to the optimal solution of the primal for arbitrarily small accuracy. Finally, the support vector regression task is obtained to arise as a particular case of the general optimization problem and the theory is specialized to this problem. We calculate several proximal points occurring when using difffferent loss functions as well as for some regularization problems applied in image restoration tasks. Numerical experiments illustrate the applicability of our approach for these types of problems.

Duality begins in the layers of intentions within this thesis. The thesis is approached from the primary need to address the questions of architecture by means of the intellect. Subjectivity is an inevitable intervention in the design process. Thus, there is the secondary need to address the questions of the self in which the intellect and its subjectivity are integral parts. Much of my identity is sculpted by the Buddhist philosophy. Thus it is the primary source which I draw from in the making of architecture. However, the philosophy and architecture are in diametrical opposition. Architecture is reliant on the tangibility of materiality and structure. Buddhism rejects that life in the tangibility. More acutely, its essence is an amorphism because of its inexplicable nature. Conversely, architecture's caliber resides in the clarity of form. This is an attempt in the construct of an architecture taken from outside of architecture while imbuing the quality of being architectural. In parallel, duality is in the dialectic of a thesis and an antithesis in search of a synthesis.
Master of Architecture

The work seeks to understand the nature of architecture where House and Institution coexist. Specific architectural constructs support the archetype of house in order to provide an instant notion of Home. The constructs range from the visual to the experiential; where reality and imagination find common ground. Even though the House in this context is meant to be inhabited for a short period of time only, the architecture seeks to provide an immediate sense of home in the moment of transition for a mother and child, who are coming out of a domestic violence scenario. In this context, the relationship between House and Institution is regarded as symbiotic. The Institution suggests a public identity, providing places for therapy, consultation, and other types of assistance to the adults. For the children, the architecture includes places for the presence of books with an invitation to read. The architecture of House is based on individuality, a kind of receptacle for idea of Home. For the mothers, the space between this duality will offer places for gathering, sharing and learning; for the children, the architecture aims to be magical.
Master of Architecture

We attack the question of BPS occupancy in a wide class of 4d N = 2 quantum field theories. We first review the Seiberg-Witten approach to finding the low energy Wilsonian effective action actions of such theories. In particular, we analyze the case of Gaiotto theories, which provide a large number of non-trivial examples in a unified framework. We then turn to understanding the massive BPS spectrum of such theories, and in particular their relation to BPS quivers. We present a purely 4d characterization of BPS quivers, and explain how a quiver's representation theory encodes the solution to the BPS occupancy problem. Next, we derive a so called mutation method, based on exploiting quiver dualities, to solve the quiver's representation theory. This method makes previously intractable calculations nearly trivial in many examples. As a particular highlight, we apply our methods to understand strongly coupled chambers in ADE SYM gauge theories with matter. Following this, we turn to the general story of quivers for theories of the Gaiotto class. We present a geometric approach to attaining quivers for the rank 2 theories, leading to a very elegant solution which includes a specification of quiver superpotentials. Finally, we solve these theories by an unrelated method based on gauging flavor symmetries in their various dual weakly coupled Lagrangian descriptions. After seeing that this method agrees in the rank 2 case, we will apply our new approach to the case of rank n.
Physics

In the present thesis, we offer an introduction to type IIB string compactifications on $\mathbb{T}^{d}/\Gamma$ toroidal orbifolds. We first describe the technical method to construct these spaces and reduce the string background on it. We will have (non)-geometrical fluxes arising from these spaces which decorate with discrete deformations our four $\mathcal{N}=1$ dimensional supergravity theory. Solving its equations of motion, we find several families of supersymmetric AdS vacua with fixed moduli, which can be related through a set of  $SL(2,\mathbb{Z})$ symmetries.

This thesis is devoted to various questions connected with duality. It is composed of two parts.The first part discusses some aspects of timelike T-duality. We explore the possibility of compactification of supergravity theories with various signatures(low energy limit of M-theories which are dual under timelike T-dualities) on parallelizable internal seven dimensional (pseudo-)spheres. We show that, beside the standard theory, only one of the dual theories known as M'-theory can admit such a solution. The effective four dimensional theory is non-supersymmetric and due to the presence of torsion the symmetry of seven dimensional (pseudo-)sphere breaks down to Spin(3, 4). In the second part, in an attempt to have a systematic discussion of gaugings in supergravity, we show the isomorphism between the space of local deformations of the appropriate zero coupling limit of the embedding tensor Lagrangian and that of the second-order scalar-vector Lagrangian, describing the bosonic sector of supergravity ignoring gravity, in a chosen duality frame determined by embedding tensors. We analyze the BV-BRST deformation of a class of scalar-vector coupled Lagrangians, which contains supergravity Lagrangians as examples, and find a set of constraints that guarantee the consistency of the deformations of the Lagrangians. We show in principle that for a large class of theories considered in this thesis, the only deformations are those of the Yang-Mills type associated with a subgroup of the rigid symmetries.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished

Soit $X$ une variété algébrique de Gorenstein à singularités rationnelles. Une résolution des singularités crépante de $X$ est souvent considérée comme une résolution des singularités minimales de $X$. Malheureusement, les résolutions crépantes sont très rares. Ainsi, les variétés déterminantielles de matrices anti-symétriques n'admettent jamais de résolution crépante des singularités. Dans cette thèse, on discutera de diverses notions de résolutions catégoriques crépantes développées par Alexander Kuznetsov. Conjecturalement, ces résolutions doivent être minimale du point de vue catégorique. On introduit dans ce manuscrit la notion de résolution magnifiques des singularités et on montre que tout variété munie d'une telle résolution admet une résolution catégorique faiblement crépante. On en déduit que toutes les variétés déterminantielles (carrées, symétriques et anti-symétriques) admettent des résolutions catégoriques faiblement crépantes. Finalement, on s'intéressera à des hypersurfaces quartiques issues du carré magique de Tits-Freudenthal. On ne peut pas construire de résolution magnifique des singularités pour de telles hypersurfaces, mais on montrera qu'elles admettent tout de même des résolutions catégorique faiblement crépantes des singularités. Ce résultat devrait s'avérer intéressant pour la construction de duales projectives homologiques de certaines Grassmaniennes symplectiques sur les algèbres de composition
Let $X$ be an algebraic variety with Gorenstein rational singularities. A crepant resolution of $X$ is often considered to be a minimal resolution of singularities for $X$. Unfortunately, crepant resolution of singularities are very rare. For instance, determinantal varieties of skew-symmetric matrices never admit crepant resolution of singularities. In this thesis, we discuss various notions of categorical crepant resolution of singularities as defined by Alexander Kuznetsov. Conjecturally, these resolutions are minimal from the categorical point of view. We introduce the notion of wonderful resolution of singularities and we prove that a variety endowed with such a resolution admits a weakly crepant resolution of singularities. As a corollary, we prove that all determinantal varieties (square, as well as symmetric and skew-symmetric) admit weakly crepant resolution of singularities. Finally, we study some quartics hypersurfaces which come from the Tits-Freudenthal magic square. Though they do no admit any wonderful resolution of singularities, we are still able to prove that they have a weakly crepant resolution of singularities. This last result should be of interest in order to construct homological projective duals for some symplectic Grassmannians over the composition algebras

Il y a deux choses. On a souvent de la difficulté à l’admettre. Ce sont deux manières de connaître, ce à quoi on résiste déjà ; mais ce sont plus encore deux manières d’être au monde et de se lier à lui en étant pour ou contre lui. La différence entre l’esprit et le corps, le dedans et le dehors, apparaît maintenant comme étant la différence entre le monde et son image, entre la réalité connue immédiatement, réalité sensible, affective, vécue, et le réel connu médiatement : connaissances symboliques, quantitatives, manières qu’a l’intelligence d’indexer un réel extérieur à partir d’un sensible immédiat. Or, ne plus reconnaître de réalité qu’en une connaissance objective, ce serait substituer l’image du réel au réel, se fermer à notre propre réalité, et donc à la vie, puisque nous sommes vie. Le travail de la raison consisterait à analyser, à découper, contrôler, dominer. Or, si ce savoir objectif, savoir de la raison, a sa praxis, le savoir subjectif aurait la sienne propre. À ces deux savoirs correspondent deux praxis, deux éthiques inscrites nécessairement dans la structure même du vivant. Ce n’est point ce que, dans la littérature contemporaine anglophone — du moins en philosophie analytique —, on semble disposé à reconnaître. On y dépeint plutôt la connaissance subjective comme étant une forme de connaissance primitive, « folklorique ». Les présentes recherches ont donc dû s’attarder à mettre en lumière l’incompréhension dont souffre le dualisme épistémique dans la philosophie de l’esprit qui répond au goût du jour, laissant surtout en friche la tâche d’élucider plus longuement le sens du dualisme éthique en tant que tel, quoique celui-ci se trouve néanmoins exposé dans le dernier chapitre du travail.

La théorie des cordes est un de nos meilleurs candidats pour une théorie quantique de la gravité. A ce jour elle n'a pas encore été conclusive à propose de ce sujet. Malgré cela, on a réalisé qu'on peut en tirer des informations sur tout une variété de sujets, dont notamment les théories de jauges supersymétriques, en étudiant la limite de basse énergie dans le volume d'univers des branes. Cette immersion des théories de jauges en théorie des cordes nous fournit un autre point de vue. Ce dernier nous permet souvent de prendre une approche plus géométrique pour obtenir de nouveaux résultats sinon inaccessible par des méthodes plus conventionnelles. Même en absence de vérification expérimentale de la supersymétrie, sa présence dans cette classe de théories de jauge nous fournit un terrain de jeux propice pour tester de nouvelles méthodes d'une manière efficace. En effet, la présence de la supersymétrie donne une structure additionnelle qui rend la théorie plus rigide. Cela simplifie les calculs et rend des résultats plus accessibles. On peut oser de dire que si on n'arrive pas à calculer un certain résultat en présence de supersymétrie, il y a très peu de chance d'y arriver sans. L'approche par la théorie des cordes le rend possible de découvrir des symétries cachées ou de comprendre des symétries connues d'une autre manière.Une classe de théories quantique intéressantes qui sont présentes en théories des cordes, c'est les théories de petites cordes. Ces dernières ont été découverte il y a deux décennies. Ces théories en six dimensions ont été construite une première fois comme théories dans le volume d'universe de branes NS5 dans le cadre de la théorie des cordes IIB en prenant la limite du couplage de la corde qui tend vers zéro. Dans cette limite, la théorie résultant reste non-trivial mais les interactions en dehors de la brane sont supprimées, notamment la gravité. Comme le nom le suggère, ces théories contiennent des cordes qu'on appelle petites cordes. La tension des petites cordes est proportionnelle à l'échelle naturelle de la corde. En plus, ces théories profitent de la T-dualité comme les théories de cordes critiques. Elle sont donc des théories quantiques non-locales. Leur complexité se situe entre celle des théories quantiques locales et celle de la théorie des cordes complète. Elles sont donc des candidates intéressantes pour étudier la dynamique dans le volume d'univers de la brane NS5. Pour des énergies inférieures à l'échelle de la corde, elles ont une description en termes de théories de jauges symétriques de type quiver. On peut donc également obtenir des informations sur ces dernières. Cette description locale n'est plus valable une fois l'échelle de la corde atteinte.Le but principal de cette thèse est d'étudier des dualités entre le théories de petites cordes en utilisant différentes constructions disponible en théorie des cordes. Cela nous permet d'attaquer le problème d'angles différents et de faire un lien avec des structures géométriques. En conséquence on peut analyser différentes relation parmi les théories de petites cordes. On confirme ensuite la validité des dualités qu'on obtient en utilisant la fonction de partition instantonique. Cet object est complètement non-perturbative et établit ces dualités comme résultat exact. Cette structure de dualités s'étend naturellement aux descriptions de basse énergie en terme de théories de jauges supersymétriques. De plus, on étudie les conséquences directes du réseaux de dualités qu'on a découvert
String theory remains one of our best candidates for a theory of quantum gravity. Until now it has not lived up to this goal. However, along the way it was realized that string theory can give us valu-able insights into a variety of subjects among which supersymmetric gauge theories by studying the low-energy worldvolume dynamics of branes. This embedding of gauge theories into string theory provides us with a different viewpoint that often allows us to use powerful geometric considerati-ons in order to obtain new results that are inaccessible from conventional methods. Even in the ab-sence of experimental confirmation of supersymmetry, its presence in this class of gauge theories provides us with a playground where different methods can be tested in an efficient way. Indeed, supersymmetry provides additional structure, rendering the underlying theory more rigid and thus simplifying computations and making results more accessible. One could dare to say that when a certain result can not be calculated in the presence of supersymmetry, there is probably not much hope of achieving it without supersymmetry. This stringy approach to gauge theories makes it pos-sible to unravel hidden dualities or to understand already known ones from a different perspective. An interesting class of quantum theories that are embedded into string theory are the so called little string theories. They have been discovered two decades ago. These six-dimensional theories were first obtained as the worldvolume theory of a stack of NS5 branes in the context of Type II string theory trough a particular decoupling limit that sends the string coupling constant to zero while kee-ping at the same time the string scale finite. In this limit, the resulting theory remains interacting but the bulk dynamics is decoupled, in particular gravity. As their name suggests, they contain strings. The tension of the little strings is proportional to the string scale, which is the only intrinsic scale in the theory. Furthermore, the little string theories enjoy T-duality similar to the critical string theory. They are thus non-local quantum theories. So the complexity of little string theory lies between that of local quantum field theories and full fledged critical string theory. This makes them interesting candidates for studying stringy phenomena in an easier setup where gravity is absent and to learn more about the worldvolume dynamics of the NS5 brane. At energies far below the string scale, they have a low-energy description in terms of quiver gauge theories, so their study can also give us insights into these kinds of theories. This local description breaks down as we reach the string scale and we must rely on the full little string theories. The main goal of this thesis is to study dualities between little string theories by using different dual constructions available in string theory. These allow us to attack the problem from different angles and they establish also a connection to geometric structures. This makes it possible to systematically analyse relations among different little string theories. We then confirm the validity of the newly found duality relations by using the so called instanton partition function. The latter is a completely non-perturbative object allowing us to establish the dualities as an exact result. This duality structure naturally extends to the low-energy description in terms of supersymmetric quiver gauge theories. Furthermore, we study the direct consequences of this duality web. We find interesting cases where the dimensional reduction from six to five dimensions simultaneously reduces the rank of the group and changes the matter content. Another result that we find is the presence of a hidden dihedral symmetry which acts in a highly non-trivial fashion on the spectrum of the underlying gauge theories

The aim of this study is to make an inquiry on the nature and the development of dualities in Bergson&rsquo
s philosophy. Since the nature of each duality differs from the others and the dualistic pattern inherent in Bergsonian philosophy is subject to change, we base our study on a chronological structure in order to comprehend better how this pattern changes. We claim that such an inquiry will yield relevant outcomes with regard to ontological and epistemological evolution of Bergson&rsquo
s thought. To state more precisely, we are of the idea that the modification in the dualistic pattern in Bergson&rsquo
s ontology is reflected in a parallel manner in his epistemology. The fundamental question that shows us the way to follow in our study is whether the elements of the dualities (whether they be ontological or epistemological) are reconciled by Bergson or they are left as absolutely distinct elements. At the end of the inquiry regarding that question, which we believe can be taken as an inspiring point in developing new approaches especially to epistemological problems, our conviction is that Bergson points out to a meeting point.

Les comportements des processus concurrents peuvent être exprimés en utilisant des calculs de processus, des langages formels simples qui permettent de démontrer des résultats mathématiques précis sur les interactions entre processus. Un exemple très simple est CCS, un autre exemple est le pi-calcul, plus expressif grâce à un mécanisme de communication de canaux. Dans ce dernier, on peut instaurer un système de types (pour raffiner l'analyse aux environnements plus contraints) et encoder le lambda-calcul (qui représente les calculs séquentiels).Certains de ces calculs, comme CCS ou des variantes du pi-calcul comme les calculs de fusions, ont une certaine propriété de symétrie. On utilise dans un premier temps cette symétrie comme un outil, pour prouver que deux encodages du lambda-calcul dans le pi-calcul sont en fait équivalents.Cette preuve nécessitant un système de types et une forme de symétrie, on se pose la question de l'existence d'un système de types pour les autres calculs symétriques, notamment les calculs de fusion, à laquelle on répond par la négative avec un théorème d'impossibilité.En analysant ce théorème, on découvre un contrainte fondamentale de ces calculs qui empêche l'utilisation des types, à savoir la présence d'une notion de relation d'équivalence entre les canaux de communication. Le relâchement de cette contrainte pour obtenir une relation de pré-ordre engendre un calcul intéressant qui recouvre des notions importantes du pi-calcul, absentes dans les calculs de fusion : les types et les noms privés. La première partie de la thèse se concentre sur l'étude de ce calcul.La deuxième partie de la thèse se concentre sur la bisimulation, une méthode pour établir l'équivalence de deux agents dans des langages d'ordre supérieur, par exemple le pi-calcul ou le lambda-calcul. Une amélioration de cette méthode est la théorie des techniques modulo, très puissante, mais qui malheureusement s'applique uniquement aux systèmes de premier ordre, comme les automates ou CCS.Cette thèse s'applique alors à décrire les langages d'ordre supérieur en tant que systèmes du premier ordre. On récupère ainsi la théorie générale des techniques modulo pour ces langages, en prouvant correctes la correspondance induite et les techniques spécifiques à chaque langage. On détaille les tenants et aboutissants de cette approche, pour fournir les outils nécessaires à son utilisation pour d'autres langages d'ordre supérieur
The behaviours of concurrent processes can be expressed using process calculi, which are simple formal languages that let us establish precise mathematical results on the behaviours and interactions between processes. A very simple example is CCS, another one is the pi-calculus, which is more expressive thanks to a name-passing mechanism. The pi-calculus supports the addition of type systems (to refine the analysis to more subtle environments) and the encoding of the lambda-calculus (which represents sequential computations).Some of these calculi, like CCS or variants of the pi-calculus such as fusion calculi, enjoy a property of symmetry. First, we use this symmetry as a tool to prove that two encodings of the lambda-calculus in the pi-calculus are in fact equivalent.This proof using a type system and a form of symmetry, we wonder if other existing symmetric calculi can support the addition of type systems. We answer negatively to this question with an impossibility theorem.Investigating this theorem leads us to a fundamental constraint of these calculi that forbids types: they induce an equivalence relation on names. Relaxing this constraint to make it a preorder relation yields another calculus that recovers important notions of the pi-calculus, that fusion calculi do not satisfy: the notions of types and of privacy of names. The first part of this thesis focuses on the study of this calculus, a pi-calculus with preorders on names.The second part of this thesis focuses on bisimulation, a proof method for equivalence of agents in higher-order languages, like the pi- or the lambda-calculi. An enhancement of this method is the powerful theory of bisimulations up to, which unfortunately only applies for first-order systems, like automata or CCS.We then proceed to describe higher-order languages as first-order systems. This way, we inherit the general theory of up-to techniques for these languages, by proving correct the translations and up-to techniques that are specific to each language. We give details on the approach, to provide the necessary tools for future applications of this method to other higher-order languages

We examine the use of branes as probes of supergravity geometries which arise in the study of gauge theory/gravity dualities. We investigate the moduh spaces of supersymmetric gauge theories through moduh spaces of brane probes in the dual gravity theories. Preferred coordinate systems emerge in which the supergravity geometries can readily be compared to the gauge theory and various gauge theory quantities such as anomalous scaling dimensions can be read off. We also consider the physics of certain expanded brane configurations, called giant gravitons. We identify supergravity solutions which represent coherent states of these objects. We find a degeneracy between giant graviton probes and massless particles in a broad class of supergravity backgrounds and uncover a close relationship with charged particle states in lower dimensions.

Après avoir exposé l'importance des dualités géométrico-algébriques dans l'histoire des mathématiques, la thèse propose de rassembler bon nombre d'entre elle sous une approche unifiée abstraite, la dualité d'Isbell. La dualité d'Isbell est formellement définie comme une adjonction entre un préfaisceau et un copréfaisceau, et permet de définir un nouveau paradigme de constructivité baptisé P3. En mathématique, nous montrons que cette dualité est présente en géométrie algébrique, en géométrie algébrique dérivée, en topologie algébrique et en analyse fonctionnelle. En logique contemporaine, nous montrons qu'elle peut être rendue explicite dans la géométrie de l'interaction de Girard. Nous montrons ensuite comment les sciences appliquées peuvent faire usage de la dualité d'Isbell, en permettant de renouveler significativement les théories. En sciences physiques, nous montrons qu'elle ouvre une perspective dans la théorie quantique des champs, vers la dualisation des représentations de Heisenberg et de Schrödinger. En sciences économiques et sociales, nous montrons qu'elle permet de renouveler la théorie de l'équilibre générale et la théorie de la valeur. En sciences de l'apprentissage, nous montrons qu'il est possible de reconsidérer la théorie de l'enquête de Dewey en termes de dualité espace-action, pour finalement dégager une dualité d'Isbell. Nous concluons en ouvrant un débat sur la notion bachelardienne d'obstacle épistémologique, pour montrer comment P3 peut avoir des difficultés à s'imposer, et en consacrant quelques développements ontologiques sur la nature kantienne et post-hégélienne de la thèse
After exposing the importance of geometric-algebraic dualities in the history of mathematics, the thesis proposes to bring together many of them under an unified abstract approach, the Isbell duality. The Isbell duality is formally defined as an adjunction between a presheaf and a copresheaf, and allows to define a new paradigm of constructivity called P3. In mathematics, we show that this duality is present in algebraic geometry, derived algebraic geometry, algebraic topology and functional analysis. In contemporary logic, we show that Isbell duality can be made explicit in the geometry of interaction of Girard. We then show how applied sciences can make use of Isbell duality, allowing to significantly renew theories. In physical sciences, we show that it opens a perspective in quantum field theory, towards the dualization of Heisenberg and Schrödinger representations. In economic and social sciences, we show that it allows to renew the general equilibrium theory and the theory of value. In learning sciences, we show that it is possible to reconsider Dewey's theory of inquiry in terms of space-action duality, ultimately to reveal an Isbell duality. We conclude by opening a debate on the Bachelardian notion of epistemological obstacle, showing how P3 can have difficulties to establish itself as reference constructive paradigm, and by devoting some ontological developments to the Kantian and post-Hegelian nature of the thesis

Notre travail est une tentative pour rapprocher les regimes partonique et hadronique de l'interaction forte a l'aide d'une matrice s effective possedant une propriete de symetrie cyclique commune aux deux regimes. La cle de notre approche est une procedure pour decomposer une amplitude arbitraire de qcd en amplitudes duales et invariantes de jauge, contenues a la limite de pente nulle, dans des amplitudes en arbre de supercordes ouvertes. Une telle decomposition facilite le calcul des sections efficaces des processus a multijets, produits dans les collisionneurs hadroniques. Les corrections virtuelles a l'approximation d'une boucle sont prises en compte par la redefinition du couplage et de la pente. Les amplitudes effectives, duales et invariantes de jauge, ainsi obtenues rendent compte des effets de coherence de couleur, sans qu'il soit necessaire de restreindre l'espace de phase. De plus, soulignons que la matrice s effective contient un secteur hadronique, d'ou une base theorique possible pour expliquer la propriete phenomenologique de dualite parton-hadron. On sait que la propriete de symetrie cyclique reliee a la dualite domine le regime hadronique, et qu'elle n'est pas affectee par une renormalisation d'intercept induite par une technique d'unitarisation topologique. Nos amplitudes ayant cette propriete de dualite, sont donc bien adaptees pour approcher le regime non perturbatif de qcd

This thesis deals with the two duality symmetries of N = 2 D = 10 supergravity theories that are descendant from the full superstring theory: fermionic T-duality and U-duality. The fermionic T-duality transformation is applied to the D-brane and ppwave solutions of type IIB supergravity. New supersymmetric solutions of complexi ed supergravity are generated. We show that the pp-wave yields a purely imaginary background after two dualities, undergoes a geometric transformation after four dualities, and is self-dual after eight dualities. Next we apply six bosonic and six fermionic T-dualities to the AdS4 CP3 background of type IIA supergravity, which is relevant to the current research in the amplitude physics. This helps to elucidate the potential obstacles in establishing the self-duality, and quite independently from that shows us that fermionic T-dualities may be degenerate under some circumstances. Finally, we make a step towards constructing a manifestly U-duality covariant action for D = 10 supergravities by deriving the generalized metric for a D1-brane. This is a single structure that treats brane wrapping coordinates on the same footing as spacetime coordinates. It turns out that the generalized metric of a D-string results from that of the fundamental string if one replaces the spacetime metric with the open string metric. We also nd an antisymmetric contribution to the generalized metric that can be interpreted as a noncommutativity parameter.

Dans leur recherche d'une théorie unifiée des interactions fondamentales, contenant en particulier un modèle quantique de la gravitation, les physiciens ont imaginé des théories de supercordes, dans lesquelles, en plus des cordes, on trouve des objets étendus de diverses dimensions, reliés par le groupe de U-dualité. De plus, on conjecture l'existence d'une théorie mère, la théorie M, dont la limite de basse énergie serait la supergravité à onze dimensions. Dans ce travail, nous montrons qu'en partant des surfaces de del Pezzo, on peut construire des superalgèbres de Kac-Moody généralisées qui contiennent les groupes de U-dualité et donnent le contenu en champs bosoniques (doublé) de la théorie M et de ses réductions dimensionnelles. On retrouve alors les équations du mouvement comme une condition d'auto-dualité, associée à une symétrie du réseau de Picard de la surface de del Pezzo correspondante. Cela permet d'expliquer la symétrie du «triangle magique» de Cremmer, Julia, Lü etPope.

Le dualisme dans la représentation artistique est le reflet des éléments de dualité présents dans la nature, dans la pensée et la condition humaine. Mon travail de création émerge naturellement des principes de dualité. J’utilise la géométrie pour représenter les réalités structurées de la pensée, de la nature et de la vie humaine. Les formes organiques, les références au corps humain en particulier, témoignent des réalités aléatoires du statut de vivant. Pour moi, ces deux types de réalités cohabitent en toute chose. La forme organique, forcément imparfaite et temporelle, est constituée d’éléments de base infiniment petits, proches de la perfection géométrique. Mais je veux aller au delà de la force expressive des antithèses entre la figuration plus ou moins affirmée du corps humain, et les formes géométriques, pures créations de l’esprit. Je cherche plutôt à unifier, à intégrer ces éléments plastiques ayant des composantes sémantiques et formelles qui semblent se contredire au départ, déterminant ainsi une sorte de dialectique entre les éléments constituants. Je ne crois pas que l’abstraction géométrique soit opposée à la figuration comme s’il s’agissait de deux entités opposées qui se nient l’une l’autre historiquement et plastiquement. Dans mon travail, ce sera le plus souvent l’utilisation de la transparence qui rendra possible la manifestation du dualisme. La transparence est un moyen privilégié pour faire apparaître les dualités du monde réel et de la représentation. La transparence pour unifier les images, moduler la lumière et les couleurs, en changer les qualités. La transparence questionne la limite. Limite du visible, limite du tangible. Voir à travers, cristallin de l’image, voir de près, microcosme, macrocosme. Dualité : verre et béton, réel et virtuel, superpositions de couches, superpositions de temps.

In der vorliegenden Arbeit wird mit Hilfe der verallgemeinerten Eichtheorie/Gravitations-Dualität, welche stark gekoppelte Eichtheorien mit schwach gekrümmten gravitativen Theorien verbindet, stark korrelierte Quantenzustände der Materie untersucht. Der Schwerpunkt liegt dabei in Anwendungen auf Systeme der kondensierten Materie, insbesondere Hochtemperatur-Supraleitung und kritische Quantenzustände bei verschwindender Temperatur. Die Eichtheorie/Gravitations-Dualität entstammt der Stringtheorie und erlaubt eine Umsetzung des holographischen Prinzips. Aus diesem Grund wird eine kurze Einführung in die Konzepte der Stringtheorie und ihre Auswirkungen auf das holographische Prinzip gegeben. Für das tiefere Verständnis der effektiven Niederenergie-Feldtheorien wird zusätzlich die Supersymmetrie benötigt. Ausgestattet mit einem robusten Stringtheorie-Hintergrund wird die unterschiedliche Interpretation der Dirichlet- oder D-Branen, ausgedehnte Objekte auf denen offene Strings/Fäden enden können, diskutiert: Zum einen als massive solitonische Lösungen der Typ II Supergravitation und auf der anderen Seite, ihre Rolle als Quelle für supersymmetrische Yang-Mills Theorien. Die Verbindung dieser unterschiedlichen Betrachtungsweise der D-Branen liefert eine explizite Konstruktion der Eichtheorie/Gravitations-Dualität, genauer der AdS_5/CFT_4 Korrespondenz zwischen der N=4 supersymmetrischen SU(N_c) Yang-Mills Theorie in vier Dimensionen mit verschwindender beta-Funktion in allen Ordnungen, also eine echte konforme Theorie, und Type IIB Supergravitation in der zehn dimensionalen AdS_5 X S^5 Raumzeit. Darüber hinaus wird das Wörterbuch, das zwischen den Operatoren der konformen Feldtheorie und den gravitativen Feldern übersetzt, im Detail eingeführt. Genauer gesagt, die Zustandssumme der stark gekoppelten N=4 supersymmetrischen Yang-Mills Theorie im Grenzwert großer N_c, ist identisch mit der Zustandssumme der Supergravitation unter Berücksichtigung der zugehörigen Lösungen der Bewegungsgleichungen, ausgewertet am Rand des AdS-Raumes. Die Anwendung der perturbativen Quantenfeldtheorie und die Verbindungen zur quantenstatistischen Zustandssumme erlaubt die Erweiterung des holographischen Wörterbuchs auf Systeme mit endlichen Dichten und endlicher Temperatur. Aus diesem Grund werden alle Aspekte der Quantenfeldtheorie behandelt, die für die Anwendung der ``Linear-Response''-Theorie, der Berechnung von Korrelationsfunktionen und die Beschreibung von kritischen Phänomenen benötigt werden, wobei die Betonung auf allgemeine Zusammenhänge zwischen Thermodynamik, statistischer Physik bzw. statistischer Feldtheorie und Quantenfeldtheorie liegt. Des Weiteren wird der Renormierungsgruppen-Formalismus zur Beschreibung von effektiven Feldtheorien und kritischen Phänomene im Kontext der verallgemeinerten Eichtheorie/Gravitations-Dualität ausführlich dargelegt. Folgende Hauptthemen werden in dieser Arbeit behandelt: Die Untersuchung der optischen Eigenschaften von holographischen Metallen und ihre Beschreibung durch das Drude-Sommerfeld Modell, ein Versuch das Homes'sche Gesetz in Hochtemperatur-Supraleitern holographisch zu beschreiben indem verschiedene Diffusionskonstanten und zugehörige Zeitskalen berechnet werden, das mesonische Spektrum bei verschwindender Temperatur und schlussendlich holographische Quantenzustände bei endlichen Dichten. Entscheidend für die Anwendung dieses Rahmenprogramms auf stark korrelierte Systeme der kondensierten Materie ist die Renormierungsgruppenfluss-Interpretation der AdS_5/CFT_4 Korrespondenz und die daraus resultierenden emergenten, holographischen Duale, welche die meisten Beschränkungen der ursprünglichen Theorie aufheben. Diese sogenannten ``Bottom-Up'' Zugänge sind besonders geeignet für Anwendungen auf Fragestellungen in der Theorie der kondensierten Materie und der ``Linear-Response''-Theorie, mittels des holographischen Fluktuations-Dissipations-Theorem. Die Hauptergebnisse der vorliegenden Arbeit umfassen eine ausführliche Untersuchung der R-Ladungs-Diffusion und der Impulsdiffusion in holographischen s- und p-Wellen Supraleitern, welche durch die Einstein-Maxwell Theorie bzw. die Einstein-Yang-Mills Theorie beschrieben werden, und eine Vertiefung des Verständnisses der universellen Eigenschaften solcher Systeme. Als zweites wurde die Stabilität der kalten holographischen Quantenzustände der Materie untersucht, wobei eine zusätzliche Diffusions-Mode entdeckt wurde. Diese Mode kann als eine Art ``R-Spin-Diffusion'' aufgefasst werden, die der Spin-Diffusion in Systemen mit frei beweglichen ``itineranten'' Elektronen ähnelt, wobei die Entkopplung der Spin-Bahn Kopplung die Spin-Symmetrie in eine globale Symmetrie überführt. Das Fehlen der Instabilitäten und die Existenz einer ``Zero-Sound'' Mode, bekannt von Fermi-Flüssigkeiten, deuten eine Beschreibung der kalten holographischen Materie durch eine effektive hydrodynamische Theorie an.
In this dissertation strongly correlated quantum states of matter are explored with the help of the gauge/gravity duality, relating strongly coupled gauge theories to weakly curved gravitational theories. The main focus of the present work is on applications to condensed matter systems, in particular high temperature superconductors and quantum matter close to criticality at zero temperature. The gauge/gravity duality originates from string theory and is a particular realization of the holographic principle. Therefore, a brief overview of the conceptual ideas behind string theory and the ramifications of the holographic principle are given. Along the way, supersymmetry and supersymmetric field theories needed to understand the low energy effective field theories of superstring theory will be discussed. Armed with the string theory background, the double life of D-branes, extended object where open strings end, is explained as massive solitonic solutions to the type II supergravity equations of motion and their role in generating supersymmetric Yang-Mills theories. Connecting these two different pictures of D-branes will give an explicit construction of a gauge/gravity duality, the AdS_5/CFT_4 correspondence between N=4 supersymmetric SU(N_c) Yang-Mills theory in four dimensions with vanishing beta-function to all orders, describing a true CFT, and type IIB supergravity in ten-dimensional AdS_5 X S^5 spacetime. Furthermore, the precise dictionary relating operators of the conformal field theory to fields in the gravitational theory is established. More precisely, the partitions functions of the strongly coupled N=4 supersymmetric Yang-Mills theory in the large N_c limit is equal to the on-shell supergravity partitionevaluated at the boundary of the AdS space. Applying the knowledge of perturbative quantum field theory and its relation to the quantum partition function the dictionary may be extended to finite temperature and finite density states. Thus, all aspects of quantum field theory relevant for the application of linear response theory, the computation of correlation functions, and the description of critical phenomena are covered with emphasis on elucidating connections between thermodynamics, statistical physics, statistical field theory and quantum field theory. Furthermore, the renormalization group formalism in the context of effective field theories and critical phenomena will be developed explaining the critical exponents in terms of hyperscaling relations. The main topics covered in this thesis are: the analysis of optical properties of holographic metals and their relation to the Drude-Sommerfeld model, an attempt to understand Homes' law of high temperature superconductors holographically by computing different diffusion constants and related timescales, the mesonic spectrum at zero temperature and holographic quantum matter at finite density. Crucially for the application of this framework to strongly correlated condensed matter systems is the renormalization flow interpretation of the AdS_5/CFT_4 correspondence and the resulting emergent holographic duals relaxing most of the constraints of the original formulation. These so-called bottom up approaches are geared especially towards applications in condensed matter physics and to linear response theory, via the central operational prescription, the holographic fluctuation-dissipation theorem. The main results of the present work are an extensive analysis of the R-charge- and momentum diffusion in holographic s- and p-wave superconductors, described by Einstein-Maxwell theory and the Einstein-Yang-Mills model, respectively, and the lessons learned how to improve the understanding of universal features in such systems. Secondly, the stability of cold holographic quantum matter is investigated. So far, there are no instabilities detected in such systems. Instead, an interesting additional diffusion mode is discovered, which can be interpreted as an ``R-spin diffusion'', resembling spin diffusion in itinerant electronic systems where the spin decouples from the orbital momenta and becomes an internal global symmetry. The lack of instabilities and the existence of a zero sound and diffusion mode indicates that cold holographic matter is closely described by an effective hydrodynamic theory.

Le thème central de la présente thèse est le contenu sémantique des quantificateurs logiques. Dans leur forme la plus simple, les quantificateurs permettent d’établir l’existence, ou la non-existence, d’individus répondant à une propriété. En tant que tels, ils incarnent la richesse et la complexité de la logique du premier ordre, par delà la logique propositionnelle. Nous contribuons à l’analyse sémantique des quantificateurs, du point de vue de la théorie de la dualité, dans trois domaines différents des mathématiques et de l’informatique théorique. D’une part, dans la théorie des langages formels à travers la logique sur les mots. D’autre part, dans la logique intuitionniste propositionnelle et dans l’étude de l’interpolation uniforme. Enfin, dans la topologie catégorique et dans la sémantique catégorique de la logique du premier ordre
The unifying theme of the thesis is the semantic meaning of logical quantifiers. In their basic form quantifiers allow to state theexistence, or non-existence, of individuals satisfying a property. As such, they encode the richness and the complexity of predicate logic, as opposed to propositional logic. We contribute to the semantic understanding of quantifiers, from the viewpoint of duality theory, in three different areas of mathematics and theoretical computer science. First, in formal language theory through the syntactic approach provided by logic on words. Second, in intuitionistic propositional logic and in the study of uniform interpolation. Third, in categorical topology and categorical semantics for predicate logic

Dans cette thèse, nous prouvons un théorème de dualité de Tannaka pour les (infini, 1)-catégories. La dualité classique de Tannaka est une dualité entre certains groupes et catégories monoïdales munies d'une structure particulière. La dualité de Tannaka supérieure renvoie, elle, à une dualité entre certains champs en groupes dérivés et certaines (infini, 1)-catégories monoïdales munies d'une structure particulière. Cette dualité supérieure est définie sur les anneaux dérivés et englobe la théorie de dualité classique. Nous comparons la dualité de Tannaka supérieure à la théorie de dualité de Tannaka classique et portons une attention particulière à la dualité de Tannaka sur les corps. Dans ce dernier cas, cette théorie a une relation étroite avec la théorie des types d'homotopie schématique de Toën. Nous décrivons également trois applications de la théorie : les complexes parfaits, les motifs et leur analogue non-commutatif dû à Kontsevich
In this thesis we prove a Tannaka duality theorem for (infini, 1)-categories. Classical Tannaka duality is a duality between certain groups and certain monoidal categories endowed with particular structure. Higher Tannaka duality refers to a duality between certain derived group stacks and certain monoidal (infini, 1)-categories endowed with particular structure. This higher duality theorem is defined over derived rings and subsumes the classical statement. We compare the higher Tannaka duality to the classical theory and pay particular attention to higher Tannaka duality over fields. In the later case this theory has a close relationship with the theory of schematic homotopy types of Toën. We also describe three applications of our theory : perfect complexes and that of both motives and its non-commutative ana­logue due to Kontsevich

Dans cette thèse, nous présentons des avancements récents en Théorie des Champs Doubles (TCD) et Théories des Champs Exceptionnels (TCE). Ces théories ont la particularité d’être des reformulations de supergravité dans lesquelles les symétries de dualité sont explicites avant toute réduction dimensionnelle. Ces reformulations se basent sur la définition d’un espace-temps étendu qui géométrise le groupe de T-dualité en TCD et les groupes exceptionnels de U-dualité en TCE. Tous les champs de cet espace sont soumis à une contrainte de section qui restreint leur dépendance en coordonnées. Il existe plusieurs solutions à la contrainte de section, qui correspondent donc à des théories différentes. Dans ce sens, le formalisme des théories des champs étendues amène à une unification de ces théories. De plus, grâce à un outil spécifique aux théories des champs étendues, l’ansatz de Scherk-Schwarz généralisé, il est possible de réécrire les ansatz compliqué de type Kaluza-Klein en supergravité sous une forme élégante et compacte: un produit matriciel en dimensions supérieures. Ici, nous présentons plusieurs exemples de l’efficacité de l’ansatz de Scherk-Schwarz généralisé. En particulier, nous prouvons deux conjectures concernant les troncations cohérentes: la réduction dite “de Pauli” de la corde bosonique ainsi que la supergravité de type IIB sur AdS5 x S5. La dernière application de cet ansatz concerne la théorie de type IIB généralisée, apparue récemment dans l’étude des système intégrables, et son plongement dans la TCE E6(6). Enfin, nous présentons la complétion supersymétrique de la TCE E8(8) bosonique
In this thesis, recent developments in Double Field Theory (DFT) and Exceptional Field Theory (EFT) are presented. They are reformulation of supergravity in which duality symmetries are made manifest before dimensional reduction. This is achieved through the definition of an extended spacetime that “geometrises” the T-duality group O(d,d) in DFT and exceptional U-duality groups in EFT. All functions on this extended space are subject to a covariant `section constraint', whose solutions then restrict the coordinates dependency of the fields. There exist different solutions to the section constraint that correspond to different theories. In this sense, different theories are unified within the formalism of extended field theories. Moreover, extended field theories possess a powerful tool to study compactifications: the generalised Scherk-Schwarz ansatz.Here, we present several examples of the effectiveness of the generalised Scherk-Schwarz ansatz. In particular, we proved two conjectures regarding consistent truncations: the so-called Pauli reduction of the bosonic string on group manifolds and type IIB supergravity on AdS5 x S5. Another application is presented on the embedding of generalised type IIB within the E6(6) EFT, which recently appeared in the study of integrable systems.Finally, we present the supersymmetric completion of the bosonic E8(8) EFT

Understanding the dynamics of strongly coupled gauge theories is one of the greatest challenges in modern theoretical physics. A new hope in attacking this problem was brought by the surprising discovery of integrability in a special four-dimensional gauge theory { the N = 4 supersymmetric Yang-Mills theory (SYM) in the limit of large number of colors. Quantum integrability manifests itself as a powerful hidden symmetry which allows to explore the theory far beyond the conventional perturbative regime, and may even lead to its exact solution. Integrability should also shed light on the striking gauge/string duality, which holographically relates N = 4 SYM with a string theory in curved geometry. In this thesis we focus on one of the key quantities in the N = 4 SYM theory { its spectrum of conformal dimensions, which correspond to string state energies. The study of integrability has culminated in reformulation of the spectral problem as a compact set of Riemann-Hilbert type equations known as the Quantum Spectral Curve (QSC). We demonstrate the power of this framework by applying it to study the spectrum in a wide variety of settings. The new methods which we present allow to explore previously unreachable regimes. We rst discuss an all-loop solution in a near-BPS limit, leading also to new strong coupling predictions. Next we describe an ecient numerical algorithm which allows to compute the nite-coupling spectrum with nearly unlimited precision (e.g. 60 digits in some important cases). We also present a universal analytic iterative method, which in particular allows to solve a longstanding open problem related to the BFKL limit in which N = 4 SYM develops close links with QCD. Finally we propose the extension of the QSC to the deformed case corresponding to a cusped Wilson line, uncovering new algebraic features of the construction. This allows to systematically study the generalized quark-antiquark potential and generate numerous new results.

Dualities evident in aspects of the visual world provide a framework for exploring how equivalence of landscape experience might be portrayed through the medium of paint. Research into micro/macro, past/present and here/there is driven by the text/image duality. This duality encompasses ekphrasis and gives rise to discussion on the process of making a visual but non-illustrative response to a poem. The writings of Gilles Deleuze (1925 – 1995) on Francis Bacon (1909 – 1992) are instrumental in investigating the function of paint to transmit feeling and act as a catalyst for the painting process itself. Each duality comprises two opposing or paired concepts or themes and in attempting to find accordance between each pair a tension is created that leads to a flow of ideas. This is exemplified by the micro/macro duality that invites comparison of small structures observed in rock fragments to large scale landscape formations and generates an imaginative response. Through juxtaposition of dualities new links between seemingly disparate subject matter are formed: geology and poetry, for instance, are brought together in a project that combines a particular rock type found in Cumbria with poems by Wordsworth (1770 – 1850). This provides a basis for autobiographical reflection on the here/there and past/present. Such juxtaposition acts to enrich the substance and scope of project material, forming a matrix of possibilities for further research. This process can be seen as a precursor towards a rhizomic approach, one where connections can be made between similar and dissimilar subjects with no barriers or hierarchical boundaries.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2008.
Includes bibliographical references (p. 179-194).
At present, there are no non-perturbative analytic methods available for investigating gauge theories at large couplings. Consequently, it is desirable to explore more avenues to gain qualitative and quantitative insights. The gauge/gravity (AdS/CFT) correspondences provides a unique opportunity to study gauge theories both at finite and zero temperatures in the strong coupling regime, potentially leading to insights into QCD. In this regime, where both the 't Hooft coupling and the number of colors is large, the dynamics of the gauge theory is described by super gravity solutions, which describe the low-energy limit of systems of D-branes. In this dissertation, I use the AdS/CFT correspondence to study the dispersion relations of mesons in a particular hot, strongly coupled, supersymmetric gauge theory plasma. This plasma arises by placing a probe D7-brane in the near-horizon geometry of non-extremal D3-branes. I find the large momentum dispersion relations of scalar mesons and extract from them their limiting velocity vo, which depends only on the ratio of the temperature to the quark mass. I use vo to find that the temperature above which no meson bound states with velocity v exist is Tdiss(V) - (1 - v2)1/4Tdiss(v = 0). This agrees with results inferred indirectly via analysis of the screening length between a static quark and anti quark in a moving plasma. Although these calculations are not done in QCD, I argue that the qualitative features of my results may apply to bottomonium and charmonium mesons propagating in the strongly coupled QCD plasma. To aid further investigations of the effect of fundamental matter on hot gauge theory plasmas I find the non-extremal, localized D2/D6-brane solution in its linear limit. Pursuing a different direction, I consider a stack of N D3-branes and M D5-branes wrapped at the apex of a cone over Sasaki- Einstein spaces yp,q. Replacing the D-branes by their fluxes, I construct asymptotic solutions for all p and q in the form of warped products of the cone and R" 3. These theories are not conformal, and the solutions describe cascading renormalization group flows.
by Qudsia Jabeen Ejaz.
Ph.D.