Дисертації з теми "Exact results"

In this thesis we discuss supersymmetric gauge theories, focusing on exact results achieved using methods of integrability. For the larger portion of the thesis we study the N = 4 super Yang-Mills theory in the planar limit, a recurring topic being the Konishi anomalous dimension, which is roughly the analogue for the mass of the proton in quantum chromodynamics. The N = 4 supersymmetric Yang-Mills theory is known to be integrable in the planar limit, which opens up a wealth of techniques one can employ in order to nd results in this limit valid at any value of the coupling. We begin with perturbation theory where the integrability of the theory rst manifests itself. Here we showcase the rst exact result, the so-called slope function, which is the linear small spin expansion coecient of the generalized Konishi anomalous dimension. We then move on to exact results mainly achieved using the novel quantum spectral curve approach, the method allowing one to nd scaling dimensions of operators at arbitrary values of the coupling. As an example we nd the second coecient in the small spin expansion after the slope, which we call the curvature function. This allows us to extract non-trivial information about the Konishi operator. Methods of integrability are also applicable to other supersymmetric gauge theories such as ABJM, which in fact shares many similarities with N = 4 super Yang-Mills. We briefly review these parallel developments in the last chapter of the thesis.

This thesis examines N = 2 Super-Yang-Mills theory where the low-energy effective action of the theory is governed by a holomorphic function called the prepotential. The Seiberg-Witten solution of the theory determines the prepotential in terms of an complex curve and, once we compactify the theory on a circle, we will examine the identification of this complex curve with the spectral curve of the Calogero-Moser integrable system. Since the supersymmetry restricts the perturbative contributions to the prepotential, the results we gain are exact. Further, they are independent of the compactification radius. The generalization to the quiver models, with gauge group SU(N)k, is introduced along with the spin generalization of the integrable system. The massive vacua of these theories have been determined previously, here we examine the case of a specific gauge group in order to determine the complete phase structure, including the massless vacua. We then move on to determining contributions coming from instantons to the prepotential of the theory with gauge group SU(N). We see how by lifting the theory onto 5 dimensions the functional integral on the instanton moduli space is realized as a quantum mechanical sigma-model with the moduli space as a target. However, just such a model is shown to calculate a particular index of the manifold, in this case a particular equivariant index since the space has isometries. We account for the non-compact nature of the moduli space by removing boundary terms and then calculate explicit results in the case of SU(2).

In this thesis we study the gauge/gravity duality and exact results in supersymmetric quantum field theories obtained using localization. We construct the gravity duals to a broad class of N = 2 supersymmetric gauge theories defined on a general class of three-manifolds. The gravity backgrounds are supersymmetric solutions of gauged four-dimensional supergravity and encompass all known examples of such solutions. We find that the holographically renormalized on-shell action agrees with the free energy of the field theory, which has previously been computed via localization of the partition function. Next, we study the Casimir energy of four-dimensional N = 1 supersymmetric field theories in the context of the rigid limit of new minimal supergravity. We revisit the computation of the localized partition function on S1 S3, and consider the same theories in the Hamiltonian formalism on R S3. We compute the vacuum expectation value of the canonical Hamiltonian using zeta function regularization, and show that this interpolates between the supersymmetric Casimir energy and the ordinary Casimir energy of supersymmetric field theories. In general, the Casimir energy depends on the regularization scheme and is therefore ambiguous. However, we show that for N = 1 supersymmetric field theories on the cylinder R S3, the supersymmetric Casimir energy is well-defined and scheme-independent, provided the regularization scheme preserves supersymmetry. Finally, we investigate the gravity duals of such N = 1 theories on R S3. Specifically, we study supersymmetric solutions of five-dimensional minimal gauged supergravity using a known classification. We propose an ansatz based on a four-dimensional local orthotic Kahler metric and reduce the problem to a single sixth-order equation for two functions, each of one variable. We find an analytic, asymptotically locally AdS solution comprising five parameters. For a conformally at boundary, this reduces to a previously known solution with three parameters, representing the most general solution of this type known in minimal gauged supergravity. We discuss the relevance for this solution to account for the supersymmetric Casimir energy, finding the answer to be in the negative.

The holographic duality between gauge theories and string theories has opened a new door to access the strongly coupled regime of quantum field theories and offers, at the same time, a completely new way to understand the elusive nature of quantum gravity and the non-perturbative regime of string theory. After almost two decades of research, the current status of the correspondence is that of a solid conjecture that has passed a great number of nontrivial tests, to the point that it is generally believed to be true. The present thesis includes a collection of four papers published in peer-reviewed scientific journals, all of them in the context of the AdS/CFT correspondence and with a particular focus on studying gauge theories by inserting heavy external probes, following prescribed trajectories and transforming under various representations of the gauge group. Each of these works reports a little step forward in the development of new strategies for capturing correc- tions beyond the leading order as well as in using exact results available in quantum field theory in order to derive exact expressions for other relevant observables and new non-trivial string theory predictions. In chapters 2 and 3 we use the AdS/CFT correspondence in order to compute several observables of N = 4 SU (N ) super Yang-Mills theory related with the presence of an infinitely heavy particle transforming in the k-symmetric or the k-antisymmetric representations of the gauge group and following particular trajectories. This is achieved by means of adding certain D-brane probes with electric fluxes turned on and reaching the boundary of AdS on the very trajectories followed by the dual particles. For the antisymmetric case we consider D5-branes reaching the boundary at arbitrary time-like trajectories, while for the symmetric case, we consider a D3-brane fully embedded in AdS5 that reaches the boundary at either a straight line or a hyperbola. This generalizes previous computations that used fundamental strings, which are claimed to be dual to infinitely heavy point particles transforming in the fundamental. Besides the intrinsic interest of these generalizations, our main motivation in studying them is that, as it happens in the computation of certain Wilson loops, the results obtained with D3-branes give an all- orders series of corrections in 1/N to the leading order result for the fundamental representation obtained by means of fundamental strings. It is important to remark, one more time, that we can not really extrapolate up to k = 1, since this is beyond the regime of validity of the supergravity approximation. Therefore, it is not justified a priori to set k = 1 in our results. Nevertheless, when compared with the exact results available, we find that the D3-brane computation reproduces the correct result in the large N , λ limit and with k = 1. This better than expected performance suggests the exciting possibility that certain D3-branes with electric fluxes might capture correctly all the 1/N corrections, but it is fair to say that we still lack of a precise string-theoretic argument to prove this.
Durant les darreres dues dècades ha aparegut un nou paradigma que permet reformular completament certes teories quàntiques de camps i ens aporta una nova eina que ens permet realitzar càlculs analítics en règims fins ara inaccessibles. Aquest nou paradigma sorgeix del descobriment d’una correspondència o dualitat exacta entre dues teories aparentment molt diferents. Per una banda de la dualitat tenim certes teories quàntiques de camps, com per exemple les denominades teories de Yang-Mills, similars a les teories del Model Estàndard. Aquestes descriuen partícules interactuant en un espai pla d-dimensional sense gravetat. A l’altra banda de la dualitat trobem teories que inclouen la gravetat, com ara la Teoria de la Relativitat General d’Einstein o les seves generalitzacions en el marc de la Teoria de Cordes. Aquestes teories de gravetat estan definides sobre espais de dimensió més alta que d, i és per això que aquesta correspondència rep sovint l’adjectiu de “hologràfica”. Depenent del context, aquesta rep el nom de dualitat gauge/gravetat, dualitat gauge/corda o AdS/CFT (acrònim anglès per la correspondència particular entre teoria de cordes a espais d’Anti-de Sitter i teories de camps conformes). Fins ara, una de les correspondències més ben estudiades i que comprenem millor (i sobre la qual es centra la present tesi) és la dualitat entre la teoria quatre-dimensional N = 4 super Yang-Mills amb grup de gauge SU (N ) i teoria de cordes tipus IIB en un espai deu-dimensional AdS5 × S5 . Aquesta tesi presenta una recopilació de quatre articles publicats en revistes científiques d’alt impacte, tots ells en el camp de la correspondència AdS/CFT i centrats en l’estudi de teories gauge supersimètriques mitjançant la inserció de partícules de prova infinitament massives, seguint trajectòries determinades i transformant sota diverses representacions del grup de gauge. Cadascun d’aquests treballs aporta un pas endavant en el desenvolupament de noves estratègies per calcular correccions més enllà del primer ordre així com en l’ús de resultats exactes accessibles a la Teoria Quàntica de Camps per tal de derivar expressions exactes d’altres observables rellevants de la teoria i realitzar prediccions de Teoria de Cordes.

The Interacting Resonant Level Model (IRLM) is studied by making use of different standard techniques common to the study strongly correlated quantum systems. Exact results are computed both analytically and numerically, in order to obtain a full consistent theory of the model in equilibrium thermodynamics, opening the path towards a future treatment of the model.

In this thesis, certain exact results in supersymmetric gauge theories are discussed. In these theories, holomorphic gauge invariant operators play a central role in understanding the structure of the space of solutions to vacuum equations, known as the moduli space. We focus on a technique to count such operators with various quantum numbers. The counting can be done by computing a partition function, known as the Hilbert series, which counts all holomorphic gauge invariant operators carrying a speci ed set of global U(1) charges. The Hilbert series can be computed exactly for various gauge theories. In Part I of this thesis, we compute the Hilbert series of four dimensional N = 1 supersymmetric QCD with classical gauge groups. In part II, we count chiral operators on the one instanton moduli space on R4 and study the hypermultiplet moduli spaces of a large class of N = 2 supersymmetric gauge theories in four dimensions. We demonstrate that the Hilbert series not only contains information about the spectrum of operators in the theory, but it also carries geometrical properties of the moduli space, e.g. the dimension. It is also an indicator of whether the moduli space is Calabi-Yau. Moreover, Hilbert series can be used as a primary tool to test various dualities in gauge theories and in string theory.

The study of supersymmetry has led us to a better understanding of field theories, specially in the strong coupling regime. In this thesis we have tried to show this through several examples. These are: - The first of these examples has been the application of localization techniques in supersymmetric theories. Specifically, we have used the partition function of N=2 supersymmetric Chern-Simons theory with gauge group U(N) and 2Nf flavors. To regularize the theory, it is necessary to make the computation in a three sphere whose radius, R, serves as an IR regulator which can be taken to infinity at the end of the computation. Once we have the exact partition function in terms of a matrix integral, we can solve the integral by means of a saddle-point approximation. This approximation becomes exact in the large N limit. The saddle-point equation can be solved exactly and in the decompactification limit it shows different phases depending on the value of the 't Hooft coupling. We have also computed the free energy and the vacuum expectation value of a Wilson loop for a big circle of the three sphere. Both of them show discontinuities in their derivatives, in particular, the discontinuity in the free energy appears in the third derivative and thus, both phase transitions are third order. - Other application that we have seen consists of the use of the gauge/gravity duality. In particular, starting from the gravity dual to N=1 super Yang-Mills, proposed by Maldacena and Núñez, we have reviewed how to add flavors (quarks) to this theory, without mass first and with mass later. We have also seen how to extract information about the field theory from these gravity duals, we have paid special attention to how the beta-function of the field theory dual is obtained from the gravity background proposed by Conde, Gaillard and Ramallo, dual to N=1 super Yang-Mills field theory with Nf massive flavors and a quartic superpotential. The main result from the point of view of the field theory is that, in the case Nf=2N, the beta-function shows a non-trivial UV fixed point, which hints on possible IR fixed point as proposed by Seiberg in the conformal window picture. No evidence of non-trivial fixed points is found for Nf different from 2N. - Again, in the context of the gauge/gravity duality, we have studied how to generate new supergravity solutions applying T-duality and how this affects the G-structures that describe the supersymmetry of these solutions. We have applied T-duality to the IIB supergravity solution of Klebanov and Witten with flavors. The supersymmetry of these backgrounds can be described by an SU(3)-structure and an SU(2)-structure before and after T-dualizing, respectively. - Finally, we have presented an N=1 supersymmetric model that exhibits a superconducting phase transition. This model is based on a quartic Kähler potential for a chiral multiplet and no superpotential. The main difference with standard superconductivity is that the phase transition becomes first order rather than second order. Another difference is that, as it is typical in supersymmetric theories, the dependence on the cut-off is softened in our model.
El estudio de supersimetría nos ha permitido un mejor entendimiento de las teorías de campos, especialmente en el régimen de acoplamiento fuerte. En esta tesis hemos tratado de mostrar esto a través de varios ejemplos. A saber: - Primero hemos mostrado cómo calcular de manera exacta con técnicas de localización la función de partición de la teoría de Chern-Simons supersimétrica N=2 con grupo gauge U(N) y 2Nf sabores en una tres esfera. Una vez que tenemos la función de partición exacta en términos de una integral de matrices, podemos resolverla por medio de la aproximación de punto silla. Esta aproximación se vuelve exacta cuando tomamos el límite de N grande. En el límite de descompactificación calculamos la energía libre y el valor de expectación de un lazo de Wilson correspondiente a un círculo máximo de la tres-esfera, mostrando transiciones de fase de tercer orden. - Otra aplicación ha consistido en el uso de la dualidad gravedad/gauge. Hemos revisado como construir una solución de supergravedad dual a la teoría N=1 super Yang-Mills con Nf sabores masivos y un superpotencial cuártico. El principal resultado desde el punto de vista de la teoría de campos que corresponde a dicha solución, consiste en que, en el caso Nf=2N, la función beta exhibe un punto fijo UV no trivial que da lugar a indicios acerca de un punto fijo IR adicional tal y como propone Seiberg en su propuesta de la ventana conforme. - También en el contexto de la dualidad gravedad/gauge hemos estudiado cómo generar nuevas soluciones de supergravedad por medio de la aplicación de T-dualidad a la solución de supergravedad de Klebanov y Witten con sabores. La supersimetría de estas soluciones pueden ser descritas en términos de una SU(3) estructura antes de T-dualizar, o una SU(2)-estructura, después de T-dualizar. - Finalmente, hemos presentado un modelo supersimétrico que exhibe una transición de fase superconductora basado en un potencial de Kähler cuártico para un multiplete quiral y sin superpotencial. Encontramos dos diferencias con superconductores usuales: nuestro modelo exhibe una transición de fase de primer orden, en lugar de ser de segundo orden, y la dependencia con la energía de corte es más suave.

Gauge theories are one of the corner stones of modern theoretical physics. They describe the nature of all fundamental interactions and have been applied in multiple branches of physics. The most challenging problem of gauge theories, which has not been solved yet, is their strong coupling dynamics. A class of gauge theories that admits simplifications allowing to deal with the strong coupling regime are supersymmetric ones. For example, recently proposed method of supersymmetric localization allows to reduce expectation values of supersymmetric observables, expressed through the path integral, to finite-dimensional matrix integral. The last one is usually easier to deal with compared to the original infinite-dimensional integral. This thesis deals with the matrix models obtained from the localization of different 5D gauge theories. The focus of our study is N=1 super Yang-Mills theory with different matter content as well as N=1 Chern-Simons-Matter theory with adjoint hypermultiplets. Both theories are considered on the five-spheres. We make use of the saddle-point approximation of the matrix integrals, obtained from localization, to evaluate expectation values of different observables in these theories. This approximation corresponds to the large-N limit of the localized gauge theory. We derive behavior for the free energy of 5D N=1* super Yang-Mills theory at strong coupling. This result is important in light of the relation between 5D theory and the world-volume theories of M5-branes, playing a significant role in string theory. We have also explored rich phase structure of 5D SU(N) N=1 super Yang-Mills theory coupled to massive matter in different representations of the gauge group. We have shown that in the case of the massive adjoint hypermultiplet theory undergoes infinite chain of the third order phase transitions while interpolating between weak and strong coupling in the decompactification limit. Finally, we obtain several interesting results for 5D Chern-Simons theory, suggesting existence of the holographic duals to this theory. In particular, we derive  behavior of the free energy of this theory, which reproduces the behavior of the free energy for 5D theories with known  holographic duals.

Mathematical models for the spread of infectious diseases between living organisms are crucial to humanity's endeavour to understand and control its environment. The threat posed by communicable diseases is great. For example, the 1918 flu pandemic resulted in the deaths of over 50 million people and the HIV/AIDS pandemic is still under way with 2.3 million new cases in 2012. Mathematical models allow us to make predictions about the likelihood, impact and time scale of possible epidemics, and to devise effective intervention programmes, e.g. mass vaccination. This thesis considers various stochastic models of disease propagation which utilise the concept of a finite contact (social) network. For such models, we investigate ways in which important information can be extracted without a full mathematical `solution' (often unavailable) or numerous time consuming simulations (often inefficient and uninformative). For example, we consider the probability that a large scale outbreak will occur when a single infected individual is introduced to a susceptible population, and the expected number of infected individuals at time t. Although we focus on the context of epidemiology, the models under investigation are elementary and will be applicable to other domains, such as the spread of computer viruses, the spread of ideas, chemical reactions, and interacting particle systems.

In this thesis I show a triple new connection we found between quantum integrability, N=2 supersymmetric gauge theories and black holes perturbation theory. I use the approach of the ODE/IM correspondence between Ordinary Differential Equations (ODE) and Integrable Models (IM), first to connect basic integrability functions - the Baxter’s Q, T and Y functions - to the gauge theory periods. This fundamental identification allows several new results for both theories, for example: an exact non linear integral equation (Thermodynamic Bethe Ansatz, TBA) for the gauge periods; an interpretation of the integrability functional relations as new exact R-symmetry relations for the periods; new formulas for the local integrals of motion in terms of gauge periods. This I develop in all details at least for the SU(2) gauge theory with Nf=0,1,2 matter flavours. Still through to the ODE/IM correspondence, I connect the mathematically precise definition of quasinormal modes of black holes (having an important role in gravitational waves’ obervations) with quantization conditions on the Q, Y functions. In this way I also give a mathematical explanation of the recently found connection between quasinormal modes and N=2 supersymmetric gauge theories. Moreover, it follows a new simple and effective method to numerically compute the quasinormal modes - the TBA - which I compare with other standard methods. The spacetimes for which I show these in all details are in the simplest Nf=0 case the D3 brane in the Nf=1,2 case a generalization of extremal Reissner-Nordström (charged) black holes. Then I begin treating also the Nf=3,4 theories and argue on how our integrability-gauge-gravity correspondence can generalize to other types of black holes in either asymptotically flat (Nf=3) or Anti-de-Sitter (Nf=4) spacetime. Finally I begin to show the extension to a 4-fold correspondence with also Conformal Field Theory (CFT), through the renowned AdS/CFT correspondence.

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

This thesis deals with superconformal and supersymmetric field theories on curved spaces with a view toward applications to holography and localisation. It contains two fairly different parts. In the first (and main) part we classify Euclidean and Lorentzian four-manifolds with some preserved N=1,2 supersymmetry, and Euclidean three-manifolds with some preserved N=2 supersymmetry. We take a holographic approach, starting with manifolds that preserve superconformal symmetries. Preserved supersymmetry for asymptotically locally AdS solutions implies the existence of a certain (generalised) ``conformal Killing spinor'' on the boundary. In the non-conformal case a closely related spinor exists, which also will be discussed. In this thesis we classify the manifolds in three and four dimensions that admit such spinors. In particular we find for the case with four supercharges that supersymmetry can be preserved in four dimensions on every Euclidean complex manifold and on any Lorentzian space-time with a null conformal Killing vector. In three Euclidean dimensions we find a condition very similar to complexity in four dimensions. When the field theory has eight supercharges, supersymmetry can generically be preserved on manifolds with time-like conformal Killing vectors; there are singular cases depending on the signature, in the Lorentzian there is a degenerate case reducing to the N=1 analysis, in the Euclidean there is a degenerate case corresponding to the topological twist. The supersymmetric curvature couplings are systematically understood in the rigid limit of supergravity. We give explicit formulae for the background fields that one needs to turn on in order to preserve some supersymmetry. Putting supersymmetric field theories on curved manifolds has led to interesting results over the past years. In the second part of the thesis we analyse a matrix model for the partition function of three dimensional field theories on S^3, which was obtained by supersymmetric localisation. In the large N limit one can evaluate the matrix model, allowing us to perform a non-classical and non-perturbative check of the AdS_4/CFT_3 correspondence and Seiberg duality. In particular, we compute the large-N free energy of various three dimensional quiver gauge theories with arbitrary R-charges, which are dual to M-theory on Sasaki-Einstein seven-manifolds. In particular, we check that the free energy functional depending on the R-charges is minimised for the exact R-symmetry, an extremisation that is dual to the volume-minimisation of the Sasaki-Einstein manifold in the gravity sector.

L'agrégation de classements consiste à établir un consensus entre un ensemble de classements (éléments ordonnés). Bien que ce problème ait de très nombreuses applications (consensus entre les votes d'utilisateurs, consensus entre des résultats ordonnés différemment par divers moteurs de recherche...), calculer un consensus exact est rarement faisable dans les cas d'applications réels (problème NP-difficile). De nombreux algorithmes d'approximation et heuristiques ont donc été conçus. Néanmoins, leurs performances (en temps et en qualité de résultat produit) sont très différentes et dépendent des jeux de données à agréger. Plusieurs études ont cherché à comparer ces algorithmes mais celles-ci n’ont généralement pas considéré le cas (pourtant courant dans les jeux de données réels) des égalités entre éléments dans les classements (éléments classés au même rang). Choisir un algorithme de consensus adéquat vis-à-vis d'un jeu de données est donc un problème particulièrement important à étudier (grand nombre d’applications) et c’est un problème ouvert au sens où aucune des études existantes ne permet d’y répondre. Plus formellement, un consensus de classements est un classement qui minimise le somme des distances entre ce consensus et chacun des classements en entrés. Nous avons considérés (comme une grande partie de l’état-de-art) la distance de Kendall-Tau généralisée, ainsi que des variantes, dans nos études. Plus précisément, cette thèse comporte trois contributions. Premièrement, nous proposons de nouveaux résultats de complexité associés aux cas que l'on rencontre dans les données réelles où les classements peuvent être incomplets et où plusieurs éléments peuvent être classés à égalité. Nous isolons les différents « paramètres » qui peuvent expliquer les variations au niveau des résultats produits par les algorithmes d’agrégation (par exemple, utilisation de la distance de Kendall-Tau généralisée ou de variantes, d’un pré-traitement des jeux de données par unification ou projection). Nous proposons un guide pour caractériser le contexte et le besoin d’un utilisateur afin de le guider dans le choix à la fois d’un pré-traitement de ses données mais aussi de la distance à choisir pour calculer le consensus. Nous proposons finalement une adaptation des algorithmes existants à ce nouveau contexte. Deuxièmement, nous évaluons ces algorithmes sur un ensemble important et varié de jeux de données à la fois réels et synthétiques reproduisant des caractéristiques réelles telles que similarité entre classements, la présence d'égalités, et différents pré-traitements. Cette large évaluation passe par la proposition d’une nouvelle méthode pour générer des données synthétiques avec similarités basée sur une modélisation en chaîne Markovienne. Cette évaluation a permis d'isoler les caractéristiques des jeux de données ayant un impact sur les performances des algorithmes d'agrégation et de concevoir un guide pour caractériser le besoin d'un utilisateur et le conseiller dans le choix de l'algorithme à privilégier. Une plateforme web permettant de reproduire et étendre ces analyses effectuée est disponible (rank-aggregation-with-ties.lri.fr). Enfin, nous démontrons l'intérêt d'utiliser l'approche d'agrégation de classements dans deux cas d'utilisation. Nous proposons un outil reformulant à-la-volé des requêtes textuelles d'utilisateur grâce à des terminologies biomédicales, pour ensuite interroger de bases de données biologiques, et finalement produire un consensus des résultats obtenus pour chaque reformulation (conqur-bio.lri.fr). Nous comparons l'outil à la plateforme de références et montrons une amélioration nette des résultats en qualité. Nous calculons aussi des consensus entre liste de workflows établie par des experts dans le contexte de la similarité entre workflows scientifiques. Nous observons que les consensus calculés sont très en accord avec les utilisateurs dans une large proportion de cas
The rank aggregation problem is to build consensus among a set of rankings (ordered elements). Although this problem has numerous applications (consensus among user votes, consensus between results ordered differently by different search engines ...), computing an optimal consensus is rarely feasible in cases of real applications (problem NP-Hard). Many approximation algorithms and heuristics were therefore designed. However, their performance (time and quality of product loss) are quite different and depend on the datasets to be aggregated. Several studies have compared these algorithms but they have generally not considered the case (yet common in real datasets) that elements can be tied in rankings (elements at the same rank). Choosing a consensus algorithm for a given dataset is therefore a particularly important issue to be studied (many applications) and it is an open problem in the sense that none of the existing studies address it. More formally, a consensus ranking is a ranking that minimizes the sum of the distances between this consensus and the input rankings. Like much of the state-of-art, we have considered in our studies the generalized Kendall-Tau distance, and variants. Specifically, this thesis has three contributions. First, we propose new complexity results associated with cases encountered in the actual data that rankings may be incomplete and where multiple items can be classified equally (ties). We isolate the different "features" that can explain variations in the results produced by the aggregation algorithms (for example, using the generalized distance of Kendall-Tau or variants, pre-processing the datasets with unification or projection). We propose a guide to characterize the context and the need of a user to guide him into the choice of both a pre-treatment of its datasets but also the distance to choose to calculate the consensus. We finally adapt existing algorithms to this new context. Second, we evaluate these algorithms on a large and varied set of datasets both real and synthetic reproducing actual features such as similarity between rankings, the presence of ties and different pre-treatments. This large evaluation comes with the proposal of a new method to generate synthetic data with similarities based on a Markov chain modeling. This evaluation led to the isolation of datasets features that impact the performance of the aggregation algorithms, and to design a guide to characterize the needs of a user and advise him in the choice of the algorithm to be use. A web platform to replicate and extend these analyzes is available (rank-aggregation-with-ties.lri.fr). Finally, we demonstrate the value of using the rankings aggregation approach in two use cases. We provide a tool to reformulating the text user queries through biomedical terminologies, to then query biological databases, and ultimately produce a consensus of results obtained for each reformulation (conqur-bio.lri.fr). We compare the results to the references platform and show a clear improvement in quality results. We also calculate consensus between list of workflows established by experts in the context of similarity between scientific workflows. We note that the computed consensus agree with the expert in a very large majority of cases

Nous obtenons des solutions exactes pour un systeme de physique statistique hors d'equilibre, le modele d'exclusion asymetrique. A l'aide d'une representation operatorielle, nous pouvons calculer exactement la mesure stationnaire de certains modeles qui presentent une transition de phase a la limite thermodynamique. La meme technique nous permet de calculer les fluctuations de grandeurs physiques dans l'etat stationnaire, comme la constante de diffusion d'un traceur, ou la variance du courant a travers un lien. A l'aide d'un passage a la limite continue, nous deduisons de nos calculs une fonction d'echelle pour l'equation de kardar, parisi et zhang, qui decrit la croissance aleatoire d'une surface. Nous exposons enfin les liens entre notre methode et la theorie des systemes integrables.

L'estimation non parametrique adaptative de la densite de probabilite de n observations independantes, identiquement distribuees fait l'objet de cette these. Dans un premier chapitre, nous introduisons le cadre minimax asymptotique d'estimation et, en particulier, l'estimation adaptative. L'adaptivite nous permet d'etudier et de comparer a l'aide d'un critere d'appreciation fixe, le comportement asymptotique des estimateurs, sur une reunion de classes tres larges de fonctions. Le deuxieme chapitre presente le calcul de deux vitesses adaptatives de convergence sur deux classes de type l p sobolev. Dans le cas des classes l 2 sobolev, le troisieme chapitre demontre un resultat de constante exacte, ou le risque adaptatif, asymptotique est explicitment calcule, ainsi que la procedure d'estimation asymptotiquement efficace qui nous permet d'atteindre ce risque. Le dernier chapitre etudie cette procedure exacte de point de vue applique, a l'aide de simulations numeriques.

On presente une solution exacte d'un modele de matrice de gravite quantique bi-dimensionelle avec un terme d'interaction proportionelle au carre de la courbure scalaire. Un reseau regulier correspond a une metrique plate. Les defauts du reseau genere une courbure locale. Pour toute constante de couplage finie, la limite infra-rouge de la theorie est decrite par la gravite pure quantique. Dans la limite ou la constante de couplage tend vers l'infinie on trouve une fonction de partition qui interpole entre la gravite pure quantique et un gaz dilue de defauts de courbure sur un fond d'espace plat. Pour resoudre ce modele on developpe une puissante methode qui s'applique aux modeles de matrices de la gravite quantique et aux theories de jauge a deux dimensions. La methode permet de calculer certaines sommes d'interet sur les representations du groupe u(n) et leur comportement pour n grand. En particulier on determine la forme asymptotique pour n grand de la derivee logarithmique des caracteres. Une introduction simple a cette methode est fournie par l'etude de la qcd a deux dimensions sur le disque. On analyse la transition de phase et on determine sa dependance vis-a-vis de l'holonomie du bord.

Doutoramento em Bioquímica
Os polissacarídeos são os componentes maioritários dos grãos de café verde e torrado e da bebida de café. Os mais abundantes são as galactomananas, seguindo-se as arabinogalactanas. Durante o processo de torra, as galactomananas e arabinogalactanas sofrem modificações estruturais, as quais estão longe de estar completamente elucidadas devido à sua diversidade e à complexidade estrutural dos compostos formados. Durante o processo de torra, as galactomananas e arabinogalactanas reagem com proteínas, ácidos clorogénicos e sacarose, originando compostos castanhos de alto peso molecular contendo nitrogénio, designados de melanoidinas. As melanoidinas do café apresentam diversas atividades biológicas e efeitos benéficos para a saúde. No entanto, a sua estrutura exata e os mecanismos envolvidos na sua formação permanecem desconhecidos, bem como a relação estrutura-atividade biológica. A utilização de sistemas modelo e a análise por espectrometria de massa permitem obter uma visão global e, simultaneamente, detalhada das modificações estruturais nos polissacarídeos do café promovidas pela torra, contribuindo para a elucidação das estruturas e mecanismos de formação das melanoidinas. Com base nesta tese, oligossacarídeos estruturalmente relacionados com a cadeia principal das galactomananas, (β1→4)-Dmanotriose (Man3), e as cadeias laterais das arabinogalactanas, (α1→5)-Larabinotriose (Ara3), isoladamente ou em misturas com ácido 5-Ocafeoilquínico (5-CQA), o ácido clorogénico mais abundante nos grãos de café verde, e péptidos compostos por tirosina e leucina, usados como modelos das proteínas, foram sujeitos a tratamento térmico a seco, mimetizando o processo de torra. A oxidação induzida por radicais hidroxilo (HO•) foi também estudada, uma vez que estes radicais parecem estar envolvidos na modificação dos polissacarídeos durante a torra. A identificação das modificações estruturais induzidas por tratamento térmico e oxidativo dos compostos modelo foi feita por estratégias analíticas baseadas principalmente em espectrometria de massa, mas também em cromatografia líquida. A cromatografia de gás foi usada na análise de açúcares neutros e ligações glicosídicas. Para validar as conclusões obtidas com os compostos modelo, foram também analisadas amostras de polissacarídeos do café obtidas a partir de resíduo de café e café instantâneo. Os resultados obtidos a partir dos oligossacarídeos modelo quando submetidos a tratamento térmico (seco), assim como à oxidação induzida por HO• (em solução), indicam a ocorrência de despolimerização, o que está de acordo com estudos anteriores que reportam a despolimerização das galactomananas e arabinogalactanas do café durante a torra. Foram ainda identificados outros compostos resultantes da quebra do anel de açúcares formados durante o tratamento térmico e oxidativo da Ara3. Por outro lado, o tratamento térmico a seco dos oligossacarídeos modelo (individualmente ou quando misturados) promoveu a formação de oligossacarídeos com um maior grau de polimerização, e também polissacarídeos com novos tipos de ligações glicosídicas, evidenciando a ocorrência de polimerização através reações de transglicosilação não enzimática induzidas por tratamento térmico a seco. As reações de transglicosilação induzidas por tratamento térmico a seco podem ocorrer entre resíduos de açúcares provenientes da mesma origem, mas também de origens diferentes com formação de estruturas híbridas, contendo arabinose e manose como observado nos casos dos compostos modelo usados. Os resultados obtidos a partir de amostras do resíduo de café e de café instantâneo sugerem a presença de polissacarídeos híbridos nestas amostras de café processado, corroborando a ocorrência de transglicosilação durante o processo de torra. Além disso, o estudo de misturas contendo diferentes proporções de cada oligossacarídeo modelo, mimetizando regiões do grão de café com composição distinta em polissacarídeos, sujeitos a diferentes períodos de tratamento térmico, permitiu inferir que diferentes estruturas híbridas e não híbridas podem ser formadas a partir das arabinogalactanas e galactomananas, dependendo da sua distribuição nas paredes celulares do grão e das condições de torra. Estes resultados podem explicar a heterogeneidade de estruturas de melanoidinas formadas durante a torra do café. Os resultados obtidos a partir de misturas modelo contendo um oligossacarídeo (Ara3 ou Man3) e 5-CQA sujeitas a tratamento térmico a seco, assim como de amostras provenientes do resíduo de café, mostraram a formação de compostos híbridos compostos por moléculas de CQA ligadas covalentemente a um número variável de resíduos de açúcar. Além disso, os resultados obtidos a partir da mistura contendo Man3 e 5-CQA mostraram que o CQA atua como catalisador das reações de transglicosilação. Por outro lado, nas misturas modelo contendo um péptido, mesmo contendo também 5-CQA e sujeitas ao mesmo tratamento, observou-se uma diminuição na extensão das reações transglicosilação. Este resultado pode explicar a baixa extensão das reações de transglicosilação não enzimáticas durante a torra nas regiões do grão de café mais ricas em proteínas, apesar dos polissacarídeos serem os componentes maioritários dos grãos de café. A diminuição das reações de transglicosilação na presença de péptidos/proteínas pode dever-se ao facto de os resíduos de açúcares redutores reagirem preferencialmente com os grupos amina de péptidos/proteínas por reação de Maillard, diminuindo o número de resíduos de açúcares redutores disponíveis para as reações de transglicosilação. Além dos compostos já descritos, uma diversidade de outros compostos foram formados a partir dos sistemas modelo, nomeadamente derivados de desidratação formados durante o tratamento térmico a seco. Em conclusão, a tipificação das modificações estruturais promovidas pela torra nos polissacarídeos do café abre o caminho para a compreensão dos mecanismos de formação das melanoidinas e da relação estrutura-atividade destes compostos.
Polysaccharides are the major components of green and roasted coffee beans, and coffee brew. The most abundant ones are galactomannans, followed by arabinogalactans. During the roasting process, galactomannans and arabinogalactans undergo structural modifications that are far to be completely elucidated due to their diversity and complexity of the compounds formed. During the roasting process, galactomannans and arabinogalactans react with proteins, chlorogenic acids, and sucrose, originating high molecular weight brown compounds containing nitrogen, known as melanoidins. Several biological activities and beneficial health effects have been attributed to coffee melanoidins. However, their exact structures and the mechanisms involved in their formation remain unknown, as well as the structure-biological activity relationship. The use of model systems and mass spectrometry analysis allow to obtain an overall view and, simultaneously, detailed, of the structural modifications in coffee polysaccharides promoted by roasting, contributing to the elucidation of the structures and formation mechanisms of melanoidins. Based on this thesis, oligosaccharides structurally related to the backbone of galactomannans, (β1→4)-D-mannotriose, and the side chains of arabinogalactans, (α1→5)-Larabinotriose, alone or in mixtures with 5-O-caffeoylquinic acid, the most abundant chlorogenic acid in green coffee beans, and dipeptides composed by tyrosine and leucine, used as models of proteins, were submitted to dry thermal treatments, mimicking the coffee roasting process. The oxidation induced by hydroxyl radicals (HO•) was also studied, since these radicals seem to be involved in the modification of the polysaccharides during roasting. The identification of the structural modifications induced by thermal and oxidative treatment of the model compounds was performed mostly by mass spectrometry-based analytical strategies, but also using liquid chromatography. Gas chromatography was used in the analysis of neutral sugars and glycosidic linkages. To validate the conclusions achieved with the model compounds, coffee polysaccharide samples obtained from spent coffee grounds and instant coffee were also analysed. The results obtained from the model oligosaccharides when submitted to thermal treatment (dry) or oxidation induced by HO• (in solution) indicate the occurrence of depolymerization, which is in line with previous studies reporting the depolymerization of coffee galactomannans and arabinogalactans during roasting. Compounds resulting from sugar ring cleavage were also formed during thermal treatment and oxidative treatment of Ara3. On the other hand, the dry thermal treatment of the model oligosaccharides (alone or when mixed) promoted the formation of oligosaccharides with a higher degree of polymerization, and also polysaccharides with new type of glycosidic linkages, evidencing the occurrence of polymerization via non-enzymatic transglycosylation reactions induced by dry thermal treatment. The transglycosylation reactions induced by dry thermal treatment can occur between sugar residues from the same origin, but also of different origins, with formation of hybrid structures, containing arabinose and mannose in the case of the model compounds used. The results obtained from spent coffee grounds and instant coffee samples suggest the presence of hybrid polysaccharides in these processed coffee samples, corroborating the occurrence of transglycosylation during the roasting process. Furthermore, the study of mixtures containing different proportions of each model oligosaccharide, mimicking coffee bean regions with distinct polysaccharide composition, subjected to different periods of thermal treatment, allowed to infer that different hybrid and non-hybrid structures may be formed from arabinogalactans and galactomannans, depending on their distribution in the bean cell walls and on roasting conditions. These results may explain the heterogeneity of melanoidins structures formed during coffee roasting. The results obtained from model mixtures containing an oligosaccharide (Ara3 or Man3) and 5-CQA and subjected to dry thermal treatment, as well as samples derived from spent coffee grounds, showed the formation of hybrid compounds composed by CQA molecules covalently linked to a variable number of sugar residues. Moreover, the results obtained from the mixture containing Man3 and 5-CQA showed that CQA acts as catalyst of transglycosylation reactions. On the other hand, in the model mixtures containing a peptide, even if containing 5-CQA and subjected to the same treatment, it was observed a decrease in the extent of transglycosylation reactions. This outcome can explain the low extent of non-enzymatic transglycosylation reactions during roasting in coffee bean regions enriched in proteins, although polysaccharides are the major components of the coffee beans. The decrease of transglycosylation reactions in the presence of peptides/proteins can be related with the preferential reactivity of reducing residues with the amino groups of peptides/proteins by Maillard reaction, decreasing the number of reducing residues available to be directly involved in the transglycosylation reactions. In addition to the compounds already described, a diversity of other compounds were formed from model systems, namely dehydrated derivatives formed during dry thermal treatment. In conclusion, the identification of the structural modifications in coffee polysaccharides promoted by roasting pave the way to the understanding of the mechanisms of formation of melanoidins and structure-activity relationship of these compounds.

碩士
國立暨南國際大學
資訊工程學系
96
In this paper, we introduce two algorithms for stringology. One is to solve the exact string matching problem and the other is to find the string cycle (period of a string). In the exact string matching problem, we are given two strings T=t1t2...tn and P=p1p2...pm. We are asked to find all occurrences of P in T. Our searching method is based upon a single character rule. Consider a location i in P. Suppose that pi is aligned with tj and pi <> tj. We then must move P in such a way that some pk=tj will be aligned with tj. In the thesis, we propose a method to find the optimal location i. We also modified a string matching approach, named wide window approach, which divides the text into n/m overlapping windows of size 2m-1. In the windows, the approach attempts m possible occurrence positions in parallel. It firstly searches pattern suffixes from middle to right with the modified convolution method, shifts the window directly when it fails, otherwise, scans the corresponding prefixes backward with the modified convolution method again. For the period of string problem, we are given a string T and we are asked to find the periods of T. Our algorithm is based upon the bit parallel approach.

In a growing number of information-processing applications, such as network-traffic monitoring, sensor networks, financial analysis, data mining for e-commerce, etc., data takes the form of continuous data streams rather than traditional stored databases/relational tuples. These applications have some common features like the need for real time analysis, huge volumes of data, and unpredictable and bursty arrivals of stream elements. In all of these applications, it is infeasible to process queries over data streams by loading the data into a traditional database management system (DBMS) or into main memory. Such an approach does not scale with high stream rates. As a consequence, systems that can manage streaming data have gained tremendous importance. The need to process a large number of continuous queries over bursty, high volume online data streams, potentially in real time, makes it imperative to design algorithms that should use limited resources. This dissertation focuses on processing exact results for join queries over high speed data streams using limited resources, and proposes several novel techniques for processing join queries incorporating secondary storages and non-dedicated computers. Existing approaches for stream joins either, (a) deal with memory limitations by shedding loads, and therefore can not produce exact or highly accurate results for the stream joins over data streams with time varying arrivals of stream tuples, or (b) suffer from large I/O-overheads due to random disk accesses. The proposed techniques exploit the high bandwidth of a disk subsystem by rendering the data access pattern largely sequential, eliminating small, random disk accesses. This dissertation proposes an I/O-efficient algorithm to process hybrid join queries, that join a fast, time varying or bursty data stream and a persistent disk relation. Such a hybrid join is the crux of a number of common transformations in an active data warehouse. Experimental results demonstrate that the proposed scheme reduces the response time in output results by exploiting spatio-temporal locality within the input stream, and minimizes disk overhead through disk-I/O amortization. The dissertation also proposes an algorithm to parallelize a stream join operator over a shared-nothing system. The proposed algorithm distributes the processing loads across a number of independent, non-dedicated nodes, based on a fixed or predefined communication pattern; dynamically maintains the degree of declustering in order to minimize communication and processing overheads; and presents mechanisms for reducing storage and communication overheads while scaling over a large number of nodes. We present experimental results showing the efficacy of the proposed algorithms.

We report on the first systematic study of correlation functions in N=4 super Yang-Mills theory using integrability techniques. In particular, we show how to compute three- and four- point functions of single-trace gauge-invariant operators at tree level in the SU(2) sector of the theory. Using the AdS/CFT correspondence, the correlation functions that we compute can be thought of as the joining or splitting of strings moving in AdS5 × S5. We show that when one (two) of the operators in the three-(four-)point function are taken to be small BPS operators, our weak coupling results match perfectly with the strong coupling results in the Frolov-Tseytlin limit. We conclude by presenting some results that will be needed to extend the methods presented in this thesis beyond the SU(2) sector of N=4 super Yang-Mills.

碩士
國立臺灣大學
物理研究所
99
A two-level system which decays spontaneously into field vacuum is studied through the Jaynes-Cummings model in the rotating wave approximation (RWA).When at most one excitation is considered, this model is exactly solvable. Here we evaluate the non-Markovian two-time correlation functions (CF''s) of system operators for this model in two ways: one by directly solving the system-environment evolution,and the other by using the perturbative time-convolutionless non-Markovian master equation approach. We derive valid to fourth order in system-bath coupling strength a non-Markovain evolution equation for the two-time CF''s of system operators. We use the derived evolution equation to calculate a two-time CF for the two-level model and compare it with the exact result obtained by direct evaluation. Another numerical series acceleration method is applied to the calculation of the perturbation decay, and this method is found to improve the accuracy of the evolution equation. The result obtained by the derived perturbative two-time evolution equation is much better than those by the perturbative Markovian Quantum regression theorem(QRT), the non-Markovian QRT and exact QRT as it agrees more closely with the exact result even when the model is in the regime where the bath correlation time is comparable to the system relaxation time.This demonstrates the validity and usefulness of our derived non-Markovain two-time evolution equation. The exact spontaneous emission spectrum is also calcukated, and it has very different behaviours in the strong coupling an the weak coupling regions.