Дисертації з теми "Matrix String Theory"

Cette thèse traite de plusieurs façons de construire une théorie quantiques des champs en quatre dimensions à partir de la théorie des cordes.

Dans une première partie nous étudions la construction d'une théorie Yang-Mills supersymétrique, couplée à un superchamp chiral dans la représentation adjointe, à partir de la théorie des cordes de type IIB sur une variété Calabi-Yau non compacte, avec des D-branes qui enroulent certaines sousvariétés. Les propriétés de
la théorie de jauge sont alors reflétées dans la structure
géométrique de la variété Calabi-Yau. En particulier, on peut calculer en principe le superpotentiel effectif de basse énergie qui décrit la structure des vides de la théorie de jauge en utilisant la théorie des cordes (topologiques). Malheureusement, en pratique, ceci n'est pas faisable. Il est remarquable qu'on puisse cependant montrer que la dynamique de basse énergie de la
théorie de jauge est codée par la géométrie d'une autre variété Calabi-Yau non compacte, reliée à la première par une transition géométrique. La théorie des cordes de type IIB sur cette deuxième variété, dans laquelle sont allumés des flux de fond appropriés, génère une théorie de jauge en quatre dimensions, qui n'est d'autre que la théorie effective de basse énergie de la théorie de
jauge originale. Ainsi, pour obtenir le superpotentiel effectif de basse énergie il suffit simplement de calculer certaines intégrales dans la deuxième géométrie Calabi-Yau, ce qui est faisable, au moins perturbativement. On trouve alors que le problème extrêmement difficile d'étudier la dynamique de basse
énergie d'une théorie de jauge non Abelienne a été réduit à celui de calculer certaines intégrales dans une géométrie connue. On peut démontrer que ces intégrales sont intimement reliées à certaines quantités dans un modèle de matrices holomorphes, et on peut alors calculer le superpotentiel effectif comme fonction de
certaines expressions du model de matrices. Il est remarquable que la série perturbative du modèle de matrices calcule alors le superpotentiel effectif non-perturbatif.

Ces relations étonnantes ont été découvertes et élaborée par plusieurs auteurs au cours des dernières années. Les résultats originaux de cette thèse comprennent la forme précise des relations de la ``géométrie spéciale" sur une variété Calabi-Yau
non compacte. Nous étudions en détail comment ces intégrales géométriques dépendent du cut-off, et leur relation à l'énergie libre du modèle de matrices. En particulier, sur une variété Calabi-Yau non compacte nous proposons une forme bilineaire sur le
produit direct de l'espace des formes avec l'espace des cycles, qui élimine toutes les divergences, sauf la divergence logarithmique. Notre analyse détaillée du modèle de matrices holomorphes clarifie aussi plusieurs aspects reliés à la méthode du col de ce modèle de matrices. Nous montrons en particulier qu'exiger une densité spectrale réelle restreint la forme de la
courbe Riemannienne qui apparaît dans la limite planaire du modèle de matrices. Çela nous donne des contraintes sur la forme du contour sur lequel les valeurs propres sont intégrées. Tous ces
résultats sont utilisés pour calculer explicitement l'énergie libre planaire d'un modèle de matrices avec un potentiel cubique.

La deuxième partie de cette thèse concerne la génération de théories de jauge supersymétriques en quatre dimensions comportant des aspects caractéristiques du modèle standard à partir de
compactifications de la supergravité en onze dimensions sur une variété G_2. Si cette variété contient une singularité conique, des fermions chiraux apparaissent dans la théorie de jauge en quatre dimensions ce qui conduit potentiellement à des anomalies. Nous montrons que, localement à chaque singularité, les anomalies
correspondantes sont annulées par une non-invariance de l'action classique au singularités (``anomaly inflow"). Malheureusement, aucune métrique d'une variété G_2 compacte n'est connue explicitement. Nous construisons ici des familles de métriques sur des variétés compactes faiblement G_2, qui contiennent deux singularités coniques. Les variétés faiblement G_2 ont des propriétés semblables aux propriétés des variétés G_2, et alors ces exemples explicites pourraient être utiles pour mieux comprendre la situation générique. Finalement, nous regardons la
relation entre la supergravité en onze dimensions et la théorie des cordes hétérotiques E_8\times E_8. Nous étudions en détail les anomalies qui apparaissent si la supergravité est formulée sur le produit d'un espace de dix dimensions et un intervalle. Encore une fois nous trouvons que les anomalies s'annulent localement sur
chaque bord de l'intervalle si on modifie l'action classique d'une façon appropriée.

The goal of this thesis is to study non-singlet sectors of the c=1 matrix model in order to determine their connection with string theory in two space-time dimensions. It is well understood that the singlet sector of this matrix model is related to the closed string sector of the string theory. The adjoint sector has been connected to long strings stretching from an FZZT brane at infinity. The goal is to find a sector which includes a large number of these long strings which condense to form an FZZT brane in line with a proposal due to Gaiotto. Sectors corresponding to symmetric and antisymmetric tensor products of the fundamental and antifundamental representation of the U(N) symmetry of the matrix model are studied. Partition functions of the matrix model with a harmonic oscillator potential as well as bases for the Hilbert spaces are found for all these sectors. The first two representations are studied using a collective field approach with the inverted oscillator potential applicable to the c=1 model and are found to contain large numbers of long strings which form a free bosonic gas in the large N limit and so do not condense. On the other hand, in the third sector the degrees of freedom are fermionic and a shift in the ground state energy is found in the harmonic oscillator potential, which points to the existence of a Fermi sea and the condensation of the long strings. A collective field that can be used to study this sector is proposed and the technical difficulties presented by the study of this sector are discussed, but its study is left for future work.

This thesis is devoted to the study of two important applications of gauge-gravity duality: the cosmological singularity problem and conformal fluid dynamics. Gauge-gravity duality is a concrete dual relationship between a gauge theory (such as electromagnetism, the theories of weak and strong interactions), and a theory of strings which contains gravity. The most concrete application of this duality is the AdS/CFT correspondence, where the theory containing gravity lives in the bulk of an asymptotically anti-de-Sitter space-time, while the dual gauge theory is a deformation of a conformal field theory which lives on the boundary of anti-de-Sitter space-time(AdS). Our first application of gauge-gravity duality is to the cosmological singularity problem in string gravity. A cosmological singularity is defined as a spacelike region of space-time which is highly curved so that Einstein’s gravity theory can be no longer applied. In our setup the bulk space-time has low curvature in the far past and the physics is well described by supergravity (which is an extension of standard Einstein gravity). The cosmological singularity is driven by a time dependent string coupling in the bulk theory. The rate of change of the coupling is slow, but the net change of the coupling can be large. The dual description of this is a time dependent coupling of the boundary gauge theory. The coupling has a profile which is a constant in the far past and future and attains a small but finite value at intermediate times. We construct the supergravity solution, with the initial condition that the bulk space-time is pure AdS in the far past and show that the solution remains smooth in a derivative expansion without formation of black holes. However when the intermediate value of the string coupling becomes weak enough, space-time becomes highly curved and the supergravity approximation breaks down, mimicking a spacelike singularity. The resulting dynamics is analyzed in the dual gauge theory with a time dependent coupling constant which varies slowly. We develop an appropriate adiabatic expansion in the gauge theory in terms of coherent states and show that the time evolution continues to be smooth. We cannot, however, arrive at a definitive conclusion about the fate of the system at very late times when the coupling has again risen and supergravity again applies. One possibility is that the energy which has been supplied to the universe is simply extracted out and the space-time goes back to its initial state. This could provide a model for a bouncing cosmology. A second possibility is that dissipation leads to a thermal state at late time. If this possibility holds, we show that such a thermal state will be described either by a gas of strings or by a small black hole, but not by a big black hole. This means that in either case, the future space-time is close to AdS. We then apply gauge-gravity duality to conformal fluid dynamics. The long wavelength behavior of any strongly coupled system with a finite mean free path is described by an appropriate fluid dynamics. The bulk dual of a fluid flow in the boundary theory is a black hole with a slowly varying horizon. In this work we consider certain fluid flows which become supersonic in some regions. It is well known that such flows present acoustic analogs of ergoregions and horizons, where acoustic waves cannot propagate in certain directions. Such acoustic horizons are expected to exhibit thermal radiation of acoustic waves with temperature essentially given by the gradient of the velocity at the acoustic horizon. We find acoustic analogs of black holes in charged conformal fluids and use gauge-gravity duality to construct dual gravity solutions. A certain class of gravitational quasinormal wave modes around these gravitational backgrounds perceives a horizon. Upon quantization, this implies that these gravitational modes should have a thermal spectrum. The final issue that we study is fluid-gravity duality at zero temperature. The usual way of constructing gravity duals of fluid flows is by means of a small derivative expansion, in which the derivatives are much smaller than the temperature of the background black hole. Recently, it has been reported that for charged fluids, this procedure breaks down in the zero temperature limit. More precisely, corrections to the small derivative expansion in the dual gravity of charged fluid at zero temperature have singularities at the black hole horizon. In this case, fluid-gravity duality is not understood precisely. We explore this problem for a zero temperature charged fluid driven by a low frequency, small amplitude and spatially homogeneous external force. In the gravity dual, this force corresponds to a time dependent boundary value of the dilaton field. We calculate the bulk solution for the dilaton and the leading backreaction using a modified low frequency expansion. The resulting solutions are regular everywhere, establishing fluid-gravity duality to this order.

String theory is a unified framework for general relativity and quantum mechanics, thus being a theory of quantum gravity. In this thesis we discuss various aspects of quantum gravity for particular systems, having in common the existence of horizons. The main motivation is that one major challenge in theoretical physics today is in trying to understanding how time dependent backgrounds, with its resulting horizons and space-like singularities, should be described in a controlled way. One such system of particular importance is our own universe.

We begin by discussing the information puzzle in de Sitter space and consequences thereof. A typical time-scale is encountered, which we interpreted as setting the thermalization time for the system. Then the question of closed time-like curves is discussed in the combined setting where we have a rotating black hole in a Gödel-like universe. This gives a unified picture of what previously was considered as independent systems. The last three projects concerns $c=1$ matrix models and their applications. First in relation to the RR-charged two dimensional type 0A black hole. We calculate the ground state energy on both sides of the duality and find a perfect agreement. Finally, we relate the 0A model at self-dual radius to the topological string on the conifold. We find that an intriguing factorization of the theory previously observed for the topological string is also present in the 0A matrix model.

O objetivo desta dissertação é revisar os operadores laços de Wilson no contexto da correspondência AdS/CFT. Estes operadores, presentes em qualquer teoria de calibre, são importantes por nos fornecer um parâmetro de ordem para a transição de fase confinante/desconfinante. Além disso, eles são particularmente importantes no estudo da correspondência AdS/ CFT pois: i) Eles nos dão, em alguns casos, resultados exatos graças ao fato de poderem ser localizados em um modelo de matrizes, desta forma nos permitindo fazer testes altamente não triviais da correspondência; ii) Eles são os objetos da teoria de calibre que são duais as cordas propagando no interior do espaço, nos dando um rico dicionário entre quantidades no interior (AdS) e na borda do espaço (CFT). Depois de revisarmos os laços de Wilson em teorias de calibre e a correspondência Ads/CFT, introduziremos a definição dos laços de Wilson supersimétricos 1/2 BPS. Calcularemos eles para o caso de um acoplamento fraco e para qualquer outro valor da constante de acoplamento usando técnicas de modelos de matrizes. Finalmente, compararemos nossos resultados com computações de superfícies minimais no interior do espaço, encontrando uma concordância perfeita.
The aim of this thesis is to review Wilson loop operators in the contexto f the AdS/CFT correspondence. These operators, wich are present in any gauge theory, are important because they furnish an order parameter for confinement/deconfinement phase transitions. Besides this, they are particularly relevant in the study of the AdS/CFT correspondence because: i) they allow, in some cases, for exact results thanks to localization to matrix models and make it possible to perform highly non-trivial tests of the correspondence; ii) they are the gauge theory objects dual to strings propagating in the bulk of the space and give a rich dictionary between bulk (AdS) and boundary (CFT) quantities. After reviews of Wilson loops in gauge theories and of the Ads/CFT correspondence, we will introduce the definition of 1/2 BPS supersymmetric Wilson loops, we will compute them at weak coupling and then at any order in the coupling constant via matrix model techniques, and finally we will compare our results with minimal surface computations in the bulk, finding perfect agreement.

A complete non trivial supersymmetric deformation of the maximal supergravity in two dimensions is achieved by the gauging of a SO(9) group. The resulting theory describes the reduction of type IIA supergravity on an AdS_2 x S^8 background and is of first importance in the Domain-Wall / Quantum Field theory correspondence for the D0-brane case. To prepare the construction of the SO(9) gauged maximal supergravity, we focus on the eleven dimensional supergravity and the maximal supergravity in three dimensions since they give rise to important off-shell inequivalent formulations of the ungauged theory in two dimensions. The embedding tensor formalism is presented, allowing for a general desciption of the gaugings consistent with supersymmetry. The SO(9) supergravity is explicitly constructed and applications are considered. In particular, an embedding of the bosonic sector of the two-dimensional theory into type IIA supergravity is obtained. Hence, the Cartan truncation of the SO(9) supergravity is proved to be consistent. This motivated holographic applications. Therefore, correlation functions for operators in dual Matrix models are derived from the study of gravity side excitations around half BPS backgrounds. These results are fully discussed and outlooks are presented.

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.

Two applications of Category Theory are considered. The link between them is applications to Physics and more specifically to Entropy. The first research chapter is broader in scope and not explicitly about Physics, although connections to Statistical Mechanics are made towards the end of the chapter. Matroids are abstract structures that describe dependence, and strong maps are certain structure-preserving functions between them with desirable properties. We examine properties of various categories of matroids and strong maps: we compute limits and colimits; we find free and cofree constructions of various subcategories; we examine factorisation structures, including a translation principle from geometric lattices; we find functors with convenient properties to/from vector spaces, multisets of vectors, geometric lattices, and graphs; we determine which widely used operations on matroids are functorial (these include deletion, contraction, series and parallel connection, and a simplification monad); lastly, we find a categorical characterisation of the greedy algorithm. In conclusion, this project determines which aspects of Matroid Theory are most and least conducive to categorical treatment. The purpose of the second research chapter is to provide a categorical framework for generalising and unifying notions of Entropy in various settings, exploiting the fact that Entropy is a monotone subadditive function. A categorical characterisation of Entropy through a category of thermodynamical systems and adiabatic processes is found. A modelling perspective (adiabatic categories) that directly generalises an existing model is compared to an axiomatisation through topological and linear structures (topological weak semimodules), where the latter is based on a categorification of semimodules. Properties of each class of categories are examined; most notably a cancellation property of adiabatic categories generalising an existing result, and an adjunction between the categories of weak semimodules and symmetric monoidal categories. An adjunction between categories of adiabatic categories and topological weak semimodules is found. We examine in which cases each of these classes of categories constitutes a traced monoidal category. Lastly, examples of physical applications are provided. In conclusion, this project uncovers a way of, and makes progress towards, retrieving the statistical formulation of Entropy from simple axioms.

Yes
Thermoplastic polyurethane (TPU)/multi-walled carbon nanotubes (CNT) nanocomposites were prepared by twin-screw extrusion and micro injection molding. The electrical conductivity of micro injection molded polymer nanocomposites exhibits a low value and uneven distribution in the micromolded samples. Real-time tracing of electrical conductivity was conducted to investigate the post thermal treatment on the electrical conductivity of microinjection molded composites. The results show that postmolding thermal treatment leads to a significant increase in the electrical conductivity by over three orders of magnitude for 5 wt % CNT-filled TPU composites. In-situ Transmission electron microscopy confirms the conductive CNT network does not change at the micron/sub-micron scale during thermal treatment. TEM image analysis by a statistical method was used to determine the spatial distribution of CNT in the sample and showed that the average distance between adjacent CNT reduced slightly at the nanometer scale after postmolding thermal treatment. A new conductive mechanism is proposed to explain the enhancement of electrical conductivity after thermal treatment, i.e. micro-contact reconstruction of adjacent CNT in the polymer matrix through annealing-induced relaxation of interfacial residual stress and strain. Raman spectra and small angle X-ray scattering curve of annealed samples provide supporting evidence for the proposed new conductive mechanism. The electron tunneling model was used to understand the effect of inter-particle distance on the conductivity of polymer composites.
Chinese Ministry of Education. Grant Number: 313036; National Natural Science Foundation of China. Grant Number: 51433006