Aspects of Integrability in Gauge/String Correspondence

In this thesis, we present our work about some generalisations of ideas, techniques and physical interpretations typical for integrable models to one of the most outstanding advances in theoretical physics of nowadays: the AdS/CFT correspondences. We have undertaken the problem of testing this conjectured duality under various points of view, but with a clear starting point - the integrability - and with a clear ambitious task in mind: to study the finite-size effects in the energy spectrum of certain string solutions on a side and in the anomalous dimensions of the gauge theory on the other. Of course, the final desire woul be the exact comparison between these two faces of the gauge/string duality. In few words, the original part of this work consists in application of well known integrability technologies, in large parte borrowed by the study of relativistic (1+1)-dimensional integrable quantum field theories, to the highly non-relativisic and much complicated case of the thoeries involved in the recent conjectures of AdS5/CFT4 and AdS4/CFT3 corrspondences. In details, exploiting the spin chain nature of the dilatation operator of N = 4 Super-Yang-Mills theory, we concentrated our attention on one of the most important sector, namely the SL(2) sector - which is also very intersting for the QCD understanding - by formulating a new type of nonlinear integral equation (NLIE) based on a previously guessed asymptotic Bethe Ansatz. The solutions of this Bethe Ansatz are characterised by the length L of the correspondent spin chain and by the number s of its excitations. A NLIE allows one, at least in principle, to make analytical and numerical calculations for arbitrary values of these parameters. The results have been rather exciting. In the important regime of high Lorentz spin, the NLIE clarifies how it reduces to a linear integral equations which governs the subleading order in s, o(s0). This also holds in the regime with L ! 1, L/ ln s finite (long operators case). This region of parameters has been particularly investigated in literature especially because of an intriguing limit into the O(6) sigma model defined on the string side. One of the most powerful methods to keep under control the finite-size spectrum of an integrable relativistic theory is the so called thermodynamic Bethe Ansatz (TBA). We proposed a highly non-trivial generalisation of this technique to the non-relativistic case of AdS5/CFT4 and made the first steps in order to determine its full spectrum - of energies for the AdS side, of anomalous dimensions for the CFT one - at any values of the coupling constant and of the size. At the leading order in the size parameter, the calculation of the finite-size corrections is much simpler and does not necessitate the TBA. It consists in deriving for a nonrelativistc case a method, invented for the first time by L¨uscher to compute the finite-size effects on the mass spectrum of relativisic theories. So, we have formulated a new version of this approach to adapt it to the case of recently found classical string solutions on AdS4 × CP3, inside the new conjecture of an AdS4/CFT3 correspondence. Our results in part confirm the string and algebraic curve calculations, in part are completely new and then could be better understood by the rapidly evolving developments of this extremely exciting research field.

Falegnami e muratori a Bologna nel Medioevo: statuti e matricole (1248-1377)

La finalità della tesi è stata quella d'improntare una riflessione sulle società legate all'edilizia nel momento di apice politico ed economico di una città, attiva e importante nel basso Medioevo, come Bologna. Questo tipo di studio, già intrapreso per alcune città dell'Italia centro settentrionale, risulta necessario per comprendere alcune dinamiche politiche, economiche e produttive della realtà cittadina. In particolare, gli obiettivi sono stati quelli di coniugare la vita corporativa dell'arte all'organizzazione lavorativa, individuando gli aspetti legati all'ambito produttivo e valutando, l'evoluzione della società nel passaggio dal Duecento al Trecento. Gli statuti e le matricole, principali fonti compilate dalle due società delle arti e giunte ai nostri giorni, salvo rare eccezioni, completamente inedite sono stati il principale punto di partenza dello studio. Il risultato è la sintesi di alcuni spunti di riflessione emersi dai documenti, quali la mentalità degli uomini che si approcciavano alle arti, le gerarchie nate all'interno delle stesse società, il loro cambiamento e quello legato alla percezione di comunità nel corso dei secoli oltre agli aspetti più prettamente lavorativi e artigianali.

Über die Existenz invarianter Tori in Hamiltonschen Systemen, die bis auf eine endlich oft differenzierbare Störung analytisch und integrabel sind

Es wird die Existenz invarianter Tori in Hamiltonschen Systemen bewiesen, die bis auf eine 2n-mal stetig differenzierbare Störung analytisch und integrabel sind, wobei n die Anzahl der Freiheitsgrade bezeichnet. Dabei wird vorausgesetzt, dass die Stetigkeitsmodule der 2n-ten partiellen Ableitungen der Störung einer Endlichkeitsbedingung (Integralbedingung) genügen, welche die Hölderbedingung verallgemeinert. Bisher konnte die Existenz invarianter Tori nur unter der Voraussetzung bewiesen werden, dass die 2n-ten Ableitungen der Störung hölderstetig sind.

Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

Quantum Integrability in Non-Linear Sigma Models related to Gauge/String Correspondences

The Thermodynamic Bethe Ansatz analysis is carried out for the extended-CP^N class of integrable 2-dimensional Non-Linear Sigma Models related to the low energy limit of the AdS_4xCP^3 type IIA superstring theory. The principal aim of this program is to obtain further non-perturbative consistency check to the S-matrix proposed to describe the scattering processes between the fundamental excitations of the theory by analyzing the structure of the Renormalization Group flow. As a noteworthy byproduct we eventually obtain a novel class of TBA models which fits in the known classification but with several important differences. The TBA framework allows the evaluation of some exact quantities related to the conformal UV limit of the model: effective central charge, conformal dimension of the perturbing operator and field content of the underlying CFT. The knowledge of this physical quantities has led to the possibility of conjecturing a perturbed CFT realization of the integrable models in terms of coset Kac-Moody CFT. The set of numerical tools and programs developed ad hoc to solve the problem at hand is also discussed in some detail with references to the code.

On the flavor problem in strongly coupled theories

rnThis thesis is on the flavor problem of Randall Sundrum modelsrnand their strongly coupled dual theories. These models are particularly wellrnmotivated extensions of the Standard Model, because they simultaneously address rntherngauge hierarchy problem and the hierarchies in the quarkrnmasses and mixings. In order to put this into context, special attention is given to concepts underlying therntheories which can explain the hierarchy problem and the flavor structure of the Standard Model (SM). ThernAdS/CFTrnduality is introduced and its implications for the Randall Sundrum model withrnfermions in the bulk andrngeneral bulk gauge groups is investigated. It will be shown that the differentrnterms in the general 5D propagator of a bulk gauge field can be related tornthe corresponding diagrams of the strongly coupled dual, which allows for arndeeperrnunderstanding of the origin of flavor changing neutral currents generated by thernexchange of the Kaluza Klein excitations of these bulk fields.rnIn the numerical analysis, different observables which are sensitive torncorrections from therntree-levelrnexchange of these resonances will be presented on the basis of updatedrnexperimental data from the Tevatron and LHC experiments. This includesrnelectroweak precision observables, namely corrections to the S and Trnparameters followed by corrections to the Zbb vertex, flavor changingrnobservables with flavor changes at one vertex, viz. BR (Bd -> mu+mu-) and BR (Bs -> mu+mu-), and two vertices,rn viz. S_psiphi and |eps_K|, as well as bounds from direct detectionrnexperiments. rnThe analysis will show that all of these bounds can be brought in agreement withrna new physics scale Lambda_NP in the TeV range, except for the CPrnviolating quantity |eps_K|, which requires Lambda_NP= Ord(10) TeVrnin the absencernof fine-tuning. The numerous modifications of the Randall Sundrum modelrnin the literature, which try to attenuate this bound are reviewed andrncategorized.rnrnSubsequently, a novel solution to this flavor problem, based on an extendedrncolor gauge group in the bulk and its thorough implementation inrnthe RS model, will be presented, as well as an analysis of the observablesrnmentioned above in the extended model. This solution is especially motivatedrnfromrnthe point of view of the strongly coupled dual theory and the implications forrnstrongly coupled models of new physics, which do not possess a holographic dual,rnare examined.rnFinally, the top quark plays a special role in models with a geometric explanation ofrnflavor hierarchies and the predictions in the Randall-Sundrum model with andrnwithout the proposed extension for the forward-backward asymmetryrnA_FB^trnin top pair production are computed.

The geometry of Lagrangian fibres

If the generic fibre f−1(c) of a Lagrangian fibration f : X → B on a complex Poisson– variety X is smooth, compact, and connected, it is isomorphic to the compactification of a complex abelian Lie–group. For affine Lagrangian fibres it is not clear what the structure of the fibre is. Adler and van Moerbeke developed a strategy to prove that the generic fibre of a Lagrangian fibration is isomorphic to the affine part of an abelian variety.rnWe extend their strategy to verify that the generic fibre of a given Lagrangian fibration is the affine part of a (C∗)r–extension of an abelian variety. This strategy turned out to be successful for all examples we studied. Additionally we studied examples of Lagrangian fibrations that have the affine part of a ramified cyclic cover of an abelian variety as generic fibre. We obtained an embedding in a Lagrangian fibration that has the affine part of a C∗–extension of an abelian variety as generic fibre. This embedding is not an embedding in the category of Lagrangian fibrations. The C∗–quotient of the new Lagrangian fibration defines in a natural way a deformation of the cyclic quotient of the original Lagrangian fibration.

Centro-sinistra e autonomia speciale. La DC trentina tra il 1955 e il 1968

La tesi è una ricerca di storia politica che affianca due diverse “storie” di centro-sinistra, quella nazionale e quella che vide protagonista la Democrazia Cristiana del Trentino. Lo studio analizza i fatti attraverso il filtro delle DC come se quello trentino e quello nazionale fossero due partiti, per poi tentare di capire ciò che accadeva alla loro sinistra alla ricerca dei diversi pesi e dei differenti equilibri che al centro e alla periferia si manifestavano nei rapporti con il PSI e con il PCI, e per osservare le reazioni della Chiesa così da valutare se le gerarchie romane e quelle trentine interagirono in modo differente sugli sviluppi delle rispettive esperienze politiche di quegli anni. Il testo è organizzato in quattro capitoli. Il primo e il secondo (speculari e dedicati allo stesso lustro: 1955-1960) rappresentano un confronto tra i differenti iter d’avvicinamento al centro-sinistra che la politica nazionale e quella trentina sperimentarono nella seconda metà degli anni Cinquanta. Nel terzo capitolo (1960-1964) e nel quarto (1964-1968) le vicende nazionali e quelle locali sono invece raccontate in modo intrecciato, ripercorrendo le diverse fasi dell’alleanza tra Democrazia Cristiana e Partito Socialista, e nel contempo dando conto della trasformazione del Trentino da una realtà di tipo agricolo ad una di tipo industriale, del passaggio da una comunità di tipo cattolico tradizionale ad una che si accinge a vivere in un contesto secolarizzato, e da una società che si autopercepisce come periferica ad una che ospita una delle contestazioni studentesche più peculiari, incisive e note d’Italia.

Learning Methods and Algorithms for Semantic Text Classification across Multiple Domains

Information is nowadays a key resource: machine learning and data mining techniques have been developed to extract high-level information from great amounts of data. As most data comes in form of unstructured text in natural languages, research on text mining is currently very active and dealing with practical problems. Among these, text categorization deals with the automatic organization of large quantities of documents in priorly defined taxonomies of topic categories, possibly arranged in large hierarchies. In commonly proposed machine learning approaches, classifiers are automatically trained from pre-labeled documents: they can perform very accurate classification, but often require a consistent training set and notable computational effort. Methods for cross-domain text categorization have been proposed, allowing to leverage a set of labeled documents of one domain to classify those of another one. Most methods use advanced statistical techniques, usually involving tuning of parameters. A first contribution presented here is a method based on nearest centroid classification, where profiles of categories are generated from the known domain and then iteratively adapted to the unknown one. Despite being conceptually simple and having easily tuned parameters, this method achieves state-of-the-art accuracy in most benchmark datasets with fast running times. A second, deeper contribution involves the design of a domain-independent model to distinguish the degree and type of relatedness between arbitrary documents and topics, inferred from the different types of semantic relationships between respective representative words, identified by specific search algorithms. The application of this model is tested on both flat and hierarchical text categorization, where it potentially allows the efficient addition of new categories during classification. Results show that classification accuracy still requires improvements, but models generated from one domain are shown to be effectively able to be reused in a different one.

Internally contracted multireference coupled-cluster methods

This thesis details the development of quantum chemical methods for the accurate theoretical description of molecular systems with a complicated electronic structure. In simple cases, a single Slater determinant, in which the electrons occupy a number of energetically lowest molecular orbitals, offers a qualitatively correct model. The widely used coupled-cluster method CCSD(T) efficiently includes electron correlation effects starting from this determinant and provides reaction energies in error by only a few kJ/mol. However, the method often fails when several electronic configurations are important, as, for instance, in the course of many chemical reactions or in transition metal compounds. Internally contracted multireference coupled-cluster methods (ic-MRCC methods) cure this deficiency by using a linear combination of determinants as a reference function. Despite their theoretical elegance, the ic-MRCC equations involve thousands of terms and are therefore derived by the computer. Calculations of energy surfaces of BeH2, HF, LiF, H2O, N2 and Be3 unveil the theory's high accuracy compared to other approaches and the quality of various hierarchies of approximations. New theoretical advances include size-extensive techniques for removing linear dependencies in the ic-MRCC equations and a multireference analog of CCSD(T). Applications of the latter method to O3, Ni2O2, benzynes, C6H7NO and Cr2 underscore its potential to become a new standard method in quantum chemistry.

Efficient parallel proximity queries and an application to highly complex motion planning problems with many narrow passages

In vielen Bereichen der industriellen Fertigung, wie zum Beispiel in der Automobilindustrie, wer- den digitale Versuchsmodelle (sog. digital mock-ups) eingesetzt, um die Entwicklung komplexer Maschinen m ̈oglichst gut durch Computersysteme unterstu ̈tzen zu k ̈onnen. Hierbei spielen Be- wegungsplanungsalgorithmen eine wichtige Rolle, um zu gew ̈ahrleisten, dass diese digitalen Pro- totypen auch kollisionsfrei zusammengesetzt werden k ̈onnen. In den letzten Jahrzehnten haben sich hier sampling-basierte Verfahren besonders bew ̈ahrt. Diese erzeugen eine große Anzahl von zuf ̈alligen Lagen fu ̈r das ein-/auszubauende Objekt und verwenden einen Kollisionserken- nungsmechanismus, um die einzelnen Lagen auf Gu ̈ltigkeit zu u ̈berpru ̈fen. Daher spielt die Kollisionserkennung eine wesentliche Rolle beim Design effizienter Bewegungsplanungsalgorith- men. Eine Schwierigkeit fu ̈r diese Klasse von Planern stellen sogenannte “narrow passages” dar, schmale Passagen also, die immer dort auftreten, wo die Bewegungsfreiheit der zu planenden Objekte stark eingeschr ̈ankt ist. An solchen Stellen kann es schwierig sein, eine ausreichende Anzahl von kollisionsfreien Samples zu finden. Es ist dann m ̈oglicherweise n ̈otig, ausgeklu ̈geltere Techniken einzusetzen, um eine gute Performance der Algorithmen zu erreichen.rnDie vorliegende Arbeit gliedert sich in zwei Teile: Im ersten Teil untersuchen wir parallele Kollisionserkennungsalgorithmen. Da wir auf eine Anwendung bei sampling-basierten Bewe- gungsplanern abzielen, w ̈ahlen wir hier eine Problemstellung, bei der wir stets die selben zwei Objekte, aber in einer großen Anzahl von unterschiedlichen Lagen auf Kollision testen. Wir im- plementieren und vergleichen verschiedene Verfahren, die auf Hu ̈llk ̈operhierarchien (BVHs) und hierarchische Grids als Beschleunigungsstrukturen zuru ̈ckgreifen. Alle beschriebenen Verfahren wurden auf mehreren CPU-Kernen parallelisiert. Daru ̈ber hinaus vergleichen wir verschiedene CUDA Kernels zur Durchfu ̈hrung BVH-basierter Kollisionstests auf der GPU. Neben einer un- terschiedlichen Verteilung der Arbeit auf die parallelen GPU Threads untersuchen wir hier die Auswirkung verschiedener Speicherzugriffsmuster auf die Performance der resultierenden Algo- rithmen. Weiter stellen wir eine Reihe von approximativen Kollisionstests vor, die auf den beschriebenen Verfahren basieren. Wenn eine geringere Genauigkeit der Tests tolerierbar ist, kann so eine weitere Verbesserung der Performance erzielt werden.rnIm zweiten Teil der Arbeit beschreiben wir einen von uns entworfenen parallelen, sampling- basierten Bewegungsplaner zur Behandlung hochkomplexer Probleme mit mehreren “narrow passages”. Das Verfahren arbeitet in zwei Phasen. Die grundlegende Idee ist hierbei, in der er- sten Planungsphase konzeptionell kleinere Fehler zuzulassen, um die Planungseffizienz zu erh ̈ohen und den resultierenden Pfad dann in einer zweiten Phase zu reparieren. Der hierzu in Phase I eingesetzte Planer basiert auf sogenannten Expansive Space Trees. Zus ̈atzlich haben wir den Planer mit einer Freidru ̈ckoperation ausgestattet, die es erlaubt, kleinere Kollisionen aufzul ̈osen und so die Effizienz in Bereichen mit eingeschr ̈ankter Bewegungsfreiheit zu erh ̈ohen. Optional erlaubt unsere Implementierung den Einsatz von approximativen Kollisionstests. Dies setzt die Genauigkeit der ersten Planungsphase weiter herab, fu ̈hrt aber auch zu einer weiteren Perfor- mancesteigerung. Die aus Phase I resultierenden Bewegungspfade sind dann unter Umst ̈anden nicht komplett kollisionsfrei. Um diese Pfade zu reparieren, haben wir einen neuartigen Pla- nungsalgorithmus entworfen, der lokal beschr ̈ankt auf eine kleine Umgebung um den bestehenden Pfad einen neuen, kollisionsfreien Bewegungspfad plant.rnWir haben den beschriebenen Algorithmus mit einer Klasse von neuen, schwierigen Metall- Puzzlen getestet, die zum Teil mehrere “narrow passages” aufweisen. Unseres Wissens nach ist eine Sammlung vergleichbar komplexer Benchmarks nicht ̈offentlich zug ̈anglich und wir fan- den auch keine Beschreibung von vergleichbar komplexen Benchmarks in der Motion-Planning Literatur.

Entanglement entropy in 1-d quantum spin chains

In questa tesi abbiamo studiato il comportamento delle entropie di Entanglement e dello spettro di Entanglement nel modello XYZ attraverso delle simulazioni numeriche. Le formule per le entropie di Von Neumann e di Renyi nel caso di una catena bipartita infinita esistevano già, ma mancavano ancora dei test numerici dettagliati. Inoltre, rispetto alla formula per l'Entropia di Entanglement di J. Cardy e P. Calabrese per sistemi non critici, tali relazioni presentano delle correzioni che non hanno ancora una spiegazione analitica: i risultati delle simulazioni numeriche ne hanno confermato la presenza. Abbiamo inoltre testato l'ipotesi che lo Schmidt Gap sia proporzionale a uno dei parametri d'ordine della teoria, e infine abbiamo simulato numericamente l'andamento delle Entropie e dello spettro di Entanglement in funzione della lunghezza della catena di spin. Ciò è stato possibile solo introducendo dei campi magnetici ''ad hoc'' nella catena, con la proprietà che l'andamento delle suddette quantità varia a seconda di come vengono disposti tali campi. Abbiamo quindi discusso i vari risultati ottenuti.

Spectral theory of differential operators on finite metric graphs and on bounded domains

Die vorliegende Arbeit widmet sich der Spektraltheorie von Differentialoperatoren auf metrischen Graphen und von indefiniten Differentialoperatoren auf beschränkten Gebieten. Sie besteht aus zwei Teilen. Im Ersten werden endliche, nicht notwendigerweise kompakte, metrische Graphen und die Hilberträume von quadratintegrierbaren Funktionen auf diesen betrachtet. Alle quasi-m-akkretiven Laplaceoperatoren auf solchen Graphen werden charakterisiert, und Abschätzungen an die negativen Eigenwerte selbstadjungierter Laplaceoperatoren werden hergeleitet. Weiterhin wird die Wohlgestelltheit eines gemischten Diffusions- und Transportproblems auf kompakten Graphen durch die Anwendung von Halbgruppenmethoden untersucht. Eine Verallgemeinerung des indefiniten Operators $-tfrac{d}{dx}sgn(x)tfrac{d}{dx}$ von Intervallen auf metrische Graphen wird eingeführt. Die Spektral- und Streutheorie der selbstadjungierten Realisierungen wird detailliert besprochen. Im zweiten Teil der Arbeit werden Operatoren untersucht, die mit indefiniten Formen der Art $langlegrad v, A(cdot)grad urangle$ mit $u,vin H_0^1(Omega)subset L^2(Omega)$ und $OmegasubsetR^d$ beschränkt, assoziiert sind. Das Eigenwertverhalten entspricht in Dimension $d=1$ einer verallgemeinerten Weylschen Asymptotik und für $dgeq 2$ werden Abschätzungen an die Eigenwerte bewiesen. Die Frage, wann indefinite Formmethoden für Dimensionen $dgeq 2$ anwendbar sind, bleibt offen und wird diskutiert.

Fontaine modules and F-T-crystals, a dévissage

Das Hauptziel der Arbeit ist es, die Beziehung zwischen Fontaine Modulen und F-T-Kristall zu studieren. Im ersten Kapitel wird die Definition von Fontaine Modulen, die auf die inversen Cartier Transform setzt erinnern wir von Ogus und Vologodsky errichtet. Neben der Erinnerung an die urspruengliche Konstruktion des inversen Cartier Transform, eine direktere Konstruktion, die wir auch vorstellen von G.T. Lan, M. Sheng und K. Zuo. Darueber hinaus beweisen wir diernGleichwertigkeit der beiden Konstruktion.rnrnrnIm zweiten Kapitel werden wir uns daran erinnern, den Konstruktion von inversen Cartier Transform in der Log Einstellung von D. Schepler und verallgemeinern die Lan-Sheng-Zuo Konstruktion an dieser Einstellung. Darueber hinaus geben wir eine Definition von Log FontainernModulen. Im dritten Kapitel werden wir erinnern an die Definition von F-T-Kristall und beweisen das wichtigste Ergebnis dieser Arbeit: Sei $Y$ eine glatte $S_{nu}$-Schema, wobei $S_{nu}$ ist eine flache $W_{nu+1}(k)$-Schema, $nugeq1$, und $X/S_0$ seine Reduction modulo $p$ sein. Bei einem F-T-Kristall $(E,Phi,B)$ auf $Y$ der Breite von weniger als $p$ und let $(E_Y ,B_Y ,nabla_Y)$ die entsprechende gefilterte $O_Y$-modulen mit einer integrierbar Zusammenhang ausgestattet. Anschliesend wird die Reduktion dieses Objekt modulo $p$ definiert eine Fontaine Modulen auf $X/S_0$ im dem Sinnernder Ogus und Vologodsky.

Hunting for warped extra dimensions via loop-induced processes

This thesis is on loop-induced processes in theories with warped extra dimensions where the fermions and gauge bosons are allowed to propagate in the bulk, while the Higgs sector is localized on or near the infra-red brane. These so-called Randall-Sundrum (RS) models have the potential to simultaneously explain the hierarchy problem and address the question of what causes the large hierarchies in the fermion sector of the Standard Model (SM). The Kaluza-Klein (KK) excitations of the bulk fields can significantly affect the loop-level processes considered in this thesis and, hence, could indirectly indicate the existence of warped extra dimensions. The analytical part of this thesis deals with the detailed calculation of three loop-induced processes in the RS models in question: the Higgs production process via gluon fusion, the Higgs decay into two photons, and the flavor-changing neutral current b → sγ. A comprehensive, five-dimensional (5D) analysis will show that the amplitudes of the Higgs processes can be expressed in terms of integrals over 5D propagators with the Higgs-boson profile along the extra dimension, which can be used for arbitrary models with a compact extra dimension. To this end, both the boson and fermion propagators in a warped 5D background are derived. It will be shown that the seemingly contradictory results for the gluon fusion amplitude in the literature can be traced back to two distinguishable, not smoothly-connected incarnations of the RS model. The investigation of the b → sγ transition is performed in the KK decomposed theory. It will be argued that summing up the entire KK tower leads to a finite result, which can be well approximated by a closed, analytical expression.rnIn the phenomenological part of this thesis, the analytic results of all relevant Higgs couplings in the RS models in question are compared with current and in particular future sensitivities of the Large Hadron Collider (LHC) and the planned International Linear Collider. The latest LHC Higgs data is then used to exclude significant portions of the parameter space of each RS scenario. The analysis will demonstrate that especially the loop-induced Higgs couplings are sensitive to KK particles of the custodial RS model with masses in the multi tera-electronvolt range. Finally, the effect of the RS model on three flavor observables associated with the b → sγ transition are examined. In particular, we study the branching ratio of the inclusive decay B → X_s γ