Mathematical aspects of Feynman integrals

In the present dissertation we consider Feynman integrals in the framework of dimensional regularization. As all such integrals can be expressed in terms of scalar integrals, we focus on this latter kind of integrals in their Feynman parametric representation and study their mathematical properties, partially applying graph theory, algebraic geometry and number theory. The three main topics are the graph theoretic properties of the Symanzik polynomials, the termination of the sector decomposition algorithm of Binoth and Heinrich and the arithmetic nature of the Laurent coefficients of Feynman integrals.rnrnThe integrand of an arbitrary dimensionally regularised, scalar Feynman integral can be expressed in terms of the two well-known Symanzik polynomials. We give a detailed review on the graph theoretic properties of these polynomials. Due to the matrix-tree-theorem the first of these polynomials can be constructed from the determinant of a minor of the generic Laplacian matrix of a graph. By use of a generalization of this theorem, the all-minors-matrix-tree theorem, we derive a new relation which furthermore relates the second Symanzik polynomial to the Laplacian matrix of a graph.rnrnStarting from the Feynman parametric parameterization, the sector decomposition algorithm of Binoth and Heinrich serves for the numerical evaluation of the Laurent coefficients of an arbitrary Feynman integral in the Euclidean momentum region. This widely used algorithm contains an iterated step, consisting of an appropriate decomposition of the domain of integration and the deformation of the resulting pieces. This procedure leads to a disentanglement of the overlapping singularities of the integral. By giving a counter-example we exhibit the problem, that this iterative step of the algorithm does not terminate for every possible case. We solve this problem by presenting an appropriate extension of the algorithm, which is guaranteed to terminate. This is achieved by mapping the iterative step to an abstract combinatorial problem, known as Hironaka's polyhedra game. We present a publicly available implementation of the improved algorithm. Furthermore we explain the relationship of the sector decomposition method with the resolution of singularities of a variety, given by a sequence of blow-ups, in algebraic geometry.rnrnMotivated by the connection between Feynman integrals and topics of algebraic geometry we consider the set of periods as defined by Kontsevich and Zagier. This special set of numbers contains the set of multiple zeta values and certain values of polylogarithms, which in turn are known to be present in results for Laurent coefficients of certain dimensionally regularized Feynman integrals. By use of the extended sector decomposition algorithm we prove a theorem which implies, that the Laurent coefficients of an arbitrary Feynman integral are periods if the masses and kinematical invariants take values in the Euclidean momentum region. The statement is formulated for an even more general class of integrals, allowing for an arbitrary number of polynomials in the integrand.

Deriving Deligne-Mumford stacks with perfect obstruction theories

Intersection theory on moduli spaces has lead to immense progress in certain areas of enumerative geometry. For some important areas, most notably counting stable maps and counting stable sheaves, it is important to work with a virtual fundamental class instead of the usual fundamental class of the moduli space. The crucial prerequisite for the existence of such a class is a two-term complex controlling deformations of the moduli space. Kontsevich conjectured in 1994 that there should exist derived version of spaces with this specific property. Another hint at the existence of these spaces comes from derived algebraic geometry. It is expected that for every pair of a space and a complex controlling deformations of the space their exists, under some additional hypothesis, a derived version of the space having the chosen complex as cotangent complex. In this thesis one version of these additional hypothesis is identified. We then show that every space admitting a two-term complex controlling deformations satisfies these hypothesis, and we finally construct the derived spaces.

Displacement Analysis of Under-Constrained Cable-Driven Parallel Robots

This dissertation studies the geometric static problem of under-constrained cable-driven parallel robots (CDPRs) supported by n cables, with n ≤ 6. The task consists of determining the overall robot configuration when a set of n variables is assigned. When variables relating to the platform posture are assigned, an inverse geometric static problem (IGP) must be solved; whereas, when cable lengths are given, a direct geometric static problem (DGP) must be considered. Both problems are challenging, as the robot continues to preserve some degrees of freedom even after n variables are assigned, with the final configuration determined by the applied forces. Hence, kinematics and statics are coupled and must be resolved simultaneously. In this dissertation, a general methodology is presented for modelling the aforementioned scenario with a set of algebraic equations. An elimination procedure is provided, aimed at solving the governing equations analytically and obtaining a least-degree univariate polynomial in the corresponding ideal for any value of n. Although an analytical procedure based on elimination is important from a mathematical point of view, providing an upper bound on the number of solutions in the complex field, it is not practical to compute these solutions as it would be very time-consuming. Thus, for the efficient computation of the solution set, a numerical procedure based on homotopy continuation is implemented. A continuation algorithm is also applied to find a set of robot parameters with the maximum number of real assembly modes for a given DGP. Finally, the end-effector pose depends on the applied load and may change due to external disturbances. An investigation into equilibrium stability is therefore performed.

On differential operators of Calabi-Yau type

This thesis is devoted to the study of Picard-Fuchs operators associated to one-parameter families of $n$-dimensional Calabi-Yau manifolds whose solutions are integrals of $(n,0)$-forms over locally constant $n$-cycles. Assuming additional conditions on these families, we describe algebraic properties of these operators which leads to the purely algebraic notion of operators of CY-type. rnMoreover, we present an explicit way to construct CY-type operators which have a linearly rigid monodromy tuple. Therefore, we first usernthe translation of the existence algorithm by N. Katz for rigid local systems to the level of tuples of matrices which was established by M. Dettweiler and S. Reiter. An appropriate translation to the level of differential operators yields families which contain operators of CY-type. rnConsidering additional operations, we are also able to construct special CY-type operators of degree four which have a non-linearly rigid monodromy tuple. This provides both previously known and new examples.

Absolute Hodge cycles and Deligne's principle b

I cicli di Hodge assoluti sono stati utilizzati da Deligne per dividere la congettura di Hodge in due sotto-congetture. La prima dice che tutte le classi di Hodge su una varietà complessa proiettiva liscia sono assolute, la seconda che le classi assolute sono algebriche. Deligne ha dato risposta affermativa alla prima sottocongettura nel caso delle varietà abeliane. La dimostrazione si basa su due teoremi, conosciuti rispettivamente come Principio A e Principio B. In questo lavoro vengono presentate la teoria delle classi di Hodge assolute e la dimostrazione del Principio B.

Quantum Einstein gravity - advancements of heat kernel-based renormalization group studies

The asymptotic safety scenario allows to define a consistent theory of quantized gravity within the framework of quantum field theory. The central conjecture of this scenario is the existence of a non-Gaussian fixed point of the theory's renormalization group flow, that allows to formulate renormalization conditions that render the theory fully predictive. Investigations of this possibility use an exact functional renormalization group equation as a primary non-perturbative tool. This equation implements Wilsonian renormalization group transformations, and is demonstrated to represent a reformulation of the functional integral approach to quantum field theory.rnAs its main result, this thesis develops an algebraic algorithm which allows to systematically construct the renormalization group flow of gauge theories as well as gravity in arbitrary expansion schemes. In particular, it uses off-diagonal heat kernel techniques to efficiently handle the non-minimal differential operators which appear due to gauge symmetries. The central virtue of the algorithm is that no additional simplifications need to be employed, opening the possibility for more systematic investigations of the emergence of non-perturbative phenomena. As a by-product several novel results on the heat kernel expansion of the Laplace operator acting on general gauge bundles are obtained.rnThe constructed algorithm is used to re-derive the renormalization group flow of gravity in the Einstein-Hilbert truncation, showing the manifest background independence of the results. The well-studied Einstein-Hilbert case is further advanced by taking the effect of a running ghost field renormalization on the gravitational coupling constants into account. A detailed numerical analysis reveals a further stabilization of the found non-Gaussian fixed point.rnFinally, the proposed algorithm is applied to the case of higher derivative gravity including all curvature squared interactions. This establishes an improvement of existing computations, taking the independent running of the Euler topological term into account. Known perturbative results are reproduced in this case from the renormalization group equation, identifying however a unique non-Gaussian fixed point.rn

Investigations on asymptotic safety of metric, tetrad and Einstein-Cartan gravity

Among the different approaches for a construction of a fundamental quantum theory of gravity the Asymptotic Safety scenario conjectures that quantum gravity can be defined within the framework of conventional quantum field theory, but only non-perturbatively. In this case its high energy behavior is controlled by a non-Gaussian fixed point of the renormalization group flow, such that its infinite cutoff limit can be taken in a well defined way. A theory of this kind is referred to as non-perturbatively renormalizable. In the last decade a considerable amount of evidence has been collected that in four dimensional metric gravity such a fixed point, suitable for the Asymptotic Safety construction, indeed exists. This thesis extends the Asymptotic Safety program of quantum gravity by three independent studies that differ in the fundamental field variables the investigated quantum theory is based on, but all exhibit a gauge group of equivalent semi-direct product structure. It allows for the first time for a direct comparison of three asymptotically safe theories of gravity constructed from different field variables. The first study investigates metric gravity coupled to SU(N) Yang-Mills theory. In particular the gravitational effects to the running of the gauge coupling are analyzed and its implications for QED and the Standard Model are discussed. The second analysis amounts to the first investigation on an asymptotically safe theory of gravity in a pure tetrad formulation. Its renormalization group flow is compared to the corresponding approximation of the metric theory and the influence of its enlarged gauge group on the UV behavior of the theory is analyzed. The third study explores Asymptotic Safety of gravity in the Einstein-Cartan setting. Here, besides the tetrad, the spin connection is considered a second fundamental field. The larger number of independent field components and the enlarged gauge group render any RG analysis of this system much more difficult than the analog metric analysis. In order to reduce the complexity of this task a novel functional renormalization group equation is proposed, that allows for an evaluation of the flow in a purely algebraic manner. As a first example of its suitability it is applied to a three dimensional truncation of the form of the Holst action, with the Newton constant, the cosmological constant and the Immirzi parameter as its running couplings. A detailed comparison of the resulting renormalization group flow to a previous study of the same system demonstrates the reliability of the new equation and suggests its use for future studies of extended truncations in this framework.

A commutative higher cycle map into Deligne-Beilinson cohomology

Das Ziel dieser Arbeit ist die Konstruktion eines Homomorphismus von partiell definierten, graduiert-kommutativen Algebren, der nach Ubergang zu rationalen Kohomologiegruppen mit der Regulatorabbildung reg zwischen motivischer und Deligne-Beilinson Kohomologie übereinstimmt.rnZu Beginn der Arbeit werden verschiedene Komplexe beschrieben, mit denen sich die motivische und die Deligne-Beilinson Kohomologie berechnen lassen.rnIm ersten Kapitel wird der Komplex der höheren Chow Ketten und der Unterkomplex der "alternierenden" Ketten "in guter Lage" eingeführt, die beide die motivische Kohomologie berechnen (letzterer mit rationalen Koeffizienten).rnIn den folgenden beiden Kapiteln werden Komplexe C_D und P_D beschrieben, mit denen sich die (rationale) Deligne-Beilinson Kohomologie berechnen lässt. Diese sind aufgebaut aus sogenannten Strömen, die im zweiten Kapitel eingeführt werden. Verknüpft sind die beiden Komplexe durch eine Auswertungsabbildung ev, die für rationale Koeffizienten zu einem Quasi-Isomorphismus wird. Auf beiden Komplexen lassen sich (Schnitt-)Produkte definieren, von denen jedoch nur das Produkt auf P_D gleichzeitig assoziativ und graduiert-kommutativ ist.rnIm vierten Kapitel wird ganz allgemein für eine Familie von Komplexen, die einer Reihe an Anforderungen genügt, ein (partiell definierter) Homomorphismus (der Regulator) von dem Komplex der höheren Chow Ketten in eben diese Komplexe konstruiert. Die beiden oben genannten Komplexe erfüllen diese Anforderungen und liefern daher Regulatoren reg_C und reg_P

Testing of subgrid scale (SGS) models for large-eddy simulation (LES) of turbulent channel flow

Sub-grid scale (SGS) models are required in order to model the influence of the unresolved small scales on the resolved scales in large-eddy simulations (LES), the flow at the smallest scales of turbulence. In the following work two SGS models are presented and deeply analyzed in terms of accuracy through several LESs with different spatial resolutions, i.e. grid spacings. The first part of this thesis focuses on the basic theory of turbulence, the governing equations of fluid dynamics and their adaptation to LES. Furthermore, two important SGS models are presented: one is the Dynamic eddy-viscosity model (DEVM), developed by \cite{germano1991dynamic}, while the other is the Explicit Algebraic SGS model (EASSM), by \cite{marstorp2009explicit}. In addition, some details about the implementation of the EASSM in a Pseudo-Spectral Navier-Stokes code \cite{chevalier2007simson} are presented. The performance of the two aforementioned models will be investigated in the following chapters, by means of LES of a channel flow, with friction Reynolds numbers $Re_\tau=590$ up to $Re_\tau=5200$, with relatively coarse resolutions. Data from each simulation will be compared to baseline DNS data. Results have shown that, in contrast to the DEVM, the EASSM has promising potentials for flow predictions at high friction Reynolds numbers: the higher the friction Reynolds number is the better the EASSM will behave and the worse the performances of the DEVM will be. The better performance of the EASSM is contributed to the ability to capture flow anisotropy at the small scales through a correct formulation for the SGS stresses. Moreover, a considerable reduction in the required computational resources can be achieved using the EASSM compared to DEVM. Therefore, the EASSM combines accuracy and computational efficiency, implying that it has a clear potential for industrial CFD usage.

Equations for the Drag Force and Aerodynamic Roughness Length of Urban Areas with Random Building Heights

We use a conceptual model to investigate how randomly varying building heights within a city affect the atmospheric drag forces and the aerodynamic roughness length of the city. The model is based on the assumptions regarding wake spreading and mutual sheltering effects proposed by Raupach (Boundary-Layer Meteorol 60:375-395, 1992). It is applied both to canopies having uniform building heights and to those having the same building density and mean height, but with variability about the mean. For each simulated urban area, a correction is determined, due to height variability, to the shear stress predicted for the uniform building height case. It is found that u (*)/u (*R) , where u (*) is the friction velocity and u (*R) is the friction velocity from the uniform building height case, is expressed well as an algebraic function of lambda and sigma (h) /h (m) , where lambda is the frontal area index, sigma (h) is the standard deviation of the building height, and h (m) is the mean building height. The simulations also resulted in a simple algebraic relation for z (0)/z (0R) as a function of lambda and sigma (h) /h (m) , where z (0) is the aerodynamic roughness length and z (0R) is z (0) found from the original Raupach formulation for a uniform canopy. Model results are in keeping with those of several previous studies.

On Groups with Two Isomorphism Classes of Derived Subgroups

The structure of groups which have at most two isomorphism classes of derived subgroups (D-2-groups) is investigated. A complete description of D-2-groups is obtained in the case where the derived subgroup is finite: the solution leads an interesting number theoretic problem. In addition, detailed information is obtained about soluble D-2-groups, especially those with finite rank, where algebraic number fields play an important role. Also, detailed structural information about insoluble D-2-groups is found, and the locally free D-2-groups are characterized.

IL28B alleles associated with poor hepatitis C virus (HCV) clearance protect against inflammation and fibrosis in patients infected with non-1 HCV genotypes

Genetic polymorphisms near IL28B are associated with spontaneous and treatment-induced clearance of hepatitis C virus (HCV), two processes that require the appropriate activation of the host immune responses. Intrahepatic inflammation is believed to mirror such activation, but its relationship with IL28B polymorphisms has yet to be fully appreciated. We analyzed the association of IL28B polymorphisms with histological and follow-up features in 2335 chronically HCV-infected Caucasian patients. Assessable phenotypes before any antiviral treatment included necroinflammatory activity (n = 1,098), fibrosis (n = 1,527), fibrosis progression rate (n = 1,312), and hepatocellular carcinoma development (n = 1,915). Associations of alleles with the phenotypes were evaluated by univariate analysis and multivariate logistic regression, accounting for all relevant covariates. The rare G allele at IL28B marker rs8099917-previously shown to be at risk of treatment failure-was associated with lower activity (P = 0.04), lower fibrosis (P = 0.02) with a trend toward lower fibrosis progression rate (P = 0.06). When stratified according to HCV genotype, most significant associations were observed in patients infected with non-1 genotypes (P = 0.003 for activity, P = 0.001 for fibrosis, and P = 0.02 for fibrosis progression rate), where the odds ratio of having necroinflammation or rapid fibrosis progression for patients with IL28B genotypes TG or GG versus TT were 0.48 (95% confidence intervals 0.30-0.78) and 0.56 (0.35-0.92), respectively. IL28B polymorphisms were not predictive of the development of hepatocellular carcinoma.

Genome-wide association study identifies variants associated with progression of liver fibrosis from HCV infection

Polymorphisms in IL28B were shown to affect clearance of hepatitis C virus (HCV) infection in genome-wide association (GWA) studies. Only a fraction of patients with chronic HCV infection develop liver fibrosis, a process that might also be affected by genetic factors. We performed a 2-stage GWA study of liver fibrosis progression related to HCV infection.