This thesis concerns topics both in low-energy effective field theories from type IIB superstring flux compactifications and in fourdimensional, rigidly supersymmetric gauge theories. We introduce flux compactifications with so-called “warped throat” regions, which lead to large hierarchies of scales in the effective four-dimensional theory. The correspondence between a particular such throat and a five-dimensional Randall-Sundrum-like model is established. We shown how certain string-theoretic features of the compactification, such as moduli stabilization by fluxes or the presence of an unstabilized Kähler modulus, are incorporated in the five-dimensional picture. The KKLT construction for metastable de Sitter vacua is reviewed, as well as some possible modifications involving spontaneous F-term supersymmetry breaking. For KKLT-like models with their hidden sector localized inside a throat, the mediation of supersymmetry breaking to the visible sector is investigated. We review the mechanism of mixed modulus-anomaly mediation, and show that there can be additional equally important gravity-mediated contributions. We finally turn to the ISS model of metastable dynamical supersymmetry breaking in four dimensions, and present a renormalizable extension which generates a large hierarchy naturally. We also recapitulate how the ISS model may be obtained from a type IIB superstring model.
We employ worldline numerics to study Casimir effect and Gross-Neveu model. In this approach, the quantum fluctuations are mapped onto quantum mechanical path integrals, which are evaluated with Monte Carlo methods. For the Casimir effect, this allows the precise computation of the interaction energy for a Dirichlet scalar in Casimir geometries inaccessible to other methods. We study geometries involving curvature and edges, both are important for experiments and applications in nanotechnology, respectively. Significant reduction of numerical cost is gained by exploiting the symmetries of the worldline ensemble in combination with those of the configurations. Our results reveal the tight validity bounds of the commonly used proximity force approximation (PFA) and provide first insight into the effect of edges of finite plates on the Casimir force. In the Gross-Neveu model, we compute the trace over the fermion fluctuations using a worldline path integral, whose numerical evaluation is demonstrated for various configurations in the two dimensional model. We incorporate temperature and chemical potential in our formalism and perform first worldline numeric computations at finite values of these quantities. We thereby rediscover aspects of the established phase diagram. The methods employed can be extended to higher dimensions, to study the existence of a spatially inhomogeneous ground state beyond the two dimensional Gross-Neveu model.
We study learning and generalisation ability of a specific two-layer feed-forward neural network and compare its properties to that of a simple perceptron. The input patterns are mapped nonlinearly onto a hidden layer, much larger than the input layer, and this mapping is either fixed or may result from an unsupervised learning process. Such preprocessing of initially uncorrelated random patterns results in the correlated patterns in the hidden layer. The hidden-to-output mapping of the network is performed by a simple perceptron, trained using a supervised learning process. We investigate the effects of the correlations on the learning and generalisation properties as opposed to those of a simple perceptron with uncorrelated patterns. As it turns out, this architecture has some advantages over a simple perceptron.
Das zweidimensionale Hubbard Modell gilt als vielversprechendes, effektives Modell zur Beschreibung der Freiheitsgrade der Elektronen in den Kupferoxidschichten von Hochtemperatursupraleitern. Wir stellen einen Zugang zu diesem Modell mithilfe der funktionalen Renormierungsgruppe mit Fokus auf Antiferromagnetismus und d-Wellensupraleitung vor. Um die relevanten Freiheitsgrade auf allen Längenskalen explizit zugänglich zu machen, führen wir zusammengesetzte bosonische Felder ein, welche die Wechselwirkung zwischen den Fermionen vermitteln. Das spontane Brechen einer Symmetrie spiegelt sich in einem nichtverschwindenden Erwartungswert eines bosonischen Feldes wieder. Wir zeigen wie durch Spinwellenfluktuationen eine d-Wellenkopplung erzeugt wird. Des Weiteren berechnen wir die höchste Temperatur, bei der die Wechselwirkungsstärke der Elektronen im Renormierungsgruppenfluss divergiert sowohl für Antiferromagnetismus als auch für d-Wellensupraleitung über einen weiten Dotierungsbereich. Diese "pseudokritische" Temperatur signalisiert das Einsetzen von lokaler Ordnung. Außerdem wird die Temperaturabhängigkeit der d-Wellensupraleitung in einem vereinfachten Modell, welches sich durch eine Kopplung lediglich im d-Wellenkanal auszeichnet, studiert. Wir finden einen Phasenübergang vom Kosterlitz-Thouless-Typ.