Gauge Theories on Deformed Spaces

The aim of this review is to present an overview over available models and approaches to non-commutative gauge theory. Our main focus thereby is on gauge models formulated on flat Groenewold-Moyal spaces and renormalizability, but we will also review other deformations and try to point out common features. This review will by no means be complete and cover all approaches, it rather reflects a highly biased selection.Comment: v2 references added; v3 published versio

Quantum Corrections for Translation-Invariant Renormalizable Non-Commutative Phi^4 Theory

In this paper we elaborate on the translation-invariant renormalizable Phi^4 theory in 4-dimensional non-commutative space which was recently introduced by the Orsay group. By explicitly performing Feynman graph calculations at one loop and higher orders we illustrate the mechanism which overcomes the UV/IR mixing problem and ultimately leads to a renormalizable model. The obtained results show that the IR divergences are also suppressed in the massless case, which is of importance for the gauge field theoretic generalization of the scalar field model.Comment: 18 pages, v2: slightly extended version including a new section on one-loop renormalization, v3: minor revisio

Translation-invariant models for non-commutative gauge fields

Motivated by the recent construction of a translation-invariant renormalizable non-commutative model for a scalar field (see arXiv:0802.0791 [math-ph]), we introduce models for non-commutative U(1) gauge fields along the same lines. More precisely, we include some extra terms into the action with the aim of getting rid of the UV/IR mixing.Comment: 9 page

Identification of regulatory variants associated with genetic susceptibility to meningococcal disease.

Non-coding genetic variants play an important role in driving susceptibility to complex diseases but their characterization remains challenging. Here, we employed a novel approach to interrogate the genetic risk of such polymorphisms in a more systematic way by targeting specific regulatory regions relevant for the phenotype studied. We applied this method to meningococcal disease susceptibility, using the DNA binding pattern of RELA - a NF-kB subunit, master regulator of the response to infection - under bacterial stimuli in nasopharyngeal epithelial cells. We designed a custom panel to cover these RELA binding sites and used it for targeted sequencing in cases and controls. Variant calling and association analysis were performed followed by validation of candidate polymorphisms by genotyping in three independent cohorts. We identified two new polymorphisms, rs4823231 and rs11913168, showing signs of association with meningococcal disease susceptibility. In addition, using our genomic data as well as publicly available resources, we found evidences for these SNPs to have potential regulatory effects on ATXN10 and LIF genes respectively. The variants and related candidate genes are relevant for infectious diseases and may have important contribution for meningococcal disease pathology. Finally, we described a novel genetic association approach that could be applied to other phenotypes

The Mehler kernel

Zsfassung in dt. SpracheIm Rahmen der Untersuchung von BRST - invarianten Eichfeldmodellen, die einmal ultimativ zu einer zu allen Ordnungen renormierbaren nicht-kommutativen Eichfeldtheorie führen sollen, wird unter anderem ein Modell mit einem Oszillator - ähnlichen Zusatzterm diskutiert, welcher bereits wie von Grosse und Wulkenhaar gezeigt im skalaren Fall zu einer vollständig renormierbaren Theorie geführt hat.Um den Propagator solcher Modelle anschreiben zu können braucht man grob gesprochen das Inverse des Operators "p 2+x 2". Das ist der Mehler Kern, welcher in dieser Diplomarbeit untersucht werden soll. Es werden nicht nur eine vollständige Herleitung und diverse Eigenschaften desselben gegeben, sondern auch in einem speziellen Modell explizite Loop Rechnungen durchgeführt wo der Mehler Kern gebraucht wird.In the framework of studying noncommutative gauge field theory a model with an additional oscillator-like term in the action is discussed, which already like Grosse and Wulkenhaar showed is renormalizable to all orders in the case of a scalar phi**4 model. To be able to write down the propagator of such models one needs roughly speaking the inverse of the operator "p 2+x 2". This is the Mehler kernel, which will be discussed in this diploma thesis. We will not only give a full derivation and different properties of the latter, but we will also explicitly execute some loop calculations in a special model where the Mehler kernel is needed.8

Models with oscillator terms in noncommutative quantum field theory

Der Hauptfokus dieser Dissertationsarbeit liegt auf nichtkommutativen Modellen mit Oszillatortermen in der Wirkung. Das historisch gesehen erste dieser Modelle ist das erfolgreiche Grosse-Wulkenhaar (G.W.) Modell, von welchem bereits gezeigt wurde, dass es zu allen Ordnungen der Störungstheorie renormierbar ist.Bemerkenswerterweise löst es außerdem das Landau Geist Problem.In einem ersten Schritt haben wir das G.W. Modell direkt auf Eichtheorien verallgemeinert, wobei die Wirkung BRS invariant ist und die guten dämpfenden Eigenschaften der Skalartheorie beibehält, indem es denselben Propagator nutzt, den sogenannten Mehler Kern. Um manche aufwändigere Einschleifenrechnungen bewältigen zu können, haben wir ein Mathematica Paket programmiert, welches in der Lage ist, Feynman Graphen mit vielen Termen analytisch zu berechnen. Das Ergebnis dieser Betrachtungen war, dass neue Terme, die ursprünglich nicht in der Wirkung vorhanden waren, entstehen, was uns zu dem Schluss führte, dass wir besser von einer Theorie wegstarten sollten bei der diese Terme bereits eingebaut sind.Glücklicherweise gibt es eine Wirkung die diese vollständige Menge von Termen enthält. Sie kann erhalten werden, indem man ein Eichfeld an das Skalarfeld der G.W. Wirkung koppelt und dann Letzteres ausintegriert.Auf diese Art und Weise ``induziert'' man eine Eichfeldtheorie, welche deswegen Induzierte Eichfeldtheorie genannt wird. Trotz des Vorteils, dass sie per Konstruktion eichinvariant ist, enthält sie auch einige unphysikalische Terme, welche linear im Eichfeld sind.Vorteilhafterweise konnten wir diese Terme durch eine Eichung, die für diesen Zweck konstruiert wurde, loswerden. In dieser Eichung konnten wir wieder den Mehler Kern als Propagator für das Eichfeld etablieren.Weiters schafften wir es, den Geistpropagator zu berechnen, was sich als sehr aufwändig herausstellte.Schließlich war uns deswegen die Möglichkeit gegeben, mit den ersten Schleifenrechnungen anzufangen, welche auch das erwartete Verhalten zeigten. Der nächste Schritt ist die Renormierbarkeit des Modells zu zeigen, wobei einige Hinweise in diese Richtung auch gegeben werden.The main focus of this Ph.D. thesis is on noncommutative models involving oscillator terms in the action. The first one historically is the successful Grosse-Wulkenhaar (G.W.) model which has already been proven to be renormalizable to all orders of perturbation theory.Remarkably it is furthermore capable of solving the Landau ghost problem.In a first step, we have generalized the G.W. model to gauge theories in a very straightforward way, where the action is BRS invariant and exhibits the good damping properties of the scalar theory by using the same propagator, the so-called Mehler kernel. To be able to handle some more involved one-loop graphs we have programmed a powerful Mathematica package, which is capable of analytically computing Feynman graphs with many terms. The result of those investigations is that new terms originally not present in the action arise, which led us to the conclusion that we should better start from a theory where those terms are already built in.Fortunately there is an action containing this complete set of terms. It can be obtained by coupling a gauge field to the scalar field of the G.W. model, integrating out the latter, and thus ``inducing'' a gauge theory. Hence the model is called Induced Gauge Theory. Despite the advantage that it is by construction completely gauge invariant, it contains also some unphysical terms linear in the gauge field.Advantageously we could get rid of these terms using a special gauge dedicated to this purpose. Within this gauge we could again establish the Mehler kernel as gauge field propagator. Furthermore we where able to calculate the ghost propagator, which turned out to be very involved.Thus we were able to start with the first few loop computations showing the expected behavior. The next step is to show renormalizability of the model, where some hints towards this direction will also be given.10