Supersymmetric Path Integrals II: The Fermionic Integral and Pfaffian Line Bundles

The Pfaffian line bundle of the covariant derivative and the transgression of the spin lifting gerbe are two canonically given real line bundles on the loop space of an oriented Riemannian manifold. It has been shown by Prat-Waldron that these line bundles are naturally isomorphic as metric line bundles and that the isomorphism maps their canonical sections to each other. In this paper, we provide a vast generalization of his results, by showing that there are natural sections of the corresponding line bundles for any $N \in \N$, which are mapped to each other under this isomorphism (with the previously known being the one for $N=0$). These canonical sections are important to define the fermionic part of the supersymmetric path integral on the loop space.Comment: The contents of this article have been integrated into arXiv:1709.10027, which now contains a simplified presentation of the result

A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold

We give a rigorous construction of the path integral in N=1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler–Jones–Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Güneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah–Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaumé, Atiyah, Bismut and Witten

Impuls-, Drehimpuls- und Energieerhaltung beim eindimensionalen Stoß zweier Kugeln: Impressionen aus dem Praktikumsalltag

In einem Videoanalyseexperiment wurde der Stoß einer rollenden mit einer ruhenden Kugel untersucht. Da der Kraftstoß während der Kollision den Linearimpuls zwischen den Kugeln transferiert, aber kein entsprechender Drehmomentstoß auftritt, erfüllen die Kugeln nach dem Stoß die Rollbedingungen nicht. Die Kugeln werden daher durch Reibung entweder beschleunigt oder verzögert. Die Analyse des Stoßes liefert Daten zum Verhältnis der End- und Anfangsgeschwindigkeiten, zum Elastizitätsgrad des Stoßes, zum Reibungskoeffizienten und zum Anteil des beim Stoß übertragenen Drehimpulses. Der Praktikumserfolg der Studierenden bei diesem Versuch wird kurz diskutiert

Impuls-, Drehimpuls- und Energieerhaltung beim eindimensionalen Stoß zweier Kugeln: Impressionen aus dem Praktikumsalltag

In einem Videoanalyseexperiment wurde der Stoß einer rollenden mit einer ruhenden Kugel untersucht. Da der Kraftstoß während der Kollision den Linearimpuls zwischen den Kugeln transferiert, aber kein entsprechender Drehmomentstoß auftritt, erfüllen die Kugeln nach dem Stoß die Rollbedingungen nicht. Die Kugeln werden daher durch Reibung entweder beschleunigt oder verzögert. Die Analyse des Stoßes liefert Daten zum Verhältnis der End- und Anfangsgeschwindigkeiten, zum Elastizitätsgrad des Stoßes, zum Reibungskoeffizienten und zum Anteil des beim Stoß übertragenen Drehimpulses. Der Praktikumserfolg der Studierenden bei diesem Versuch wird kurz diskutiert

A relative trace formula for obstacle scattering

We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$. In this setting the absolutely continuous part of the Laplace operator $\Delta$ with Dirichlet boundary conditions and the free Laplace operator $\Delta_0$ are unitarily equivalent. For suitable functions that decay sufficiently fast we have that the difference $g(\Delta)-g(\Delta_0)$ is a trace-class operator and its trace is described by the Krein spectral shift function. In this paper we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles we consider the Laplace operators $\Delta_1$ and $\Delta_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then $g(\Delta) - g(\Delta_1) - g(\Delta_2) + g(\Delta_0)$ is a trace class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman-Krein formula. In case $g(x)=x^\frac{1}{2}$ the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using non-rigorous path integral derivations and our formula provides both a rigorous justification as well as a generalisation.Comment: 33 pages, 1 figure, revision of proof presentation relative to previous versio

Supergeometry in locally covariant quantum field theory

In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a functor A : SLoc --> S*Alg to the category of super-*-algebras which can be interpreted as a non-interacting super-quantum field theory. This construction turns out to disregard supersymmetry transformations as the morphism sets in the above categories are too small. We then solve this problem by using techniques from enriched category theory, which allows us to replace the morphism sets by suitable morphism supersets that contain supersymmetry transformations as their higher superpoints. We construct super-quantum field theories in terms of enriched functors eA : eSLoc --> eS*Alg between the enriched categories and show that supersymmetry transformations are appropriately described within the enriched framework. As examples we analyze the superparticle in 1|1-dimensions and the free Wess-Zumino model in 3|2-dimensions

The Spectral Action for Dirac Operators with skew-symmetric Torsion

We derive a formula for the gravitational part of the spectral action for Dirac operators on 4-dimensional manifolds with totally anti-symmetric torsion. We find that the torsion becomes dynamical and couples to the traceless part of the Riemann curvature tensor. Finally we deduce the Lagrangian for the Standard Model of particle physics in presence of torsion from the Chamseddine-Connes Dirac operator.Comment: Longer introduction and conclusion adde

Carcinoma cells misuse the host tissue damage response to invade the brain

The metastatic colonization of the brain by carcinoma cells is still barely understood, in particular when considering interactions with the host tissue. The colonization comes with a substantial destruction of the surrounding host tissue. This leads to activation of damage responses by resident innate immune cells to protect, repair, and organize the wound healing, but may distract from tumoricidal actions. We recently demonstrated that microglia, innate immune cells of the CNS, assist carcinoma cell invasion. Here we report that this is a fatal side effect of a physiological damage response of the brain tissue. In a brain slice coculture model, contact with both benign and malignant epithelial cells induced a response by microglia and astrocytes comparable to that seen at the interface of human cerebral metastases. While the glial damage response intended to protect the brain from intrusion of benign epithelial cells by inducing apoptosis, it proved ineffective against various malignant cell types. They did not undergo apoptosis and actually exploited the local tissue reaction to invade instead. Gene expression and functional analyses revealed that the C-X-C chemokine receptor type 4 (CXCR4) and WNT signaling were involved in this process. Furthermore, CXCR4-regulated microglia were recruited to sites of brain injury in a zebrafish model and CXCR4 was expressed in human stroke patients, suggesting a conserved role in damage responses to various types of brain injuries. Together, our findings point to a detrimental misuse of the glial damage response program by carcinoma cells resistant to glia-induced apoptosis

Methods for identification of CA125 from ovarian cancer ascites by high resolution mass spectrometry

CA125 is the most widely used tumour marker in ovarian cancer with unsatisfactory sensitivity and specificity especially at early stage. It is quantified by antibody-based immunoassays; however different molecular weight isoforms have been described in the literature which have never been validated by mass spectrometry, potentially affecting the diagnostic accuracy and clinical reliability of the test. In this study, CA125 was detected by Western blot and its identity confirmed by mass spectrometry. Two-dimensional (2D) gel electrophoresis in combination with mass spectrometry revealed that positive Western blot signals up to 500 kDa are most likely false-positive interactions of M11-like and OC125-like antibodies. Fibronectin, identified as one of these false-positive interaction partners, increased the reading for CA125 in a first generation ELISA significantly (p = 0.02). The existence of low-molecular weight isoforms of CA125 is therefore questionable and is most likely reflecting cross-reactivity of the antibodies with other proteins. This would explain the conflicting reports on the molecular structure of CA125 and also the inconsistency of CA125 levels by different ELISAs. Our results are also the first steps towards a mass spectrometric assay for CA125 quantification, which would improve sensitivity and reliability.Florian Weiland, Katarina Fritz, Martin K. Oehler and Peter Hoffman