TY - JOUR A1 - Ludewig, Matthias T1 - Vector fields with a non-degenerate source JF - Journal of geometry and physics N2 - We discuss the solution theory of operators of the form del(x) + A, acting on smooth sections of a vector bundle with connection del over a manifold M, where X is a vector field having a critical point with positive linearization at some point p is an element of M. As an operator on a suitable space of smooth sections Gamma(infinity)(U, nu), it fulfills a Fredholm alternative, and the same is true for the adjoint operator. Furthermore, we show that the solutions depend smoothly on the data del, X and A. KW - Recursive transport equations KW - First order PDE KW - Fredholm alternative KW - Heat kernel coefficients KW - WKB expansion KW - Semiclassical analysis Y1 - 2014 U6 - https://doi.org/10.1016/j.geomphys.2014.01.014 SN - 0393-0440 SN - 1879-1662 VL - 79 SP - 59 EP - 76 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Ludewig, Matthias A1 - Roos, Saskia T1 - The chiral anomaly of the free fermion in functorial field theory JF - Annales Henri Poincaré : a journal of theoretical and mathematical physics N2 - When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, but an element of a complex line. In functorial field theory language, this means that the theory is twisted, which gives rise to an anomaly theory. In this paper, we give a detailed construction of this anomaly theory, as a functor that sends manifolds to infinite-dimensional Clifford algebras and bordisms to bimodules. Y1 - 2020 U6 - https://doi.org/10.1007/s00023-020-00893-6 SN - 1424-0637 SN - 1424-0661 VL - 21 IS - 4 SP - 1191 EP - 1233 PB - Springer International Publishing AG CY - Cham (ZG) ER - TY - THES A1 - Ludewig, Matthias T1 - Path integrals on manifolds with boundary and their asymptotic expansions T1 - Pfadintegrale auf Mannigfaltigkeiten mit Rand und ihre asymptotischen Entwicklungen N2 - It is "scientific folklore" coming from physical heuristics that solutions to the heat equation on a Riemannian manifold can be represented by a path integral. However, the problem with such path integrals is that they are notoriously ill-defined. One way to make them rigorous (which is often applied in physics) is finite-dimensional approximation, or time-slicing approximation: Given a fine partition of the time interval into small subintervals, one restricts the integration domain to paths that are geodesic on each subinterval of the partition. These finite-dimensional integrals are well-defined, and the (infinite-dimensional) path integral then is defined as the limit of these (suitably normalized) integrals, as the mesh of the partition tends to zero. In this thesis, we show that indeed, solutions to the heat equation on a general compact Riemannian manifold with boundary are given by such time-slicing path integrals. Here we consider the heat equation for general Laplace type operators, acting on sections of a vector bundle. We also obtain similar results for the heat kernel, although in this case, one has to restrict to metrics satisfying a certain smoothness condition at the boundary. One of the most important manipulations one would like to do with path integrals is taking their asymptotic expansions; in the case of the heat kernel, this is the short time asymptotic expansion. In order to use time-slicing approximation here, one needs the approximation to be uniform in the time parameter. We show that this is possible by giving strong error estimates. Finally, we apply these results to obtain short time asymptotic expansions of the heat kernel also in degenerate cases (i.e. at the cut locus). Furthermore, our results allow to relate the asymptotic expansion of the heat kernel to a formal asymptotic expansion of the infinite-dimensional path integral, which gives relations between geometric quantities on the manifold and on the loop space. In particular, we show that the lowest order term in the asymptotic expansion of the heat kernel is essentially given by the Fredholm determinant of the Hessian of the energy functional. We also investigate how this relates to the zeta-regularized determinant of the Jacobi operator along minimizing geodesics. N2 - Es ist "wissenschaftliche Folklore", abgeleitet von der physikalischen Anschauung, dass Lösungen der Wärmeleitungsgleichung auf einer riemannschen Mannigfaltigkeit als Pfadintegrale dargestellt werden können. Das Problem mit Pfadintegralen ist allerdings, dass schon deren Definition Mathematiker vor gewisse Probleme stellt. Eine Möglichkeit, Pfadintegrale rigoros zu definieren ist endlich-dimensionale Approximation, oder time-slicing-Approximation: Für eine gegebene Unterteilung des Zeitintervals in kleine Teilintervalle schränkt man den Integrationsbereich auf diejenigen Pfade ein, die auf jedem Teilintervall geodätisch sind. Diese endlichdimensionalen Integrale sind wohldefiniert, und man definiert das (unendlichdimensionale) Pfadintegral als den Limes dieser (passend normierten) Integrale, wenn die Feinheit der Unterteilung gegen Null geht. In dieser Arbeit wird gezeigt, dass Lösungen der Wärmeleitungsgleichung auf einer allgemeinen riemannschen Mannigfaltigkeit tatsächlich durch eine solche endlichdimensionale Approximation gegeben sind. Hierbei betrachten wir die Wärmeleitungsgleichung für allgemeine Operatoren von Laplace-Typ, die auf Schnitten in Vektorbündeln wirken. Wir zeigen auch ähnliche Resultate für den Wärmekern, wobei wir uns allerdings auf Metriken einschränken müssen, die eine gewisse Glattheitsbedingung am Rand erfüllen. Eine der wichtigsten Manipulationen, die man an Pfadintegralen vornehmen möchte, ist das Bilden ihrer asymptotischen Entwicklungen; in Falle des Wärmekerns ist dies die Kurzzeitasymptotik. Um die endlich-dimensionale Approximation hier nutzen zu können, ist es nötig, dass die Approximation uniform im Zeitparameter ist. Dies kann in der Tat erreicht werden; zu diesem Zweck geben wir starke Fehlerabschätzungen an. Schließlich wenden wir diese Resultate an, um die Kurzzeitasymptotik des Wärmekerns (auch im degenerierten Fall, d.h. am Schittort) herzuleiten. Unsere Resultate machen es außerdem möglich, die asymptotische Entwicklung des Wärmekerns mit einer formalen asymptotischen Entwicklung der unendlichdimensionalen Pfadintegrale in Verbindung zu bringen. Auf diese Weise erhält man Beziehungen zwischen geometrischen Größen der zugrundeliegenden Mannigfaltigkeit und solchen des Pfadraumes. Insbesondere zeigen wir, dass der Term niedrigster Ordnung in der asymptotischen Entwicklung des Wärmekerns im Wesentlichen durch die Fredholm-Determinante der Hesseschen des Energie-Funktionals gegeben ist. Weiterhin untersuchen wir die Verbindung zur zeta-regularisierten Determinante des Jakobi-Operators entlang von minimierenden Geodätischen. KW - heat equation KW - heat kernel KW - path integral KW - determinant KW - asymptotic expansion KW - Laplace expansion KW - heat asymptotics KW - Wiener measure KW - Wärmeleitungsgleichung KW - Wärmekern KW - Pfadintegrale KW - asymptotische Entwicklung KW - Determinante Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-94387 ER - TY - JOUR A1 - Ludewig, Matthias A1 - Rosenberger, Elke T1 - Asymptotic eigenfunctions for Schrödinger operators on a vector bundle JF - Reviews in mathematical physics N2 - In the limit (h) over bar -> 0, we analyze a class of Schrödinger operators H-(h) over bar = (h) over bar L-2 + (h) over barW + V .id(epsilon) acting on sections of a vector bundle epsilon over a Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has a non-degenerate minimum at some point p is an element of M. We construct quasimodes of WKB-type near p for eigenfunctions associated with the low-lying eigenvalues of H-(h) over bar. These are obtained from eigenfunctions of the associated harmonic oscillator H-p,H-(h) over bar at p, acting on smooth functions on the tangent space. KW - Semi-classical analysis KW - WKB approximation KW - Schrödinger operators KW - semi-classical limit Y1 - 2020 U6 - https://doi.org/10.1142/S0129055X20500208 SN - 0129-055X SN - 1793-6659 VL - 32 IS - 7 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Ludewig, Matthias T1 - A semiclassical heat kernel proof of the Poincare-Hopf theorem JF - Manuscripta mathematica N2 - We consider the semiclassical asymptotic expansion of the heat kernel coming from Witten's perturbation of the de Rham complex by a given function. For the index, one obtains a time-dependent integral formula which is evaluated by the method of stationary phase to derive the Poincare-Hopf theorem. We show how this method is related to approaches using the Thom form of Mathai and Quillen. Afterwards, we use a more general version of the stationary phase approximation in the case that the perturbing function has critical submanifolds to derive a degenerate version of the Poincare-Hopf theorem. Y1 - 2015 U6 - https://doi.org/10.1007/s00229-015-0741-y SN - 0025-2611 SN - 1432-1785 VL - 148 IS - 1-2 SP - 29 EP - 58 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Hanisch, Florian A1 - Ludewig, Matthias T1 - A rigorous construction of the supersymmetric path integral associated to a compact spin manifold JF - Communications in mathematical physics N2 - 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 Guneysu 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-Gaume, Atiyah, Bismut and Witten. Y1 - 2022 U6 - https://doi.org/10.1007/s00220-022-04336-7 SN - 0010-3616 SN - 1432-0916 VL - 391 IS - 3 SP - 1209 EP - 1239 PB - Springer CY - Berlin ; Heidelberg ER -