@article{Ludewig2014, author = {Ludewig, Matthias}, title = {Vector fields with a non-degenerate source}, series = {Journal of geometry and physics}, volume = {79}, journal = {Journal of geometry and physics}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0393-0440}, doi = {10.1016/j.geomphys.2014.01.014}, pages = {59 -- 76}, year = {2014}, abstract = {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.}, language = {en} } @article{LudewigRoos2020, author = {Ludewig, Matthias and Roos, Saskia}, title = {The chiral anomaly of the free fermion in functorial field theory}, series = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics}, volume = {21}, journal = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics}, number = {4}, publisher = {Springer International Publishing AG}, address = {Cham (ZG)}, issn = {1424-0637}, doi = {10.1007/s00023-020-00893-6}, pages = {1191 -- 1233}, year = {2020}, abstract = {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.}, language = {en} } @phdthesis{Ludewig2016, author = {Ludewig, Matthias}, title = {Path integrals on manifolds with boundary and their asymptotic expansions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-94387}, school = {Universit{\"a}t Potsdam}, pages = {146}, year = {2016}, abstract = {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.}, language = {en} } @article{LudewigRosenberger2020, author = {Ludewig, Matthias and Rosenberger, Elke}, title = {Asymptotic eigenfunctions for Schr{\"o}dinger operators on a vector bundle}, series = {Reviews in mathematical physics}, volume = {32}, journal = {Reviews in mathematical physics}, number = {7}, publisher = {World Scientific}, address = {Singapore}, issn = {0129-055X}, doi = {10.1142/S0129055X20500208}, pages = {28}, year = {2020}, abstract = {In the limit (h) over bar -> 0, we analyze a class of Schr{\"o}dinger 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.}, language = {en} } @article{Ludewig2015, author = {Ludewig, Matthias}, title = {A semiclassical heat kernel proof of the Poincare-Hopf theorem}, series = {Manuscripta mathematica}, volume = {148}, journal = {Manuscripta mathematica}, number = {1-2}, publisher = {Springer}, address = {Heidelberg}, issn = {0025-2611}, doi = {10.1007/s00229-015-0741-y}, pages = {29 -- 58}, year = {2015}, abstract = {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.}, language = {en} } @article{HanischLudewig2022, author = {Hanisch, Florian and Ludewig, Matthias}, title = {A rigorous construction of the supersymmetric path integral associated to a compact spin manifold}, series = {Communications in mathematical physics}, volume = {391}, journal = {Communications in mathematical physics}, number = {3}, publisher = {Springer}, address = {Berlin ; Heidelberg}, issn = {0010-3616}, doi = {10.1007/s00220-022-04336-7}, pages = {1209 -- 1239}, year = {2022}, abstract = {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.}, language = {en} }