# Dr. Matthias Ludewig

### ehemaliger Mitarbeiter

Kontakt

#### Arbeitsgebiete

• Path integrals and stochastic analysis on manifolds
• Semiclassical Analysis
• Global analysis and index theory
• Geometric quantum field theory

#### Drittmittelprojekte

• Finanzierung durch Stipendium der Potsdam Graduate School

#### Verschiedenes

1 + 2 + 3 + 4 + ... = - 1/12

#### Publikationen

2017 | Path Integrals on Manifolds with Boundary | Matthias Ludewig Zeitschrift: Comm. Math. Phys. Verlag: Springer Seiten: 621-640 Band: 354, no. 2 Link zur Publikation, Link zum Preprint

### Path Integrals on Manifolds with Boundary

#### Autoren: Matthias Ludewig (2017)

We give time-slicing path integral formulas for solutions to the heat equation corresponding to a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold with boundary. More specifically, we show that such a solution can be approximated by integrals over finite-dimensional path spaces of piecewise geodesics subordinated to increasingly fine partitions of the time interval. We consider a subclass of mixed boundary conditions which includes standard Dirichlet and Neumann boundary conditions.

Zeitschrift:
Comm. Math. Phys.
Verlag:
Springer
Seiten:
621-640
Band:
354, no. 2

2017 | Supersymmetric Path Integrals II: The Fermionic Integral and Pfaffian Line Bundles | Florian Hanisch, Matthias Ludewig Link zum Preprint

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

#### Autoren: Florian Hanisch, Matthias Ludewig (2017)

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∈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.

2017 | Supersymmetric Path Integrals I: Differential Forms on the Loop Space | Florian Hanisch, Matthias Ludewig Link zum Preprint

### Supersymmetric Path Integrals I: Differential Forms on the Loop Space

#### Autoren: Florian Hanisch, Matthias Ludewig (2017)

In this paper, we construct an integral map for differential forms on the loop space of Riemannian spin manifolds. In particular, the even and odd Bismut-Chern characters are integrable by this map, with their integrals given by indices of Dirac operators. We also show that our integral map satisfies a version of the localization principle in equivariant cohomology. This should provide a rigorous background for supersymmetry proofs of the Atiyah-Singer Index theorem.

2016 | Heat Kernel Asymptotics, Path Integrals and Infinite-Dimensional Determinants | Matthias Ludewig Link zum Preprint

### Heat Kernel Asymptotics, Path Integrals and Infinite-Dimensional Determinants

#### Autoren: Matthias Ludewig (2016)

We investigate the short-time expansion of the heat kernel of a Laplace type operator on a compact Riemannian manifold and show that the lowest order term of this expansion is given by the Fredholm determinant of the Hessian of the energy functional on a space of finite energy paths. This is the asymptotic behavior to be expected from formally expressing the heat kernel as a path integral and then (again formally) using Laplace's method on the integral. We also relate this to the zeta determinant of the Jacobi operator, which is another way to assign a determinant to the Hessian of the energy functional. We consider both the near-diagonal asymptotics as well as the behavior at the cut locus.

2016 | Strong Short Time Asymptotics and Convolution Approximation of the Heat Kernel | Matthias Ludewig Link zum Preprint

### Strong Short Time Asymptotics and Convolution Approximation of the Heat Kernel

#### Autoren: Matthias Ludewig (2016)

We give a short proof of a strong version of the short time asymptotic expansion of heat kernels associated to Laplace type operators acting on sections of vector bundles over compact Riemannian manifolds, including exponential decay of the difference of the approximate heat kernel and the true heat kernel. We use this to show that repeated convolution of the approximate heat kernels can be used to approximate the heat kernel on all of $M$, which is related to expressing the heat kernel as a path integral. This scheme is then applied to obtain a short-time asymptotic expansion of the heat kernel at the cut locus.

2015 | A Semiclassical Heat Kernel Proof of the Poincaré-Hopf Theorem | Matthias Ludewig Zeitschrift: Manuscripta Math. Verlag: Springer Seiten: 29-58 Band: 148, no. 1-2 Link zur Publikation, Link zum Preprint

### A Semiclassical Heat Kernel Proof of the Poincaré-Hopf Theorem

#### Autoren: Matthias Ludewig (2015)

We treat the Witten operator on the de Rham complex with semiclassical heat kernel methods to derive the Poincar\'e-Hopf theorem and degenerate generalizations of it. Thereby, we see how the semiclassical asymptotics of the Witten heat kernel are related to approaches using the Thom form of Mathai and Quillen.

Zeitschrift:
Manuscripta Math.
Verlag:
Springer
Seiten:
29-58
Band:
148, no. 1-2

2014 | Vector Fields with a non-degenerate Source | Matthias Ludewig Zeitschrift: J. Geom. Phys. Verlag: Elsevier Seiten: 59-76 Band: 79 Link zur Publikation, Link zum Preprint

### Vector Fields with a non-degenerate Source

#### Autoren: Matthias Ludewig (2014)

We discuss the solution theory of operators of the form ∇X + A, acting on smooth sections of a vector bundle with connection ∇ over a manifold M, where X is a vector field having a critical point with positive linearization at some point p ∈ M. As an operator on a suitable space of smooth sections Γ(U, V), 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 ∇, X and A.

Zeitschrift:
J. Geom. Phys.
Verlag:
Elsevier
Seiten:
59-76
Band:
79

2013 | Asymptotic eigenfunctions for Schrödinger operators on a vector bundle | Matthias Ludewig, Elke Rosenberger Link zum Preprint

### Asymptotic eigenfunctions for Schrödinger operators on a vector bundle

#### Autoren: Matthias Ludewig, Elke Rosenberger (2013)

In the limit ℏ→0, we analyze a class of Schr\"odinger operators H = ℏ2 L + ℏ W + V idEh acting on sections of a vector bundle Eh 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 ∈ M. We construct quasimodes of WKB-type near p for eigenfunctions associated with the low lying eigenvalues of H. These are obtained from eigenfunctions of the associated harmonic oscillator Hp,ℏ at p, acting on C(TpM, Ehp).