Dr. Florian Hanisch

wissenschaftlicher Mitarbeiter

Kontakt
Raum:
2.09.0.20
Telefon:
+49 331 977-1347
...

Sprechstunde: Dienstags, 14.15-15 Uhr oder nach Vereinbarung per email 

  • Differentialgeometrie, insbes. Supergeometrie
  • unendlich-dimensionale Geometrie
  • Geometrische Analysis u. hyperbolische PDE
  • Mathematische Physik

2022 | The Fermionic integral on loop space and the Pfaffian line bundle | Florian Hanisch, Matthias LudewigZeitschrift: Journal of Mathematical PhysicsBand: 61, Issue 12Link zur Publikation , Link zum Preprint

The Fermionic integral on loop space and the Pfaffian line bundle

Autoren: Florian Hanisch, Matthias Ludewig (2022)

As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the "top degree component" of a differential form on it. In this paper, we show that a formula from finite dimensions generalizes to assign a sensible "top degree component" to certain composite forms, obtained by wedging with the exponential (in the exterior algebra) of the canonical 2-form on the loop space. The result is a section on the Pfaffian line bundle on the loop space. We then identify this with a section of the line bundle obtained by transgression of the spin lifting gerbe. These results are a crucial ingredient for defining the fermionic part of the supersymmetric path integral on the loop space.

Zeitschrift:
Journal of Mathematical Physics
Band:
61, Issue 12

2022 | A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold | Florian Hanisch, Matthias LudewigZeitschrift: Comm. Math. Phys.Verlag: SpringerSeiten: 1209–1239Band: 391Link zur Publikation , Link zum Preprint

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

Autoren: Florian Hanisch, Matthias Ludewig (2022)

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 using supersymmetric path integrals, as investigated by Alvarez-Gaumé, Atiyah, Bismut and Witten.

Zeitschrift:
Comm. Math. Phys.
Verlag:
Springer
Seiten:
1209–1239
Band:
391

2022 | A relative trace formula for obstacle scattering | Florian Hanisch, Alexander Strohmaier, Alden WatersZeitschrift: Duke Mathematical JournalVerlag: Duke University PressSeiten: 2233 - 2274Band: 171 (11)Link zur Publikation , Link zum Preprint

A relative trace formula for obstacle scattering

Autoren: Florian Hanisch, Alexander Strohmaier, Alden Waters (2022)

We consider the case of scattering of several obstacles in \(\mathbb{R}^d\) for \(d \geq 2\). Then 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.

Zeitschrift:
Duke Mathematical Journal
Verlag:
Duke University Press
Seiten:
2233 - 2274
Band:
171 (11)

2017 | Magnetic Geodesics via the Heat Flow | Volker Branding, Florian HanischZeitschrift: Asian Journal of MathematicsVerlag: International PressSeiten: 995-1014Band: 21, no. 6Link zur Publikation , Link zum Preprint

Magnetic Geodesics via the Heat Flow

Autoren: Volker Branding, Florian Hanisch (2017)

Magnetic geodesics describe the trajectory of a particle in a Riemannian manifold under the influence of an external magnetic field. In this article, we use the heat flow method to derive existence results for such curves. We first establish subconvergence of this flow to a magnetic geodesic under certain boundedness assumptions. It is then shown that these conditions are satisfied provided that either the magnetic field admits a global potential or the initial curve is sufficiently small. Finally, we discuss different examples which illustrate our results.

Zeitschrift:
Asian Journal of Mathematics
Verlag:
International Press
Seiten:
995-1014
Band:
21, no. 6

2016 | Supergeometry in locally covariant quantum field theory | Thomas Paul Hack, Florian Hanisch, Alexander SchenkelZeitschrift: Comm. Math. Phys.Verlag: SpringerSeiten: 615-673Band: 342, no. 2Link zur Publikation , Link zum Preprint

Supergeometry in locally covariant quantum field theory

Autoren: Thomas Paul Hack, Florian Hanisch, Alexander Schenkel (2016)

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.

Zeitschrift:
Comm. Math. Phys.
Verlag:
Springer
Seiten:
615-673
Band:
342, no. 2

2014 | A Supermanifold structure on Spaces of Morphisms between Supermanifolds | Florian HanischLink zum Preprint

A Supermanifold structure on Spaces of Morphisms between Supermanifolds

Autoren: Florian Hanisch (2014)

The aim of this work is the construction of a "supermanifold of morphisms X→Y", given two finite-dimensional supermanifolds X and Y. More precisely, we will define an object SC(X,Y) in the category of supermanifolds proposed by Molotkov and Sachse. Initially, it is given by the set-valued functor characterised by the adjunction formula Hom(PxX, Y) ≅ Hom(P, SC(X,Y)) where P ranges over all superpoints. We determine the structure of this functor in purely geometric terms: We show that it takes values in the set of certain differential operators and establish a bijective correspondence to the set of sections in certain vector bundles associated to X and Y. Equipping these spaces of sections with infinite-dimensional manifold structures using the convenient setting by Kriegl and Michor, we obtain at a supersmooth structure on SC(X,Y), i.e. a supermanifold of all morphisms X→Y.

2010 | The Spectral Action for Dirac Operators with skew-symmetric Torsion | Florian Hanisch, Frank Pfäffle, Christoph StephanZeitschrift: Comm. Math. Phys.Verlag: SpringerSeiten: 877-888Band: 300, no. 3Link zur Publikation , Link zum Preprint

The Spectral Action for Dirac Operators with skew-symmetric Torsion

Autoren: Florian Hanisch, Frank Pfäffle, Christoph Stephan (2010)

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.

Zeitschrift:
Comm. Math. Phys.
Verlag:
Springer
Seiten:
877-888
Band:
300, no. 3