@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}
}
@article{HackHanischSchenkel2016,
  author    = {Hack, Thomas-Paul and Hanisch, Florian and Schenkel, Alexander},
  title     = {Supergeometry in Locally Covariant Quantum Field Theory},
  series = {Communications in mathematical physics},
  volume    = {342},
  journal   = {Communications in mathematical physics},
  publisher = {Springer},
  address   = {New York},
  issn      = {0010-3616},
  doi       = {10.1007/s00220-015-2516-4},
  pages     = {615 -- 673},
  year      = {2016},
  abstract  = {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 U : 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 superquantum field theories in terms of enriched functors eU : 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 vertical bar 1-dimensions and the free Wess-Zumino model in 3 vertical bar 2-dimensions.},
  language  = {en}
}
@phdthesis{Hanisch2011,
  author    = {Hanisch, Florian},
  title     = {Variational problems on supermanifolds},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59757},
  school      = {Universit{\"a}t Potsdam},
  year      = {2011},
  abstract  = {In this thesis, we discuss the formulation of variational problems on supermanifolds. Supermanifolds incorporate bosonic as well as fermionic degrees of freedom. Fermionic fields take values in the odd part of an appropriate Grassmann algebra and are thus showing an anticommutative behaviour. However, a systematic treatment of these Grassmann parameters requires a description of spaces as functors, e.g. from the category of Grassmann algberas into the category of sets (or topological spaces, manifolds). After an introduction to the general ideas of this approach, we use it to give a description of the resulting supermanifolds of fields/maps. We show that each map is uniquely characterized by a family of differential operators of appropriate order. Moreover, we demonstrate that each of this maps is uniquely characterized by its component fields, i.e. by the coefficients in a Taylor expansion w.r.t. the odd coordinates. In general, the component fields are only locally defined. We present a way how to circumvent this limitation. In fact, by enlarging the supermanifold in question, we show that it is possible to work with globally defined components. We eventually use this formalism to study variational problems. More precisely, we study a super version of the geodesic and a generalization of harmonic maps to supermanifolds. Equations of motion are derived from an energy functional and we show how to decompose them into components. Finally, in special cases, we can prove the existence of critical points by reducing the problem to equations from ordinary geometric analysis. After solving these component equations, it is possible to show that their solutions give rise to critical points in the functor spaces of fields.},
  language  = {en}
}