Communications in Mathematical Physics

, Volume 342, Issue 2, pp 615–673 | Cite as

Supergeometry in Locally Covariant Quantum Field Theory

  • Thomas-Paul Hack
  • Florian Hanisch
  • Alexander Schenkel


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 \({{\mathfrak{A}}}\) : SLoc \({\to}\) 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 \({{\mathfrak{e}\mathfrak{A}}}\) : eSLoc \({\to}\) 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.


Natural Transformation Monoidal Category Supersymmetry Transformation Structure Sheaf Covariant Quantum 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Thomas-Paul Hack
    • 1
    • 2
  • Florian Hanisch
    • 3
  • Alexander Schenkel
    • 4
  1. 1.Dipartimento di MatematicaUniversità degli Studi di GenovaGenoaItaly
  2. 2.Institut für Theoretische PhysikUniversität LeipzigLeipzigGermany
  3. 3.Institut für MathematikUniversität PotsdamGolm (Potsdam)Germany
  4. 4.Department of MathematicsHeriot-Watt UniversityEdinburghUK

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