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Construction of TQFT-Double Functors

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1765)

Abstract

In Sect. 6.1 of this chapter we give the precise definition of an extended TQFT and state the main result of this book, namely, the existence of a large class of such TQFT’s. Then we proceed with the construction of a TQFT from a given modular category. For this we define some functors, representing surfaces with holes, as coends of expressions involving tensor products, the Hopf algebra F and the functor of invariants. In a sequence of lemmas we show that a cobordism between two surfaces determines a natural transformation between the corresponding functors. To achieve this we first associate a natural transformation to an equivalence class of tangles under ambient isotopy. Then we show that it depends on a wider equivalence class, stable under topological moves, which defines the category of cobordisms as a quotient of the category of tangles (Theorem 3.0.6).

Keywords

  • Hopf Algebra
  • Natural Transformation
  • Ambient Isotopy
  • Modular Category
  • Braided Hopf Algebra

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2001 Springer-Verlag Berlin Heidelberg

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(2001). Construction of TQFT-Double Functors. In: Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners. Lecture Notes in Mathematics, vol 1765. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44625-7_7

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  • DOI: https://doi.org/10.1007/3-540-44625-7_7

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42416-1

  • Online ISBN: 978-3-540-44625-5

  • eBook Packages: Springer Book Archive