S-Structures for k-linear categories and the definition of a modular functor

Details S-Structures for k-linear categories and the definition of a modular functor S-Structures for k-linear categories and the definition of a modular functor

1998-02-18

arxiv.org/abs/math/9802089v1

Mathematics

Geometric Topology

Accepted for publication in the Journal of the LMS, April 1996

Scientific paper

Motivated by ideas from string theory and quantum field theory new invariants of knots and 3-dimensional manifolds have been constructed from complex algebraic structures such as Hopf algebras (Reshetikhin and Turaev), monoidal categories with additional structure (Turaev and Yetter), and modular functors (Walker and Kontsevich). These constructions are very closely related. We take a unifying categorical approach based on a natural 2-dimensional generalization of a topological field theory in the sense of Atiyah and Segal, and show that the axioms defining these complex algebraic structures are a consequence of the underlying geometry of surfaces. In particular, we show that any linear category over a field with an action of the surface category is semi-simple and Artinian.

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