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

TL;DR

This paper unifies various algebraic structures used in topological quantum field theories through a categorical approach, revealing their geometric foundations and properties like semi-simplicity.

Contribution

It introduces a 2-dimensional categorical framework that explains the axioms of complex algebraic structures in TQFTs as consequences of surface geometry.

Findings

Categories with surface actions are semi-simple.

The axioms of Hopf algebras, modular functors, and related structures follow from surface geometry.

A unifying categorical approach links string theory, quantum field theory, and algebraic invariants.

Abstract

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.