Modular properties of ribbon abelian categories

TL;DR

This paper develops a categorical framework for ribbon abelian categories, constructing modular functors from labeled surface categories and ribbon categories, advancing mathematical understanding of topological quantum field theories.

Contribution

It introduces new generators and relations for categories of ribbon graphs and constructs functors to vector spaces, linking surface topology with ribbon categories in a novel way.

Findings

Constructs a functor from a category of labeled ribbon graphs to vector spaces.

Establishes a modular functor from surface categories to vector spaces.

Provides a framework for proofs in related topological quantum field theory research.

Abstract

A category N of labeled (oriented) trivalent graphs (nets) or ribbon graphs is extended by new generators called fusing, braiding, twist and switch with relations which can be called Moore--Seiberg relations. A functor to N is constructed from the category Surf of oriented surfaces with labeled boundary and their homeomorphisms. Given an (eventually non-semisimple) k-linear abelian ribbon braided category C with some finiteness conditions we construct a functor from a central extension of N with the set of labels ObC to k-vector spaces. Composing the functors we get a modular functor from a central extension of Surf to k-vector spaces. This is a mathematical paper which explains how to get proofs for its hep-th companion paper, which should be read first. Complete proofs are not given here. (Talk at Second Gauss Simposium, Munich, August 1993.)