Geometry & Topology Monographs 2 (1999), Proceedings of the Kirbyfest, paper no. 21, pages 407-453.

Group categories and their field theories

Frank Quinn

Abstract. A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1-dimensional representations are the invertible simple objects. This paper gives a detailed exploration of "topological quantum field theories" for group-categories, in hopes of finding clues to a better understanding of the general situation. Group-categories are classified in several ways extending results of Froelich and Kerler. Topological field theories based on homology and cohomology are constructed, and these are shown to include theories obtained from group-categories by Reshetikhin-Turaev constructions. Braided-commutative categories most naturally give theories on 4-manifold thickenings of 2-complexes; the usual 3-manifold theories are obtained from these by normalizing them (using results of Kirby) to depend mostly on the boundary of the thickening. This is worked out for group-categories, and in particular we determine when the normalization is possible and when it is not.

Keywords. Topological quantum field theory, braided category

AMS subject classification. Primary: 18D10. Secondary: 81R50, 55B20.

E-print: arXiv:math.GT/9811047

Submitted: 9 November 1998. (Revised: 27 January 1999.) Published: 21 November 1999.

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Frank Quinn
Department of Mathematics, Virginia Tech
Blacksburg VA 24061-0123, USA

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