An Introduction to Topological Field Theory

R.J. Lawrence

Abstract: A topological quantum field theory (TQFT) is an, almost, metric independent quantum field theory that gives rise to topological invariants of the background manifold. The best known example of a 3-dimensional TQFT is Chern-Simons-Witten theory, in which the expectation value of an observable, obtained as the product of the Wilson loops associated with a link, is the generalised Jones invariant of the link. Unfortunately the form of the invariants obtained by this procedure is that of an integral over an infinite dimensional space on which, for a mathematician, a measure has not yet been rigorously defined. Various ways of avoiding this difficulty have been developed. These fall into two main categories, namely, formal manipulations of Witten's path integral into a form which can then be rigorously defined, and axiomatic encapsulations of the properties of TQFTs. In these notes we will be concerned with the second path, demonstrating how complex categorical and algebraic structures appear, from apparently simple geometry. These structures are related to the quantum group structures which arise in other approaches.

Keywords: topological field theory, category theory, higher algebraic structures, polyhedral decompositions, manifold invariants

AMS subject classification: 58Dxx 18-xx 57M25 57R57

Length: 40 pages

Reference: Proc. Symp. Appl. Math. 51 (1996) 89-128. MR1372766 (97a:57020) (review by Gretchen Wright .)

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Last updated on June 11th, 2000.