**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.*

ruthel@ma.huji.ac.il