文档介绍：Communications in
Commun. Math. Phys. 121, 351-399 (1989) Mathematical
Physics
© Springer-Verlag 1989
Quantum Field Theory and the Jones Polynomial
Edward Witten **
School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton,
NJ 08540, USA
Abstract. It is shown that 2+1 dimensional quantum Yang-Mills theory, with
an action consisting purely of the Chern-Simons term, is exactly soluble and
gives a natural framework for understanding the Jones polynomial of knot
theory in three dimensional terms. In this version, the Jones polynomial can be
generalized from S3 to arbitrary three manifolds, giving invariants of three
manifolds that putable from a surgery presentation. These results shed
a surprising new light on conformal field theory in 1 -f-1 dimensions.
In a lecture at the Hermann Weyl Symposium last year [1], Michael Atiyah
proposed two problems for quantum field theorists. The first problem was to give
a physical interpretation to Donaldson theory. The second problem was to find an
intrinsically three dimensional definition of the Jones polynomial of knot theory.
These two problems might roughly be described as follows.
Donaldson theory is a key to understanding geometry in four dimensions.
Four is the physical dimension at least macroscopically, so one may take a slight
liberty and say that Donaldson theory is a key to understanding the geometry of
space-time. Geometers have long known that (via de Rham theory) the self-dual
and anti-self-dual Maxwell equations are related to natural topological invariants
of a four manifold, namely the second homology group and its intersection form.
For a simply connected four manifold, these are essentially the only classical
invariants, but they leave many basic questions out of reach. Donaldson's great
insight [2] was to realize that moduli spaces of solutions of the self-dual Yang-
Mills equations can be powerful tools for addressing these questions.
Donaldson the