1.1 General introduction

In recent years there has been a remarkable renais­sance in the interaction between geometry and physics. After a long fallow period in which mathematicians and physicists pursued apparently independent paths their interests have now converged in a striking manner. However, it appears that parallel problems were being investigated in the past but a common language and framework were missing. This has now been rectified with gauge theory (alias the theory of connec­tions) providing the common ground.

In earlier periods geometry and physics interacted at the classical level, as in Einstein's theory of general relativity, with gravitational force being interpreted in terms of cur­vature. The new feature of the present interaction is that quantum theory is now involved and it turns out to have significant relations with topology. Thus geometry is involved in a global and not purely local way.

A somewhat surprising feature of the new developments is that quantum field theory seems to tie up with deep properties of low-dimensional geometry, i.e. in dimensions 2, 3 and 4 [3]. Thus the exciting new results of Donaldson [10] on four- dimensional manifolds, and the associated theory of Floer [13] on three-dimensional manifolds, are intimately linked to Yang-Mills theory. This has been made even clearer by Witten [35], where the Donaldson-Floer theory is interpreted as a topological quantum field theory in 3 +1 dimensions.A slightly different case arises from the recently discovered polynomial invariants of knots by Vaughan Jones [17]. These are related to physics in various ways but the most funda­mental is due to Witten [36] who has shown that the Jones invariants have a natural interpretation in terms of a topologi­cal quantum field theory in 2 +1 dimensions. My purpose in these lectures is to present this new theory of Witten. Shortage of time and the present novelty and incompleteness of the theory mean that this is not a definitive treatment. Rather it is an introduction to Witten's ideas, presented from the mathe­matical point of view. The whole subject is still developing rapidly and a provisional account accessible to mathema­ticians may serve a useful purpose.