(抜粋) 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.
Acknowledgements. Thiswork originated with the realization that some results about conformat field theory described by G. Segal could be given a three dimensional interpretation by considering a gauge theory with Chern-Simons action. I am grateful to Segal for explaining his results, and to M. Atiyah for interesting me in and educating me about the Jones polynomial. V.F.R. Jones and L. Kauffman, and other participants at the IAMP Congress, raised many relevant questions. Finally, I must thank S. Deser and D.J. Gross for pointing out Polyakov's paper, G. Moore and N. Seiberg for explanations of their work, and the organizers of the IAMP Congress for their hospitality.
