QUANTUM GEOMETRY OF LOOPS AND THE EXACT SOLUBILITY OF NON-ABELIAN GAUGE CHERN-SIMONS THEORY .2.

We quantize non-abelian Chern-Simons gauge theory in three dimensions in the presence of Wilson lines. We determine the theory dynamically in terms of the geometry of loops and show that it is exactly soluble. Remarkably the quantum loop equations are linear for S3 and they possess a class of solutions, among which is a non-critical Fermi string theory. Using these solutions we determine various important identities relevant to knot theory discovered recently by E. Witten, in particular, we show that the loop equation yields precisely the full exact skein relation of knot theory. As a byproduct we show that the partition function of an unknotted Wilson loop on S3 is nothing but the character of SU(2) in which the rotations are SU(N)-valued fractional angles. Furthermore, we generalize our solutions to the case where the manifold M3 is oriented, closed, and non-simply connected with H1(M3)=0 (a homology 3-sphere). © 1990 Springer-Verlag.