Quantum geometry of loops and the exact solubility of non-abelian gauge Chern-Simons theory-II

Abstract

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 onS ³ is nothing but the character ofSU(2) in which the rotations areSU(N)-valued fractional angles. Furthermore, we generalize our solutions to the case where the manifoldM ₃ is oriented, closed, and non-simply connected withH ₁(M ₃)=0 (a homology 3-sphere).