The vacuum expectation values of Wilson line operators $$ in the Chern-Simons theory are computed to second order of perturbation theory. The meaning of the framing procedure for knots is analyzed in the context of the Chern-Simons field theory. The relation between $$ and the link invariant polynomials is discussed. We derive an explicit analytic expression for the second coefficient of the Alexander-Conway polynomial, which is related to the ARF and Casson invariants. We present also some new relations between the HOMFLY coefficients

The quantization of the non-abelian Chern-Simons theory in three dimensions is performed in the framework of the BRS formalism. General covariance is preserved on the physical subspace. The perturbative analysis at two loops confirms that the model is finite. To this order, the radiative corrections to the two- and three-point proper vertices vanish

May 3, 1995 hep-th/9505027, IFUM 503/FT prepared for the special issue of Journal of Math. Phys. on Quantum geometry and diffeomorphism-invariant quantum field theory P.A.C.S. 02.40, 11.15, 04.60 AbstractIn this paper we discuss topological BF theories in 3 and 4 dimensions. Observables are associated to ordinary knots and links (in 3 dimensions) and to 2-knots (in 4 dimensions). The vacuum expectation values of such observables give a wide range of invariants. Here we consider mainly the 3 dimensional case, where these invariants include Alexander polynomials, HOMFLY polynomials and Kontsevich integrals.

We describe the explicit construction of Yang-Mills instantons on Asymptotically Locally Euclidean (ALE) spaces, following the work of Kronheimer and Nakajima. For multicenter ALE metrics, we determine the abelian instanton connections which are needed for the construction in the non-abelian case.We compute the partition function of Maxwell theories on ALE manifolds and comment on the issue of electromagnetic duality. We discuss the topological characterization of the instanton bundles as well as the identification of their moduli spaces. We generalize the 't Hooft ansatz to SU(2) instantons on ALE spaces and on other hyper-Kähler manifolds. Specializing to the Eguchi-Hanson gravitational background, we explicitly solve the ADHM equations for SU(2) gauge bundles with second