A topological Chern-Simons sigma model and new invariants of three-manifolds

Abstract

We construct a topological Chern-Simons sigma model on a Riemannian three-manifold M with gauge group G whose hyperkahler target space X is equipped with a G-action. Via a perturbative computation of its partition function, we obtain new topological invariants of M that define new weight systems which are characterized by both Lie algebra structure and hyperkahler geometry. In canonically quantizing the sigma model, we find that the partition function on certain M can be expressed in terms of Chern-Simons knot invariants of M and the intersection number of certain G-equivariant cycles in the moduli space of G-covariant maps from M to X. We also construct supersymmetric Wilson loop operators, and via a perturbative computation of their expectation value, we obtain new knot invariants of M that define new knot weight systems which are also characterized by both Lie algebra structure and hyperkahler geometry.