A Topological Chern-Simons Sigma Model and New Invariants of Three-Manifolds

TL;DR

This paper introduces a new topological sigma model based on Chern-Simons theory with hyperkahler target spaces, leading to novel invariants of three-manifolds and knots that incorporate both Lie algebra and hyperkahler geometry.

Contribution

It constructs a topological Chern-Simons sigma model with hyperkahler targets, deriving new three-manifold and knot invariants characterized by Lie algebra and hyperkahler structures.

Findings

New topological invariants of three-manifolds derived from the model

Novel knot invariants related to hyperkahler geometry

Partition functions expressed via Chern-Simons knot invariants

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.