Hyper-Kahler Geometry and Invariants of Three-Manifolds

TL;DR

Rozansky and Witten introduce a twisted N=4-like 3D sigma-model with a hyper-Kähler target X, producing finite-type invariants of 3-manifolds through perturbative graph weights generated by the curvature of X. The theory parallels Chern-Simons perturbation theory but replaces Lie algebra structure constants with curvature-derived tensors, enforcing IHX via a Bianchi identity. For special X (notably the Atiyah-Hitchin moduli space AH and K3), the invariants reproduce Casson-Walker-type quantities, with explicit surgery and gluing behavior derived from SL(2,Z) actions on torus quantization and boundary-state formalisms. The paper also clarifies how these invariants simplify to classical 3-manifold invariants in various b1(M) regimes and provides explicit calculations for rational homology spheres and low-b1 manifolds, connecting to Lescop’s Casson-Walker theory. Overall, it proposes a physically motivated, geometrically rich approach to finite-type 3-manifold invariants with concrete links to known topological invariants and surgery formulas.

Abstract

We study a 3-dimensional topological sigma-model, whose target space is a hyper-Kahler manifold X. A Feynman diagram calculation of its partition function demonstrates that it is a finite type invariant of 3-manifolds which is similar in structure to those appearing in the perturbative calculation of the Chern-Simons partition function. The sigma-model suggests a new system of weights for finite type invariants of 3-manifolds, described by trivalent graphs. The Riemann curvature of X plays the role of Lie algebra structure constants in Chern-Simons theory, and the Bianchi identity plays the role of the Jacobi identity in guaranteeing the so-called IHX relation among the weights. We argue that, for special choices of X, the partition function of the sigma-model yields the Casson-Walker invariant and its generalizations. We also derive Walker's surgery formula from the SL(2,Z) action on the finite-dimensional Hilbert space obtained by quantizing the sigma-model on a two-dimensional torus.