  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.
