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Hyper-Kahler Geometry and Invariants of Three-Manifolds

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arxiv hep-th/9612216 v3 pith:3A6PXSQ7 submitted 1996-12-20 hep-th math.QAq-alg

L. Rozansky , E. Witten This is my paper · ORCID
classification hep-thmath.QAq-alg
keywords sigma-modelfunctionpartitioncalculationchern-simonsfinitehyper-kahleridentity
verification ladder T0 review T1 audit T2 compute T3 formal
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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.

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