Hyper-Kähler geometry and invariants of three-manifolds

L. Rozansky, E. Witten

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108 Scopus citations


We study a 3-dimensional topological sigma-model, whose target space is a hyper-Kähler 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 Hubert space obtained by quantizing the sigma-model on a two-dimensional torus.

Original languageEnglish (US)
Pages (from-to)401-458
Number of pages58
JournalSelecta Mathematica, New Series
Issue number3
StatePublished - 1997
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • General Physics and Astronomy


  • Casson's invariant
  • Topological sigma-models


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