Hyper-Kähler geometry and invariants of three-manifolds
-
L. Rozansky [1] ; E. Witten [2]
-
[1] University of Illinois at Chicago
University of Illinois at Chicago
City of Chicago, Estados Unidos
-
[2] Institute for Advanced Study
Institute for Advanced Study
Estados Unidos
-
- Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 3, Nº. 3, 1997, págs. 401-458
- Idioma: inglés
- DOI: 10.1007/s000290050016
- Enlaces
-
Resumen
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 Hilbert space obtained by quantizing the sigma-model on a two-dimensional torus.
