TY - JOUR

T1 - Hyper-Kähler geometry and invariants of three-manifolds

AU - Rozansky, L.

AU - Witten, E.

N1 - Funding Information:
We would like to acknowledge helpful discussions with S. Axelrod and D. Freed. The work of E. W. was supported in part by the NSF Grant PHY95-13835. The work of L. R. was supported in part by the NSF Grant DMS 9304580.

PY - 1997

Y1 - 1997

N2 - 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.

AB - 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.

KW - Casson's invariant

KW - Topological sigma-models

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U2 - 10.1007/s000290050016

DO - 10.1007/s000290050016

M3 - Article

AN - SCOPUS:0001643399

VL - 3

SP - 401

EP - 458

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

SN - 1022-1824

IS - 3

ER -