Gaudin Model, KZ Equation, and Isomonodromic Problem on Torus

Details Gaudin Model, KZ Equation, and Isomonodromic Problem on Torus Gaudin Model, KZ Equation, and Isomonodromic Problem on Torus

1997-11-10

arxiv.org/abs/hep-th/9711058v5

Lett.Math.Phys. 44 (1998) 143-156

Physics

High Energy Physics

High Energy Physics - Theory

15 pages, latex, no figures; Etingof's paper on the elliptic KZ equation is added; the setting for twisted de Rham cohomologie

Scientific paper

This paper presents a construction of isospectral problems on the torus. The construction starts from an SU(n) version of the XYZ Gaudin model recently studied by Kuroki and Takebe in the context of a twisted WZW model. In the classical limit, the quantum Hamiltonians of the generalized Gaudin model turn into classical Hamiltonians with a natural $r$-matrix structure. These Hamiltonians are used to build a non-autonomous multi-time Hamiltonian system, which is eventually shown to be an isomonodromic problem on the torus. This isomonodromic problem can also be reproduced from an elliptic analogue of the KZ equation for the twisted WZW model. Finally, a geometric interpretation of this isomonodromic problem is discussed in the language of a moduli space of meromorphic connections.

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