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JOURNALS // Teoreticheskaya i Matematicheskaya Fizika // Archive

TMF, 2004 Volume 140, Number 2, Pages 179–215 (Mi tmf95)

This article is cited in 4 papers

Hitchin System on Singular Curves

D. V. Talalaev, A. V. Chervov

Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: We study the Hitchin system on singular curves. We consider curves obtainable from the projective line by matching at several points or by inserting cusp singularities. It appears that on such singular curves, all basic ingredients of Hitchin integrable systems (moduli space of vector bundles, dualizing sheaf, Higgs field, etc.) can be explicitly described, which can be interesting in itself. Our main result is explicit formulas for the Hitchin Hamiltonians. We also show how to obtain the Hitchin integrable system on such curves by Hamiltonian reduction from a much simpler system on a finite-dimensional space. We pay special attention to a degenerate curve of genus two for which we find an analogue of the Narasimhan–Ramanan parameterization of the moduli space of $SL(2)$ bundles as well as the explicit expressions for the symplectic structure and Hitchin-system Hamiltonians in these coordinates. We demonstrate the efficiency of our approach by rederiving the rational and trigonometric Calogero–Moser systems, which are obtained from Hitchin systems on curves with a marked point and with the respective cusp and node.

Keywords: integrable systems, Hitchin systems, singular curves, Calogero–Moser system, Narasimhan–Ramanan parameterization.

Received: 02.10.2003

DOI: 10.4213/tmf95


 English version:
Theoretical and Mathematical Physics, 2004, 140:2, 1043–1072

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