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

This article is cited in 4 scientific papers (total 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

DOI: https://doi.org/10.4213/tmf95

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

Bibliographic databases:

Received: 02.10.2003

Citation: D. V. Talalaev, A. V. Chervov, “Hitchin System on Singular Curves”, TMF, 140:2 (2004), 179–215; Theoret. and Math. Phys., 140:2 (2004), 1043–1072

Citation in format AMSBIB
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\by D.~V.~Talalaev, A.~V.~Chervov
\paper Hitchin System on Singular Curves
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\pages 179--215
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\zmath{https://zbmath.org/?q=an:1178.14035}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2004TMP...140.1043T}
\transl
\jour Theoret. and Math. Phys.
\yr 2004
\vol 140
\issue 2
\pages 1043--1072
\crossref{https://doi.org/10.1023/B:TAMP.0000036537.38312.04}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. E. Mironov, I. A. Taimanov, “Orthogonal Curvilinear Coordinate Systems Corresponding to Singular Spectral Curves”, Proc. Steklov Inst. Math., 255 (2006), 169–184  mathnet  crossref  mathscinet
    2. Chervov, A, “Hitchin systems on singular curves II. Gluing subschemes”, International Journal of Geometric Methods in Modern Physics, 4:5 (2007), 751  crossref  mathscinet  zmath  isi  scopus  scopus
    3. JETP Letters, 106:3 (2017), 179–183  mathnet  crossref  crossref  isi  elib
    4. Yu. Chernyakov, S. Kharchev, A. Levin, M. Olshanetsky, A. Zotov, “Generalized Calogero and Toda models”, JETP Letters, 109:2 (2019), 136–143  mathnet  crossref  crossref  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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