zbMATH — the first resource for mathematics

Certain reductions of Hitchin systems of rank 2 and genera 2 and 3. (English. Russian original) Zbl 1432.14030
Dokl. Math. 97, No. 2, 144-146 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 479, No. 3, 254-256 (2018).
In the present paper, the author considers reductions of Hitchin systems, i.e., their restrictions to invariant subvarieties of a positive codimension. He obtains some manageable rank 2, genera 2 and 3 systems, certain relations for their dynamical variables, and some particular solutions to the original systems. Certain reductions are shown to give an integrable system of 2, resp. 3, interacting points on the line. It is shown that the reduced systems are particular cases of a certain universal integrable system related to the Lagrange interpolation polynomial. Admissibility of the reduction is proved using computer technique. The corresponding codes are given in the text.
This paper is organized as follows: Section 1 is an introduction to the subject. In Section 2, the author describes the reduction of the Hitchin systems of rank 2, genera 2 and 3, proves it to be an admissible reduction, observes that the reduced systems are completely integrable and finds some their particular solutions which are solutions to the original Hitchin system. The proof of admissibility is computational. The codes of the corresponding programs are given in Sections 4, 5. Section 3 is devoted to the integrable system related to the Lagrange interpolation polynomial.

14H70 Relationships between algebraic curves and integrable systems
53D18 Generalized geometries (à la Hitchin)
14H60 Vector bundles on curves and their moduli
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
14-04 Software, source code, etc. for problems pertaining to algebraic geometry
Full Text: DOI
[1] Buchstaber, V. M., No article title, Proc. Steklov Inst. Math., 294, 176-200, (2016) · Zbl 1360.14093
[2] Gawedzki, K.; Tran-Ngog-Bich, P., No article title, J. Math. Phys., 41, 4695-4712, (2000) · Zbl 0974.32009
[3] Geemen, B.; Previato, E., No article title, Duke Math. J., 85, 659-683, (1996) · Zbl 0879.14010
[4] Krichever, I. M., No article title, Commun. Math. Phys., 229, 229-269, (2002) · Zbl 1073.14048
[5] O. K. Sheinman, Current Algebras on Riemann Surfaces (Walter de Gruyter, Berlin, 2012). · Zbl 1258.81002
[6] Sheinman, O. K., No article title, Theor. Math. Phys., 185, 1816-1831, (2015) · Zbl 1417.37246
[7] Sheinman, O. K., No article title, Proc. Steklov Inst. Math., 290, 191-201, (2015) · Zbl 1395.17073
[8] Sheinman, O. K., No article title, Moscow Math. J., 15, 833-846, (2015)
[9] Sheinman, O. K., No article title, Russ. Math. Surv., 71, 109-156, (2016) · Zbl 1395.17047
[10] Tyurin, A. N., No article title, Am. Math. Transl. Ser. 2, 63, 245-279, (1967)
[11] O. K. Sheinman, arXiv:1709.06803 [math-ph].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.