Geodesic
Spectral Curves of the Hyperelliptic Hitchin Systems
Funkcionalʹnyj analiz i ego priloženiâ, Tome 53 (2019) no. 4, pp. 63-78.

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This paper describes a class of spectral curves and gives explicit formulas for the Darboux coordinates of the Hitchin systems of types $A_l$, $B_l$, and $C_l$ on hyperelliptic curves. The current state of the problem in the case of the systems of type $D_l$ is described. 
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O. K. Sheinman. Spectral Curves of the Hyperelliptic Hitchin Systems. Funkcionalʹnyj analiz i ego priloženiâ, Tome 53 (2019) no. 4, pp. 63-78. http://geodesic.mathdoc.fr/item/FAA_2019_53_4_a5/

[1] K. Yakobi, Lektsii po dinamike, Glavnaya redaktsiya obschetekhnicheskoi literatury, M.–L., 1936

[2] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1979 | MR

[3] O. Babelon, D. Bernard, M. Talon, Introduction to Classical Integrable Systems, Cambridge University Press, Cambridge, 2003 | MR | Zbl

[4] O. Babelon, M. Talon, “Riemann surfaces, separation of variables and classical and quantum integrability”, Phys. Lett. A, 312:1–2 (2003), 71–77, arXiv: hep-th/0209071 | DOI | MR | Zbl

[5] P. I. Borisova, “Razdelenie peremennykh dlya sistem Khitchina tipa $D_l$ na giperellipticheskoi krivoi”, UMN (to appear)

[6] R. Donagi, E. Witten, “Supersymmetric Yang–Mills theory and integrable systems”, Nuclear Phys. B, 460:2 (1996), 299–334, arXiv: hep-th/9510101 | DOI | MR | Zbl

[7] B. A. Dubrovin, I. M. Krichever, S. P. Novikov, Integrable Systems, v. 1, Encyclopaedia of Mathematical Sciences, 4, Dynamical Systems IV, Springer-Verlag, Berlin–Heidelberg, 1990 | MR | Zbl

[8] A. I. Esterov, “Galois theory for general systems of polynomial equations”, Compos. Math., 155:2 (2019), 229–245, arXiv: 1801.08260 | DOI | MR | Zbl

[9] B. Enriquez, V. Rubtsov, “Commuting families in skew fields and quantization of Beauville's fibration”, Duke Math. J., 119:2 (2003), 197–219 | DOI | MR | Zbl

[10] K. Gawedzki, P. Tran-Ngoc-Bich, “Hitchin systems at low genera”, J. Math. Phys., 41:7 (2000), 4695–4712, arXiv: hep-th/9803101 | DOI | MR | Zbl

[11] B. van Geemen, E. Previato, “On the Hitchin system”, Duke Math. J., 85:3 (1996), 659–683, arXiv: alg-geom/9410015 | DOI | MR | Zbl

[12] A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, A. Morozov, “Integrability and Seiberg-Witten exact solution”, Phys. Lett. B, 355:3–4 (1995), 466–474 | DOI | MR | Zbl

[13] A. Gorsky, N. Nekrasov, V. Rubtsov, “Hilbert Schemes, Separated Variables, and D-Branes”, Comm. Math. Phys., 222:2 (2001), 299–318 | DOI | MR | Zbl

[14] N. Hitchin, “Stable bundles and integrable systems”, Duke Math. J., 54:1 (1987), 91–112 | DOI | MR

[15] E. D'Hocker, D. H. Phong, Lax pairs and spectral curves for Calogero–Moser and spin Calogero–Moser systems, arXiv: hep-th/9903002 | MR

[16] J. Hurtubise, “Integrable systems and algebraic surfaces”, Duke. Math. J., 83:1 (1996), 19–50 | DOI | MR | Zbl

[17] I. Krichever, “Vector Bundles and Lax Equations on Algebraic Curves”, Comm. Math. Phys., 229:2 (2002), 229–269 | DOI | MR | Zbl

[18] I. M. Krichever, “Ellipticheskie resheniya uravneniya Kadomtseva–Petviashvili i integriruemye sistemy”, Funkts. analiz i ego pril., 14:4 (1980), 45–54 | MR | Zbl

[19] I. M. Krichever, O. K. Sheinman, “Algebry operatorov Laksa”, Funkts. analiz i ego pril., 41:4 (2007), 46–59 | DOI | MR | Zbl

[20] M. Schlichenmaier, Krichever–Novikov type algebras. Theory and applications, De Gruyter Studies in Mathematics, 53, Walter de Gruyter Gmbh, Berlin–Boston, 2014 | MR

[21] O. K. Sheinman, “Ierarkhii konechnomernykh uravnenii Laksa so spektralnym parametrom na rimanovoi poverkhnosti i poluprostye algebry Li”, TMF, 185:3 (2015), 527–544 | DOI | MR | Zbl

[22] O. K. Sheinman, Current Algebras on Riemann Surfaces. New Results and Applications, De Gruyter Expositions in Mathematics, 58, Walter de Gruyter GmbH, Berlin–Boston, 2012 | MR

[23] O. K. Sheinman, “Algebry operatorov Laksa i integriruemye sistemy”, UMN, 71:1(427) (2016), 117–168, arXiv: 1602.04320 | DOI | MR | Zbl

[24] O. K. Sheinman, “Nekotorye reduktsii sistem Khitchina ranga 2 i rodov 2 i 3”, Dokl. RAN, ser. matem., 479:3 (2018), 254–256 | DOI | MR | Zbl

[25] O. K. Sheinman, Some reductions of rank 2 and genera 2 and 3 Hitchin systems, arXiv: 1709.06803 | MR

[26] O. K. Sheinman, “Integriruemye sistemy algebraicheskogo proiskhozhdeniya i razdelenie peremennykh”, Funkts. analiz i ego pril., 52:4 (2018), 94–98, arXiv: 1712.04422 | DOI | MR | Zbl

[27] O. K. Sheinman, Spectral curves of the hyperelliptic Hitchin systems, arXiv: 1806.10178

[28] E. K. Sklyanin, “Separation of variables. New trends”, Prog. Theoret. Phys. Suppl., 118 (1995), 35–60, arXiv: solv-int/9504001 | DOI | MR | Zbl

[29] D. Talalaev, Riemann bilinear form and Poisson structure in Hitchin-type systems, arXiv: hep-th/0304099

[30] A. V. Tsyganov, Integriruemye sistemy v metode razdeleniya peremennykh, Nauchno-izdatelskii tsentr «Regulyarnaya i khaoticheskaya dinamika», Moskva–Izhevsk, 2013