Hitchin Systems on Hyperelliptic Curves

P. I. Borisova ; O. K. Sheinman

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and mathematical physics, Tome 311 (2020), pp. 27-40.

Résumé

We describe a class of spectral curves and find explicit formulas for the Darboux coordinates of Hitchin systems corresponding to classical simple groups on hyperelliptic curves. We consider in detail the systems with rank $2$ groups on genus $2$ curves.

Bibliographie

[1] Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts Math., 60, Springer, New York, 1989

[2] Babelon O., Talon M., “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

[3] P. I. Borisova, “Separation of variables for the type $D_l$ Hitchin systems on a hyperelliptic curve”, Usp. Mat. Nauk | MR

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

[5] Dubrovin B.A., Krichever I.M., Novikov S.P., “Integrable systems. I”, Dynamical systems IV: Symplectic geometry and its applications, Encycl. Math. Sci., 4, ed. by V.I. Arnold, S.P. Novikov, Springer, Berlin, 1990, 173–280 | MR

[6] Fay J.D., Theta functions on Riemann surfaces, Lect. Notes Math., 352, Springer, Berlin, 1973 | DOI | MR | Zbl

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

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

[9] Gorsky A., Nekrasov N., Rubtsov V., “Hilbert schemes, separated variables, and $D$-branes”, Commun. Math. Phys., 222:2 (2001), 299–318 | DOI | MR | Zbl

[10] Hitchin N., “Stable bundles and integrable systems”, Duke Math. J., 54:1 (1987), 91–114 | DOI | MR | Zbl

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

[12] Krichever I., “Vector bundles and Lax equations on algebraic curves”, Commun. Math. Phys., 229:2 (2002), 229–269 | DOI | MR | Zbl

[13] I. M. Krichever and O. K. Sheinman, “Lax operator algebras”, Funct. Anal. Appl., 41:4 (2007), 284–294 | DOI | MR | Zbl

[14] Sheinman O.K., Current algebras on Riemann surfaces: New results and applications, De Gruyter Expo. Math., 58, W. de Gruyter, Berlin, 2012 | MR | Zbl

[15] O. K. Sheinman, “Hierarchies of finite-dimensional Lax equations with a spectral parameter on a Riemann surface and semisimple Lie algebras”, Theor. Math. Phys., 185:3 (2015), 1816–1831 | DOI | MR | Zbl

[16] O. K. Sheinman, “Lax operator algebras and integrable systems”, Russ. Math. Surv., 71:1 (2016), 109–156 | DOI | MR | Zbl

[17] Sheinman O.K., Some reductions of rank 2 and genera 2 and 3 Hitchin systems, E-print, 2017, arXiv: 1709.06803 [math-ph] | MR

[18] O. K. Sheinman, “Certain reductions of Hitchin systems of rank 2 and genera 2 and 3”, Dokl. Math., 97:2 (2018), 144–146 | DOI | MR | Zbl

[19] O. K. Sheinman, “Integrable systems of algebraic origin and separation of variables”, Funct. Anal. Appl., 52:4 (2018), 316–320 | DOI | MR | Zbl

[20] O. K. Sheinman, “Spectral curves of the hyperelliptic Hitchin systems”, Funct. Anal. Appl., 53:4 (2019), 291–303 | DOI | MR | Zbl

[21] V. V. Shokurov, “Prym varieties: Theory and applications”, Math. USSR, Izv., 23:1 (1984), 83–147 | DOI | MR

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

[23] Talalaev D., Riemann bilinear form and Poisson structure in Hitchin-type systems, E-print, 2003, arXiv: hep-th/0304099.