Hitchin systems: some recent advances
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 79 (2024) no. 4, pp. 683-720 Cet article a éte moissonné depuis la source Math-Net.Ru

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We give a survey of some recent advances in parabolic Hitchin systems (parabolic Beauville–Narasimhan–Ramanan correspondence, mirror symmetry for parabolic Hitchin systems) and in exact methods of solving non-parabolic Hitchin systems. Bibliography: 55 titles.
Keywords: classical and parabolic Hitchin systems, Hitchin map and fibres, mirror symmetry, exact methods.
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O. K. Sheinman; B. Wang. Hitchin systems: some recent advances. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 79 (2024) no. 4, pp. 683-720. http://geodesic.mathdoc.fr/item/RM_2024_79_4_a3/

[1] V. I. Arnold, Mathematical methods of classical mechanics, Grad. Texts in Math., 60, Corr. reprint of the 2nd ed., Springer-Verlag, New York, 1997, xvi+520 pp. | MR

[2] M. F. Atiyah and R. Bott, “The Yang–Mills equations over Riemann surfaces”, Philos. Trans. Roy. Soc. London Ser. A, 308:1505 (1983), 523–615 | DOI | MR | Zbl

[3] O. Babelon and M. Talon, Riemann surfaces, separation of variables and classical and quantum integrability, 2002, 10 pp., arXiv: hep-th/0209071

[4] D. Baraglia and M. Kamgarpour, “On the image of the parabolic Hitchin map”, Q. J. Math., 69:2 (2018), 681–708 | DOI | MR | Zbl

[5] A. Beauville, M. S. Narasimhan, and S. Ramanan, “Spectral curves and the generalised theta divisor”, J. Reine Angew. Math., 1989:398 (1989), 169–179 | DOI | MR | Zbl

[6] I. Biswas and S. Ramanan, “An infinitesimal study of the moduli of Hitchin pairs”, J. Lond. Math. Soc. (2), 49:2 (1994), 219–331 | DOI | MR | Zbl

[7] P. I. Borisova and O. K. Sheinman, “Hitchin systems on hyperelliptic curves”, Proc. Steklov Inst. Math., 311 (2020), 22–35 | DOI | MR | Zbl

[8] M. A. de Cataldo, D. Maulik, and Junliang Shen, “Hitchin fibrations, abelian surfaces, and the $P=W$ conjecture”, J. Amer. Math. Soc., 35:3 (2022), 911–953 | DOI | MR | Zbl

[9] M. A. de Cataldo, D. Maulik, and Junliang Shen, “On the $P=W$ conjecture for $\operatorname{SL}_n$”, Selecta Math. (N. S.), 28:5 (2022), 90, 21 pp. | DOI | MR | Zbl

[10] R. Donagi and T. Pantev, “Geometric Langlands and non-abelian Hodge theory”, Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009, 85–116 | DOI | MR | Zbl

[11] R. Donagi and T. Pantev, “Langlands duality for Hitchin systems”, Invent. Math., 189:3 (2012), 653–735 | DOI | MR | Zbl

[12] B. A. Dubrovin, “Theta functions and non-linear equations”, Russian Math. Surveys, 36:2 (1981), 11–92 | DOI | MR | Zbl

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

[14] B. van Geemen and A. J. de Jong, “On Hitchin's connection”, J. Amer. Math. Soc., 11:1 (1998), 189–228 | DOI | MR | Zbl

[15] B. van Geemen and E. Previato, “On the Hitchin system”, Duke Math. J., 85:3 (1996), 659–683 | DOI | MR | Zbl

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

[17] P. B. Gothen and A. G. Oliveira, “Topological mirror symmetry for parabolic Higgs bundles”, J. Geom. Phys., 137 (2019), 7–34 | DOI | MR | Zbl

[18] M. Groechenig, D. Wyss, and P. Ziegler, “Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration”, Invent. Math., 221:2 (2020), 505–596 | DOI | MR | Zbl

[19] V. Guillemin and S. Sternberg, Geometric asymptotics, Math. Surveys, 14, Amer. Math. Soc., Providence, RI, 1977, xviii+474 pp. | MR | Zbl

[20] S. Gukov and E. Witten, “Gauge theory, ramification, and the geometric Langlands program”, Current developments in mathematics, 2006, Int. Press, Somerville, MA, 2008, 35–180 | DOI | MR | Zbl

[21] S. Gukov and E. Witten, “Rigid surface operators”, Adv. Theor. Math. Phys., 14:1 (2010), 87–178 | DOI | MR | Zbl

[22] J. D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Math., 352, Springer-Verlag, Berlin–New York, 1973, iv+137 pp. | DOI | MR | Zbl

[23] D. Friedan and S. Shenker, “The analytic geometry of two-dimensional conformal field theory”, Nuclear Phys. B, 281:3-4 (1987), 509–545 | DOI | MR

[24] T. Hausel, A. Mellit, A. Minets, and O. Schiffmann, $P=W$ via $H_2$, 2022, 54 pp., arXiv: 2209.05429

[25] T. Hausel and M. Thaddeus, “Mirror symmetry, Langlands duality, and the Hitchin system”, Invent. Math., 153:1 (2003), 197–229 | DOI | MR | Zbl

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

[27] N. J. Hitchin, “Flat connections and geometric quantization”, Comm. Math. Phys., 131:2 (1990), 347–380 | DOI | MR | Zbl

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

[29] A. Kapustin and E. Witten, “Electric-magnetic duality and the geometric Langlands program”, Commun. Number Theory Phys., 1:1 (2007), 1–236 | DOI | MR | Zbl

[30] M. Kontsevich, “Homological algebra of mirror symmetry”, Proceedings of the international congress of mathematicians (Zürich 1994), v. 1, Birkhäuser Verlag, Basel, 1995, 120–139 | DOI | MR | Zbl

[31] I. M. Krichever, “Methods of algebraic geometry in the theory of non-linear equations”, Russian Math. Surveys, 32:6 (1977), 185–213 | DOI | MR | Zbl

[32] I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry”, Funct. Anal. Appl., 11:1 (1977), 12–26 | DOI | MR | Zbl

[33] I. M. Krichever, “Nonlinear equations and elliptic curves”, J. Soviet Math., 28:1 (1985), 51–90 | DOI | MR | Zbl

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

[35] M. Logares and J. Martens, “Moduli of parabolic Higgs bundles and Atiyah algebroids”, J. Reine Angew. Math., 2010:649 (2010), 89–116 | DOI | MR | Zbl

[36] D. Maulik and Junliang Shen, “Endoscopic decompositions and the Hausel–Thaddeus conjecture”, Forum Math. Pi, 9 (2021), e8, 49 pp. | DOI | MR | Zbl

[37] D. Maulik and Junliang Shen, The $P=W$ conjecture for $\operatorname{GL}_n$, 2022, 23 pp., arXiv: 2209.02568

[38] V. B. Mehta and C. S. Seshadri, “Moduli of vector bundles on curves with parabolic structures”, Math. Ann., 248:3 (1980), 205–239 | DOI | MR | Zbl

[39] A. Mellit, “Poincaré polynomials of character varieties, Macdonald polynomials and affine Springer fibers”, Ann. of Math. (2), 192:1 (2020), 165–228 | DOI | MR | Zbl

[40] M. S. Narasimhan and S. Ramanan, “Moduli of vector bundles on a compact Riemann surface”, Ann. of Math. (2), 89 (1969), 14–51 | DOI | MR | Zbl

[41] M. S. Narasimhan and C. S. Seshadri, “Stable and unitary vector bundles on a compact Riemann surface”, Ann. of Math. (2), 82:3 (1965), 540–567 | DOI | MR | Zbl

[42] M. Oka, “Introduction to plane curve singularities. Toric resolution tower and Puiseux pairs”, Arrangements, local systems and singularities, Progr. Math., 283, Birkhäuser Verlag, Basel, 2010, 209–245 | DOI | MR | Zbl

[43] P. Scheinost and M. Schottenloher, “Metaplectic quantization of the moduli spaces of flat and parabolic bundles”, J. Reine Angew. Math., 1995:466 (1995), 145–219 | DOI | MR | Zbl

[44] O. K. Sheinman, “Lax operator algebras and integrable hierarchies”, Proc. Steklov Inst. Math., 263 (2008), 204–213 | DOI | MR | Zbl

[45] O. K. Sheinman, Current algebras on Riemann surfaces. New results and applications, De Gruyter Exp. Math., 58, Walter de Gruyter GmbH Co. KG, Berlin, 2012, xiv+150 pp. | DOI | MR | Zbl

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

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

[48] Shiyu Shen, “Mirror symmetry for parabolic Higgs bundles via $p$-adic integration”, Adv. Math., 443 (2024), 109616, 40 pp. | DOI | MR | Zbl

[49] C. T. Simpson, “Harmonic bundles on noncompact curves”, J. Amer. Math. Soc., 3:3 (1990), 713–770 | DOI | MR | Zbl

[50] Xiaoyu Su, Bin Wang, and Xueqing Wen, “Parabolic Hitchin maps and their generic fibers”, Math. Z., 301:1 (2022), 343–372 | DOI | MR | Zbl

[51] Xiaoyu Su, Bin Wang, and Xueqing Wen, Topological mirror symmetry of parabolic Hitchin systems, 2022, 17 pp., arXiv: 2206.02527

[52] D. Talalaev, Riemann bilinear form and Poisson structure in Hitchin-type systems, 2003, 15 pp., arXiv: hep-th/0304099

[53] A. N. Tyurin, “Classification of vector bundles over an algebraic curve of arbitrary genus”, Amer. Math. Soc. Transl. Ser. 2, 63, Amer. Math. Soc., Providence, RI, 1967, 245–279 | MR | Zbl

[54] A. N. Tyurin, “On the classification of $n$-dimensional vector-bundles over an algebraic curve of arbitrary genus”, Amer. Math. Soc. Transl. Ser. 2, 73, Amer. Math. Soc., Providence, RI, 1968, 196–211 | MR | Zbl

[55] K. Yokogawa, “Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves”, J. Math. Kyoto Univ., 33:2 (1993), 451–504 | DOI | MR | Zbl