Non-abelian Hodge theory and moduli spaces of Higgs bundles

Gallego , Guillermo

doi:10.15517/dg20cm45

Autores/as

Guillermo Gallego Universidad Autónoma de Madrid, Departamento de Matemáticas, Madrid, Spain. Autor/a https://orcid.org/0000-0002-3228-8534

DOI:

https://doi.org/10.15517/dg20cm45

Palabras clave:

Fibrados de Higgs, Teoría de Hodge No-abeliana, Fibración de Hitchin

Resumen

Este artículo es una introducción a la teoría de Hodge no abeliana y a los espacios de móduli de fibrados de Higgs sobre superficies de Riemann compactas. Desarrollamos la teoría de móduli de fibrados vectoriales y fibrados de Higgs, establecemos las principales correspondencias de la teoría de Hodge no abeliana, y las interpretamos en términos de la estructura hiperkähleriana del espacio de móduli de Hitchin. Estudiamos la fibración de Hitchin y sus propiedades geométricas, incluyendo la simetría espejo SYZ y la simetría espejo topológica para sistemas de Hitchin de tipo A. Como ilustración, calculamos el polinomio de Poincaré del espacio de móduli en rango 2 y verificamos la simetría espejo topológica en este caso.

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Referencias

D. Abramovich, M. Olsson, A. Vistoli, Tame stacks in positive characteristic. Ann. Inst. Fourier (Grenoble) 58(2008), no. 4, 1057–1091. doi: 10.5802/aif.2378 DOI: https://doi.org/10.5802/aif.2378

D. Arinkin et al., Proof of the geometric Langlands conjecture. List of papers. 2024. url: https://people.mpim-bonn.mpg.de/gaitsgde/GLC/.

M. F. Atiyah, Vector bundles over an elliptic curve. Proc. London Math. Soc. (3) 7(1957), 414–452. doi: 10.1112/plms/s3-7.1.414 DOI: https://doi.org/10.1112/plms/s3-7.1.414

M. F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A 308(1983), no. 1505, 523–615. doi:10.1098/rsta.1983.0017 DOI: https://doi.org/10.1098/rsta.1983.0017

D. Baraglia, L. P. Schaposnik, Higgs bundles and (A, B, A)-branes. Comm. Math. Phys. 331(2014), no. 3, 1271–1300. doi: 10.1007/s00220-014-2053-6 DOI: https://doi.org/10.1007/s00220-014-2053-6

A. Beilinson, V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves. Preprint. 1991.

D. Ben-Zvi, Y. Sakellaridis, A. Venkatesh, Relative Langlands Duality. Preprint. 2024. doi: 10.48550/arXiv.2409.04677 DOI: https://doi.org/10.4171/owr/2025/18

I. Biswas, O. García-Prada, J. Hurtubise, Higgs bundles, branes and Langlands duality. Comm. Math. Phys. 365(2019), no. 3, 1005–1018. doi: 10.1007/s00220-019-03290-1 DOI: https://doi.org/10.1007/s00220-019-03290-1

M. A. A. de Cataldo, T. Hausel, L. Migliorini, Topology of Hitchin systems and Hodge theory of character varieties: the case A1. Ann. of Math. (2) 175(2012), no. 3, 1329–1407. doi: 10.4007/annals.2012.175.3.7 DOI: https://doi.org/10.4007/annals.2012.175.3.7

E. Y. Chen, E. Hsiao, M. Yang, (BAA)-branes from higher Teichmüller theory. Preprint. 2025. doi: 10.48550/arXiv.2508.09562

T. H. Chen, B. C. Ngô, On the Hitchin morphism for higher-dimensional varieties.

Duke Math. J. 169(2020), no. 10, 1971–2004. doi: 10.1215/00127094-2019-0085 DOI: https://doi.org/10.1215/00127094-2019-0085

K. Corlette, Flat G-bundles with canonical metrics. J. Differential Geom. 28(1988), no. 3, 361–382. doi: 10.4310/jdg/1214442469 DOI: https://doi.org/10.4310/jdg/1214442469

R. Donagi, T. Pantev, Langlands duality for Hitchin systems. Invent. Math. 189(2012), no. 3, 653–735. doi: 10.1007/s00222-012-0373-8 DOI: https://doi.org/10.1007/s00222-012-0373-8

R. Y. Donagi, D. Gaitsgory, The gerbe of Higgs bundles. Transform. Groups 7(2002), no. 2, 109–153. doi: 10.1007/s00031-002-0008-z DOI: https://doi.org/10.1007/s00031-002-0008-z

S. K. Donaldson, A new proof of a theorem of Narasimhan and Seshadri. J. Differential Geom. 18(1983), no. 2, 269–277. doi: 10.4310/jdg/1214437664 DOI: https://doi.org/10.4310/jdg/1214437664

S. K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. (3) 50(1985), no. 1, 1–26. doi: 10.1112/plms/s3-50.1.1 DOI: https://doi.org/10.1112/plms/s3-50.1.1

S. K. Donaldson, Twisted harmonic maps and the self-duality equations. Proc. LondonMath. Soc. (3) 55(1987), no. 1, 127–131. doi: 10.1112/plms/s3-55.1.127 DOI: https://doi.org/10.1112/plms/s3-55.1.127

E. Franco, P. B. Gothen, A. Oliveira, A. Peón-Nieto, Narasimhan-Ramanan branes and wobbly Higgs bundles. Internat. J. Math. 35(2024), no. 9, 2441006. doi: 10.1142/S0129167X24410064 DOI: https://doi.org/10.1142/S0129167X24410064

E. Franco, R. Hanson, The Dirac-Higgs complex and categorification of (BBB) branes. Int. Math. Res. Not. IMRN 2024(2024), no. 19, 12919–12953. DOI: https://doi.org/10.1093/imrn/rnae187

D. Gaiotto, S-duality and boundary conditions and the geometric Langlands program. String-Math 2016. Vol. 98. Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 2018, 139–179. doi: 10.1090/pspum/098/01721 DOI: https://doi.org/10.1090/pspum/098/06

G. Gallego, Multiplicative Hitchin fibrations and Langlands duality. Preprint. 2025. doi: 10.48550/arXiv.2509.14364

G. Gallego, O. García-Prada, M. S. Narasimhan, Higgs bundles twisted by a vector bundle. Internat. J. Math. 35(2024), no. 9, 2441007. doi: 10.1142/S0129167X24410076 DOI: https://doi.org/10.1142/S0129167X24410076

O. Garcia-Prada, P. B. Gothen, I. M. i Riera, The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations. Preprint. 2012. doi: 10.48550/arXiv.0909.4487

O. García-Prada, J. Heinloth, A. Schmitt, On the motives of moduli of chains and Higgs bundles. J. Eur. Math. Soc. (JEMS) 16(2014), no. 12, 2617–2668. doi: 10.4171/JEMS/494 DOI: https://doi.org/10.4171/jems/494

O. García-Prada, A. Peón-Nieto, Abelianization of Higgs bundles for quasisplit real groups. Transform. Groups 28(2023), no. 1, 285–325. doi: 10.1007/s00031-021-09658-9 DOI: https://doi.org/10.1007/s00031-021-09658-9

O. García-Prada, S. Ramanan, Involutions of rank 2 Higgs bundle moduli spaces. Geometry and physics. Vol. II. Oxford Univ. Press, Oxford, 2018, 535–550. doi: 10.1093/oso/9780198802020.003.0022 DOI: https://doi.org/10.1093/oso/9780198802020.003.0022

V. Ginzburg, N. Rozenblyum, Gaiotto’s Lagrangian subvarieties via derived symplectic geometry. Algebr. Represent. Theory 21(2018), 1003–1015. doi: 10.1007/s10468-018-9801-9 DOI: https://doi.org/10.1007/s10468-018-9801-9

P. B. Gothen, The Betti numbers of the moduli space of stable rank 3 Higgs bundles on a Riemann surface. Internat. J. Math. 5(1994), 861–875. doi: 10.1142/S0129167X94000449 DOI: https://doi.org/10.1142/S0129167X94000449

M. Groechenig, D. Wyss, P. Ziegler, Geometric stabilisation via p-adic integration. J. Amer. Math. Soc. 33(2020), no. 3, 807–873. doi: 10.1090/jams/948 DOI: https://doi.org/10.1090/jams/948

M. Groechenig, D. Wyss, P. Ziegler, Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration. Invent. Math. 221(2020), 505–596. doi: 10.1007/s00222-020-00957-8 DOI: https://doi.org/10.1007/s00222-020-00957-8

A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère

de Riemann. Amer. J. Math. 79(1957), 121–138. doi: 10.2307/2372388 DOI: https://doi.org/10.2307/2372388

A. Grothendieck, Sur le mémoire de Weil: généralisation des fonctions abéliennes. Séminaire Bourbaki, Vol. 4. Soc. Math. France, Paris, 1995, Exp. No. 141, 57–71.

T. Hameister, Z. Luo, B. Morrissey, Relative Dolbeault Geometric Langlands via the Regular Quotient. 2025. doi: 10.48550/arXiv.2409.15691

T. Hameister, B. Morrissey, The Hitchin fibration for symmetric pairs. Adv. Math. 482(2025), 110560. doi: 10.1016/j.aim.2025.110560 DOI: https://doi.org/10.1016/j.aim.2025.110560

G. Harder, M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves. Math. Ann. 212(1975), 215–248. doi: 10.1007/BF01357141 DOI: https://doi.org/10.1007/BF01357141

T. Hausel, Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve. F. Bogomolov, Y. Tschinkel (Eds.). Geometric Methods in Algebra and Number Theory. Birkhäuser Boston, Boston, MA, 2005, 193–217. doi: 10.1007/0-8176-4417-2 9 DOI: https://doi.org/10.1007/0-8176-4417-2_9

T. Hausel, Enhanced mirror symmetry for Langlands dual Hitchin systems. ICM—International Congress of Mathematicians. Vol. 3. Sections 1–4. D. Beliaev, S. Smirnov (Eds.). EMS Press, Berlin, 2023, 2228–2249. DOI: https://doi.org/10.4171/icm2022/164

T. Hausel, A. Mellit, A. Minets, O. Schiffmann, P = W via H2. Preprint. 2025. doi: 10.48550/arXiv.2209.05429

T. Hausel, F. Rodriguez-Villegas, Mixed Hodge polynomials of character varieties. Invent. Math. 174(2008), 555–624. doi: 10.1007/s00222-008-0142-x DOI: https://doi.org/10.1007/s00222-008-0142-x

T. Hausel, M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153(2003), no. 1, 197–229. doi: 10.1007/s00222-003-0286-7 DOI: https://doi.org/10.1007/s00222-003-0286-7

S. Heller, L. P. Schaposnik, Branes through finite group actions. J. Geom. Phys. 129(2018), 279–293. doi: 10.1016/j.geomphys.2018.03.014 DOI: https://doi.org/10.1016/j.geomphys.2018.03.014

N. J. Hitchin, Monopoles and geodesics. Comm. Math. Phys. 83(1982), no. 4, 579–602. DOI: https://doi.org/10.1007/BF01208717

N. J. Hitchin, The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55(1987), no. 1, 59–126. doi: 10.1112/plms/s3-55.1.59 DOI: https://doi.org/10.1112/plms/s3-55.1.59

N. Hitchin, Stable bundles and integrable systems. Duke Math. J. 54(1987), no. 1, 91–114. doi: 10.1215/S0012-7094-87-05408-1 DOI: https://doi.org/10.1215/S0012-7094-87-05408-1

N. Hitchin, Lectures on special Lagrangian submanifolds. Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999). Vol. 23. AMS/IP Stud. Adv. Math. Amer. Math. Soc., Providence, RI, 2001, 151–182. doi: 10.1090/amsip/023/06 DOI: https://doi.org/10.1090/amsip/023/06

N. Hitchin, Langlands duality and G2 spectral curves. Q. J. Math. 58(2007), no. 3, 319–344. doi: 10.1093/qmath/ham016 DOI: https://doi.org/10.1093/qmath/ham016

N. Hitchin, Higgs Bundles and Characteristic Classes. Arbeitstagung Bonn 2013. W. Ballmann et al. (Eds.). Springer International Publishing, Cham, 2016, 247–264. doi: 10.1007/978-3-319-43648-7 8 DOI: https://doi.org/10.1007/978-3-319-43648-7_8

V. Hoskins, Geometric invariant theory and symplectic quotients. Lecture notes. 2012.

D. Huybrechts, Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2006, viii+307. doi: 10.1093/acprof:oso/9780199296866.001.0001 DOI: https://doi.org/10.1093/acprof:oso/9780199296866.001.0001

A. Kapustin, E. Witten, Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1(2007), no. 1, 1–236. doi:10.4310/CNTP.2007.v1.n1.a1 DOI: https://doi.org/10.4310/CNTP.2007.v1.n1.a1

S. Kobayashi, Differential geometry of complex vector bundles. Princeton Legacy Library. Princeton University Press, Princeton, NJ, 2014. doi: 10.1515/9781400858682 DOI: https://doi.org/10.1515/9781400858682

M. Kontsevich, Homological algebra of mirror symmetry. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994). Birkhäuser, Basel, 1995, 120–139. doi: 10.1007/978-3-0348-9078-6 11 DOI: https://doi.org/10.1007/978-3-0348-9078-6_11

I. G. Macdonald, The Poincaré polynomial of a symmetric product. Proc. Cambridge Philos. Soc. 58(1962), 563–568. doi: 10.1017/S0305004100040573 DOI: https://doi.org/10.1017/S0305004100040573

D. Maulik, J. Shen, Endoscopic decompositions and the Hausel-Thaddeus conjecture. Forum Math. Pi 9(2021), e8. doi: 10.1017/fmp.2021.7 DOI: https://doi.org/10.1017/fmp.2021.7

D. Maulik, J. Shen, The P = W conjecture for GLn. Ann. of Math. (2) 200(2024), no. 2, 529–556. doi: 10.4007/annals.2024.200.2.3 DOI: https://doi.org/10.4007/annals.2024.200.2.3

D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory. Third. Vol. 34. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Springer-Verlag, Berlin, 1994, xiv+292. doi: 10.1007/978-3-642-57916-5 DOI: https://doi.org/10.1007/978-3-642-57916-5

J. R. Munkres, Topology: a first course. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1975, xvi+413.

M. S. Narasimhan, S. Ramanan, Generalised Prym varieties as fixed points. J. Indian Math. Soc. (N.S.) 39(1975), 1–19.

M. S. Narasimhan, C. S. Seshadri, Holomorphic vector bundles on a compact Riemann surface. Math. Ann. 155(1964), 69–80. doi: 10.1007/BF01350891 DOI: https://doi.org/10.1007/BF01350891

A. Neitzke, Moduli of Higgs bundles. Lecture notes. 2016.

B. C. Ngô, Le lemme fondamental pour les algèbres de Lie. Publ. Math. Inst. Hautes Études Sci. (2010), no. 111, 1–169. doi: 10.1007/s10240-010-0026-7 DOI: https://doi.org/10.1007/s10240-010-0026-7

N. Nitsure, Construction of Hilbert and Quot schemes. Fundamental algebraic geometry. Vol. 123. Math. Surveys Monogr. Amer. Math. Soc., Providence, RI, 2005, 105–137. doi: 10.1090/surv/123/05 DOI: https://doi.org/10.1090/surv/123/05

A. Ramanathan, Stable principal bundles on a compact Riemann surface. Math. Ann. 213(1975), 129–152. doi: 10.1007/BF01343949 DOI: https://doi.org/10.1007/BF01343949

J.-P. Serre, Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier (Grenoble) 6(1956), 1–42. doi: 10.5802/aif.59 DOI: https://doi.org/10.5802/aif.59

S. S. Shatz, Degeneration and specialization in algebraic families of vector bundles. Bull. Amer. Math. Soc. 82(1976), no. 4, 560–562. doi: 10.1090/S0002-9904-1976-14099-2 DOI: https://doi.org/10.1090/S0002-9904-1976-14099-2

C. T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Amer. Math. Soc. 1(1988), no. 4, 867–918. doi: 10.2307/1990994 DOI: https://doi.org/10.1090/S0894-0347-1988-0944577-9

C. T. Simpson, Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75(1992), 5–95. doi: 10.1007/BF02699491 DOI: https://doi.org/10.1007/BF02699491

C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I. Inst. Hautes Études Sci. Publ. Math. 79(1994), 47–129. doi: 10.1007/BF02698887 DOI: https://doi.org/10.1007/BF02698887

C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. 80(1994), 5–79. doi: 10.1007/BF02698895 DOI: https://doi.org/10.1007/BF02698895

A. Strominger, S.-T. Yau, E. Zaslow, Mirror symmetry is T-duality. Nuclear Phys. B 479(1996), no. 1-2, 243–259. doi: 10.1016/0550-3213(96)00434-8 DOI: https://doi.org/10.1016/0550-3213(96)00434-8

K. Uhlenbeck, S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Vol. 39. no. S1. 1986, S257–S293. doi: 10.1002/cpa.3160390714 DOI: https://doi.org/10.1002/cpa.3160390714

K. K. Uhlenbeck, Connections with Lp bounds on curvature. Comm. Math. Phys. 83(1982), no. 1, 31–42. DOI: https://doi.org/10.1007/BF01947069

X. G. Wang, Multiplicative Hitchin Fibrations and the Fundamental Lemma. Preprint. 2025. doi: 10.48550/arXiv.2402.19331

E. Witten,Mirror symmetry, Hitchin’s equations, and Langlands duality. The many facets of geometry. O. Garcia-Prada, J. P. Bourguignon, S. Salamon (Eds.). Oxford Univ. Press, Oxford, 2010, 113–128. doi: 10.1093/acprof:oso/9780199534920.003.0007 DOI: https://doi.org/10.1093/acprof:oso/9780199534920.003.0007

Z. Yun, Global Springer theory. Adv. Math. 228(2011), no. 1, 266–328. doi: 10.1016/j.aim.2011.05.012 DOI: https://doi.org/10.1016/j.aim.2011.05.012