Anti-holomorphic involutions of the moduli spaces of Higgs bundles
[Involutions anti-holomorphes des espaces de modules de fibrés de Higgs]
Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 35-54.

Nous étudions les involutions anti-holomorphes des espaces de modules de G-fibrés de Higgs sur une surface de Riemann compacte X, où G est un groupe de Lie semi-simple complexe. Ces involutions sont définies en fixant des involutions anti-holomorphes à la fois sur X et G. Nous en analysons le lieu des points fixes dans l’espace de modules et leur relation avec les représentation du groupe fondamental orbifold de X muni de l’involution anti-holomorphe. Nous étudions aussi la relation avec les « branes ». Ceci généralise les travaux de Biswas–García-Prada–Hurtubise et Baraglia–Schaposnik.

We study anti-holomorphic involutions of the moduli space of G-Higgs bundles over a compact Riemann surface X, where G is a complex semisimple Lie group. These involutions are defined by fixing anti-holomorphic involutions on both X and G. We analyze the fixed point locus in the moduli space and their relation with representations of the orbifold fundamental group of X equipped with the anti-holomorphic involution. We also study the relation with branes. This generalizes work by Biswas–García-Prada–Hurtubise and Baraglia–Schaposnik.

Reçu le :
Accepté le :
DOI : 10.5802/jep.16
Classification : 14H60, 57R57, 58D29
Keywords: Higgs $G$-bundle, reality condition, branes, character variety.
Mot clés : $G$-fibré de Higgs, condition de réalité, « branes », variétés caractères.

Indranil Biswas 1 ; Oscar García-Prada 2

1 School of Mathematics, Tata Institute of Fundamental Research Homi Bhabha Road, Bombay 400005, India
2 Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM Nicolás Cabrera, 13–15, 28049 Madrid, Spain
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Indranil Biswas; Oscar García-Prada. Anti-holomorphic involutions of the moduli spaces of Higgs bundles. Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 35-54. doi : 10.5802/jep.16. https://jep.centre-mersenne.org/articles/10.5802/jep.16/

[1] M. F. Atiyah - “Complex analytic connections in fibre bundles”, Trans. Amer. Math. Soc. 85 (1957), p. 181-207 | DOI | MR | Zbl

[2] D. Baraglia - “Classification of the automorphism and isometry groups of Higgs bundle moduli spaces” (2014), arXiv:1411.2228 | Zbl

[3] D. Baraglia & L. P. Schaposnik - “Real structures on moduli spaces of Higgs bundles”, to appear in Adv. Theo. Math. Phys. | DOI | Zbl

[4] D. Baraglia & L. P. Schaposnik - “Higgs bundles and (A,B,A)-branes”, Comm. Math. Phys. 331 (2014) no. 3, p. 1271-1300 | DOI | MR | Zbl

[5] I. Biswas, O. García-Prada & J. Hurtubise - “Higgs bundles on compact Kähler manifolds”, Ann. Inst. Fourier (Grenoble) 64 (2014), p. 2527-2562 | DOI | Numdam | Zbl

[6] I. Biswas & N. Hoffmann - “A Torelli theorem for moduli spaces of principal bundles over a curve”, Ann. Inst. Fourier (Grenoble) 62 (2012) no. 1, p. 87-106 | DOI | Numdam | MR | Zbl

[7] I. Biswas & G. Schumacher - “Yang-Mills equation for stable Higgs sheaves”, Internat. J. Math. 20 (2009) no. 5, p. 541-556 | DOI | MR | Zbl

[8] S. B. Bradlow, O. García-Prada & I. Mundet i Riera - “Relative Hitchin-Kobayashi correspondences for principal pairs”, Q. J. Math. 54 (2003) no. 2, p. 171-208 | DOI | MR | Zbl

[9] É. Cartan - “Les groupes réels simples, finis et continus”, Ann. Sci. École Norm. Sup. 31 (1914), p. 263-355 | DOI | Numdam | Zbl

[10] K. Corlette - “Flat G-bundles with canonical metrics”, J. Differential Geom. 28 (1988) no. 3, p. 361-382 | DOI | MR | Zbl

[11] S. K. Donaldson - “Twisted harmonic maps and the self-duality equations”, Proc. London Math. Soc. (3) 55 (1987) no. 1, p. 127-131 | DOI | MR | Zbl

[12] O. García-Prada - “Involutions of the moduli space of SL(n,)-Higgs bundles and real forms”, in Vector bundles and low codimensional subvarieties: state of the art and recent developments, Quad. Mat., vol. 21, Dept. Math., Seconda Univ. Napoli, Caserta, 2007, p. 219-238 | MR

[13] O. García-Prada - “Higgs bundles and surface group representations”, in Moduli spaces and vector bundles, London Math. Soc. Lecture Note Ser., vol. 359, Cambridge Univ. Press, Cambridge, 2009, p. 265-310 | DOI | MR | Zbl

[14] O. García-Prada, P. B. Gothen & I. Mundet i Riera - “The Hitchin–Kobayashi correspondence, Higgs pairs and surface group representations” (2009), arXiv:0909.4487

[15] O. García-Prada & S. Ramanan - “Involutions of Higgs bundle moduli spaces”, in preparation

[16] W. M. Goldman - “The symplectic nature of fundamental groups of surfaces”, Advances in Math. 54 (1984) no. 2, p. 200-225 | DOI | MR | Zbl

[17] N. J. Hitchin - “The self-duality equations on a Riemann surface”, Proc. London Math. Soc. (3) 55 (1987) no. 1, p. 59-126 | DOI | MR | Zbl

[18] N. J. Hitchin - “Higgs bundles and characteristic classes” (2013), arXiv:1308.4603

[19] N.-K. Ho, G. Wilkin & S. Wu - “Hitchin’s equations on a nonorientable manifold” (2012), arXiv:1211.0746 | Zbl

[20] A. Kapustin & E. Witten - “Electric-magnetic duality and the geometric Langlands program”, Commun. Number Theory Phys. 1 (2007) no. 1, p. 1-236 | DOI | MR | Zbl

[21] S. Kobayashi - Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ, 1987 | MR | Zbl

[22] J. de Siebenthal - “Sur les groupes de Lie compacts non connexes”, Comment. Math. Helv. 31 (1956), p. 41-89 | DOI | MR | Zbl

[23] C. T. Simpson - “Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization”, J. Amer. Math. Soc. 1 (1988) no. 4, p. 867-918 | DOI | MR | Zbl

[24] C. T. Simpson - “Higgs bundles and local systems”, Publ. Math. Inst. Hautes Études Sci. (1992) no. 75, p. 5-95 | DOI | Numdam | MR | Zbl

[25] C. T. Simpson - “Moduli of representations of the fundamental group of a smooth projective variety. II”, Publ. Math. Inst. Hautes Études Sci. (1994) no. 80, p. 5-79 | DOI | Numdam | MR

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