Anti-holomorphic involutions of the moduli spaces of Higgs bundles

[Involutions anti-holomorphes des espaces de modules de fibrés de Higgs]

Biswas, Indranil ; García-Prada, Oscar

Journal de l'École polytechnique - Mathématiques, Tome 2 (2015), p. 35-54 / Harvested from Numdam

Résumé

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.

Abstract

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.

Détail

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/jep.16
Classification:  14H60,  57R57,  58D29
Mots clés: $G$-fibré de Higgs, condition de réalité, « branes », variétés caractères.

Bibliographie

[1] Atiyah, M. F. Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., Tome 85 (1957), pp. 181-207 | MR 86359 | Zbl 0078.16002

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

[3] Baraglia, D.; Schaposnik, L. P. Real structures on moduli spaces of Higgs bundles (to appear in Adv. Theo. Math. Phys.) | MR 3248065

[4] Baraglia, D.; Schaposnik, L. P. Higgs bundles and $(A,B,A)$-branes, Comm. Math. Phys., Tome 331 (2014) no. 3, pp. 1271-1300 | Article

[5] Biswas, I.; García-Prada, O.; Hurtubise, J. Higgs bundles on compact Kähler manifolds, Ann. Inst. Fourier (Grenoble), Tome 64 (2014), pp. 2527-2562

[6] Biswas, I.; Hoffmann, N. A Torelli theorem for moduli spaces of principal bundles over a curve, Ann. Inst. Fourier (Grenoble), Tome 62 (2012) no. 1, pp. 87-106 http://aif.cedram.org/item?id=AIF_2012__62_1_87_0 | Numdam | MR 2986266 | Zbl 1268.14010

[7] Biswas, I.; Schumacher, G. Yang-Mills equation for stable Higgs sheaves, Internat. J. Math., Tome 20 (2009) no. 5, pp. 541-556 | Article | MR 2526306 | Zbl 1169.53017

[8] Bradlow, S. B.; García-Prada, O.; Mundet I Riera, I. Relative Hitchin-Kobayashi correspondences for principal pairs, Q. J. Math., Tome 54 (2003) no. 2, pp. 171-208 | Article | MR 1989871 | Zbl 1064.53056

[9] Cartan, É. Les groupes réels simples, finis et continus, Ann. Sci. École Norm. Sup., Tome 31 (1914), pp. 263-355

[10] Corlette, K. Flat $G$-bundles with canonical metrics, J. Differential Geom., Tome 28 (1988) no. 3, pp. 361-382 http://projecteuclid.org/euclid.jdg/1214442469 | MR 965220 | Zbl 0676.58007

[11] Donaldson, S. K. Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. (3), Tome 55 (1987) no. 1, pp. 127-131 | Article | MR 887285 | Zbl 0634.53046

[12] García-Prada, O. Involutions of the moduli space of $\mathrm{SL}(n,\mathbb{C})$-Higgs bundles and real forms, Vector bundles and low codimensional subvarieties: state of the art and recent developments, Dept. Math., Seconda Univ. Napoli, Caserta (Quad. Mat.) Tome 21 (2007), pp. 219-238 | MR 2544088

[13] García-Prada, O. Higgs bundles and surface group representations, Moduli spaces and vector bundles, Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 359 (2009), pp. 265-310 | MR 2537072 | Zbl 1187.14037

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

[15] García-Prada, O.; Ramanan, S. Involutions of Higgs bundle moduli spaces (in preparation)

[16] Goldman, W. M. The symplectic nature of fundamental groups of surfaces, Advances in Math., Tome 54 (1984) no. 2, pp. 200-225 | Article | MR 762512 | Zbl 0574.32032

[17] Hitchin, N. J. The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3), Tome 55 (1987) no. 1, pp. 59-126 | Article | MR 887284 | Zbl 0634.53045

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

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

[20] Kapustin, A.; Witten, E. Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys., Tome 1 (2007) no. 1, pp. 1-236 | Article | MR 2306566 | Zbl 1128.22013

[21] Kobayashi, S. Differential geometry of complex vector bundles, Princeton University Press, Princeton, NJ, Publications of the Mathematical Society of Japan, Tome 15 (1987), pp. xii+305 | MR 909698 | Zbl 0708.53002

[22] De Siebenthal, J. Sur les groupes de Lie compacts non connexes, Comment. Math. Helv., Tome 31 (1956), pp. 41-89 | MR 94408 | Zbl 0075.01602

[23] Simpson, C. T. Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc., Tome 1 (1988) no. 4, pp. 867-918 | Article | MR 944577 | Zbl 0669.58008

[24] Simpson, C. T. Higgs bundles and local systems, Publ. Math. Inst. Hautes Études Sci. (1992) no. 75, pp. 5-95 | Numdam | MR 1179076 | Zbl 0814.32003

[25] Simpson, C. T. Moduli of representations of the fundamental group of a smooth projective variety. II, Publ. Math. Inst. Hautes Études Sci. (1994) no. 80, pp. 5-79 | Numdam | MR 1320603 | Zbl 0891.14006