Anti-holomorphic involutions of the moduli spaces of Higgs bundles

Biswas, Indranil; García-Prada, Oscar

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.

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.

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

[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

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

[5] Biswas, I.; García-Prada, O.; Hurtubise, J. Higgs bundles on compact Kähler manifolds, Ann. Inst. Fourier (Grenoble), Volume 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), Volume 62 (2012) no. 1, pp. 87-106 | Numdam | MR | Zbl

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

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

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

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

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

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

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

[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., Volume 54 (1984) no. 2, pp. 200-225 | DOI | MR | Zbl

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

[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., Volume 1 (2007) no. 1, pp. 1-236 | DOI | MR | Zbl

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

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

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

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

[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 | Zbl