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Higgs bundles, branes and Langlands duality. (English) Zbl 1410.14046

The authors study the moduli space \(\mathcal{M} (G)\) of \(G\)-Higgs bundles over a compact Riemann surface \(X\) in the setting of the geometric Langlands programme. They assume that \(X\) is equipped with an anti-holomorphic involution \(\tau_X : X \to X\) (i.e., \(X\) is a real Riemann surface) and that \(G\) is endowed with an anti-holomorphic group involution \(\tau_G : G \to G\) (i.e., a real form). They use this structure to induce a pair of natural involutions on the moduli space \(\mathcal{M} (G)\) whose fixed point sets are an \((A,A,B)\)-brane and an \((A,B,A)\)-brane.
Then the question they ask is “how the various components of \((A, A, B)\) and \((A, B, A)\) branes match up, as one passes from \(G\) to \({}^L G\).” One way this question can be posed is to ask how the real structure interacts with the dual fibres of the moduli spaces \(\mathcal{M} (G)\) and \(\mathcal{M} ({}^L G)\). The authors give a description of the intersection between a brane and a pair of dual fibres, and prove that these fibres have the same number of real components.
Another way is to interpret this question globally on the full moduli space, where the situation turns out to be very different in general. Even in the case of \(G = \mathrm{SL}(2, \mathbb{C})\) and \({}^L G = \mathrm{PGL}(2, \mathbb{C})\) (the only one considered in the paper), “the real components are far from matching”: the fixed point locus in \(\mathcal{M} ({}^L G)\) has a strictly greater number of connected components. Their technique to obtain this count is first to give a very explicit description of the effect that the involution has on the homology lattice of a given \(\mathrm{SL}(2, \mathbb{C})\) spectral curve \(S\) over \(X\), and then use it to count the number of real components of the Prym variety for \(S\).

MSC:

14P99 Real algebraic and real-analytic geometry
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
32Q15 Kähler manifolds
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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[1] Baraglia D., Schaposnik L.: Higgs bundles and (A, B, A)-branes. Commun. Math. Phys. 331, 1271-1300 (2014) · Zbl 1311.53058 · doi:10.1007/s00220-014-2053-6
[2] Baraglia D., Schaposnik L.P.: Real structures on moduli spaces of Higgs bundles. Adv. Theor. Math. Phys. 20, 525-551 (2016) · Zbl 1372.81165 · doi:10.4310/ATMP.2016.v20.n3.a2
[3] Biswas I., Garcia-Prada O.: Antiholomorphic involutions of the moduli spaces of Higgs bundles. J. Éc. Polytech. Math. 2, 35-54 (2015) · Zbl 1333.14032 · doi:10.5802/jep.16
[4] Biswas I., Garcia-Prada O., Hurtubise J.: Pseudo-real principal Higgs bundles on compact Kähler manifolds. Ann. Inst. Fourier 64, 2527-2562 (2014) · Zbl 1329.14106 · doi:10.5802/aif.2920
[5] Biswas I., Garcia-Prada O., Hurtubise J.: Pseudo-real principal G-bundles over a real curve. J. Lond. Math. Soc. 93, 47-64 (2016) · Zbl 1352.14039 · doi:10.1112/jlms/jdv055
[6] Biswas I., Gómez T.L.: Connections and Higgs fields on a principal bundle. Ann. Glob. Anal. Geom. 33, 19-46 (2008) · Zbl 1185.14032 · doi:10.1007/s10455-007-9072-x
[7] Biswas I., Huisman J., Hurtubise J.: The moduli space of stable vector bundles over a real algebraic curve. Math. Ann. 347, 201-233 (2010) · Zbl 1195.14048 · doi:10.1007/s00208-009-0442-5
[8] Cartan É.: Les groupes réels simples, finis et continus. Ann. Éc. Norm. Sup. 31, 263-355 (1914) · JFM 45.1408.03 · doi:10.24033/asens.676
[9] Curtis C.W., Reiner I.: Representation Theory of Finite Groups and Associative Algebras, Pure and Applied Mathematics. Interscience Publishers, Geneva (1962) · Zbl 0131.25601
[10] Donagi R.: Decomposition of spectral covers. Astérisque 218, 145-175 (1993) · Zbl 0820.14031
[11] Donagi R.: Spectral Covers, Current Topics in Complex Algebraic Geometry (Berkeley, CA, 1992/93), pp. 65-88. Mathematical Sciences Research Institute Publications, Cambridge University Press, Cambridge (1995)
[12] Donagi R., Gaitsgory D.: The gerbe of Higgs bundles. Transform Groups 7, 109-153 (2002) · Zbl 1083.14519 · doi:10.1007/s00031-002-0008-z
[13] Donagi R., Pantev T.: Langlands duality for Hitchin systems. Invent Math. 189, 653-735 (2012) · Zbl 1263.53078 · doi:10.1007/s00222-012-0373-8
[14] Faltings G.: Stable G-bundles and projective connections. J. Algebr. Geom. 2, 507-568 (1993) · Zbl 0790.14019
[15] García-Prada, O.: Involutions of the moduli space of \[{{\rm SL}(n, \mathbb{C})}\] SL(n,C) -Higgs bundles and real forms. In: Casnati, G., Catanese, F., Notari, R. (eds.) Vector Bundles and Low Codimensional Subvarieties: State of the Art and Recent Developments, Quaderni di Matematica (2007)
[16] García-Prada, O., Gothen, P.B., Mundet i Riera, I.: The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations (2009). arXiv:0909.4487 · Zbl 1303.14043
[17] García-Prada, O., Ramanan, S.: Involutions and higher order automorphisms of Higgs bundle moduli spaces. arXiv:1605.05143 · Zbl 1427.14068
[18] García-Prada, O., Wilkin, G.: Action of the mapping class group on character varieties and Higgs bundles. arXiv:1612.02508 · Zbl 1451.14108
[19] Groechenig, M., Wyss, D., Ziegler, P.: Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration. arXiv:1707.06417 [math.AG] · Zbl 1451.14123
[20] Hausel T., Thaddeus M.: Mirror symmetry, Langlands duality, and the Hitchin system. Invent Math. 153, 197-229 (2003) · Zbl 1043.14011 · doi:10.1007/s00222-003-0286-7
[21] Heller, S., Schaposnik, L.P.: Branes through finite group actions. arXiv:1611.00391 · Zbl 1388.81546
[22] Hitchin N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55, 59-126 (1987) · Zbl 0634.53045 · doi:10.1112/plms/s3-55.1.59
[23] Hitchin N.J.: Stable bundles and integrable systems. Duke Math. J. 54, 91-114 (1987) · Zbl 0627.14024 · doi:10.1215/S0012-7094-87-05408-1
[24] Hitchin N.J.: Langlands duality and G2 spectral curves. Q. J. Math. 58, 319-344 (2007) · Zbl 1144.14034 · doi:10.1093/qmath/ham016
[25] Kapustin A., Witten E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Thoery Phys. 1, 1-236 (2007) · Zbl 1128.22013 · doi:10.4310/CNTP.2007.v1.n1.a1
[26] Nadler D.: Perverse sheaves on real loop Grassmannians. Invent Math. 159, 1-73 (2005) · Zbl 1089.14008 · doi:10.1007/s00222-004-0382-3
[27] Scognamillo R.: An elementary approach to the abelianization of the Hitchin system for arbitrary reductive groups. Compos. Math. 110, 17-37 (1998) · Zbl 0915.14007 · doi:10.1023/A:1000235107340
[28] Strominger A., Yau S.-T., Zaslow E.: Mirror symmetry is T-duality. Nucl. Phys. B 479, 243-259 (1996) · Zbl 0896.14024 · doi:10.1016/0550-3213(96)00434-8
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