Cayley and Langlands type correspondences for orthogonal Higgs bundles

Baraglia, David; Schaposnik, Laura

doi:10.1090/tran/7587

Cayley and Langlands type correspondences for orthogonal Higgs bundles
HTML articles powered by AMS MathViewer

by David Baraglia and Laura P. Schaposnik PDF

Trans. Amer. Math. Soc. 371 (2019), 7451-7492 Request permission

Abstract:

Through Cayley and Langlands type correspondences, we give a geometric description of the moduli spaces of real orthogonal and symplectic Higgs bundles of any signature in the regular fibers of the Hitchin fibration. As applications of our methods, we complete the concrete abelianization of real slices corresponding to all quasi-split real forms, and we describe how extra components emerge naturally from the spectral data point of view.

References

M. Aparicio Arroyo, The geometry of $SO(p,q)$-Higgs bundles, 2009. Thesis (Ph.D.)–University of Salamanca.
M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451–491. MR 232406, DOI 10.2307/1970721
David Baraglia, Monodromy of the $SL(n)$ and $GL(n)$ Hitchin fibrations, Math. Ann. 370 (2018), no. 3-4, 1681–1716. MR 3770177, DOI 10.1007/s00208-017-1611-6
D. Baraglia and L. P. Schaposnik, Monodromy of rank $2$ twisted Hitchin systems and real character varieties, Trans. of the AMS, DOI: https://doi.org/10.1090/tran/7144.
David Baraglia and Laura P. Schaposnik, Real structures on moduli spaces of Higgs bundles, Adv. Theor. Math. Phys. 20 (2016), no. 3, 525–551. MR 3565861, DOI 10.4310/ATMP.2016.v20.n3.a2
Arnaud Beauville, M. S. Narasimhan, and S. Ramanan, Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398 (1989), 169–179. MR 998478, DOI 10.1515/crll.1989.398.169
Olivier Biquard, Oscar García-Prada, and Roberto Rubio, Higgs bundles, the Toledo invariant and the Cayley correspondence, J. Topol. 10 (2017), no. 3, 795–826. MR 3797597, DOI 10.1112/topo.12023
Steven B. Bradlow, Oscar García-Prada, and Peter B. Gothen, Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces, Geom. Dedicata 122 (2006), 185–213. MR 2295550, DOI 10.1007/s10711-007-9127-y
Marc Burger, Alessandra Iozzi, and Anna Wienhard, Surface group representations with maximal Toledo invariant, Ann. of Math. (2) 172 (2010), no. 1, 517–566. MR 2680425, DOI 10.4007/annals.2010.172.517
B. Collier, SO(n,n+1)-surface group representations and their Higgs bundles, arXiv:1710.01287 (2017).
Brian Collier, Finite order automorphisms of Higgs Bsundles: Theory and application, ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. MR 3624308
B. Collier, N. Tholozan, and J. Touilisse, The geometry of maximal representations of surface groups into $SO_0(2,n)$, arXiv:1702.08799 (2017).
R. Donagi and T. Pantev, Langlands duality for Hitchin systems, Invent. Math. 189 (2012), no. 3, 653–735. MR 2957305, DOI 10.1007/s00222-012-0373-8
O. García-Prada, P. B. Gothen, and I. Mundet i Riera, Higgs bundles and surface group representations in the real symplectic group, J. Topol. 6 (2013), no. 1, 64–118. MR 3029422, DOI 10.1112/jtopol/jts030
Oscar García-Prada and André G. Oliveira, Connectedness of the moduli of $\textrm {Sp}(2p,2q)$-Higgs bundles, Q. J. Math. 65 (2014), no. 3, 931–956. MR 3261975, DOI 10.1093/qmath/hat045
P. Gothen, The topology of Higgs bundle moduli spaces, 1995. Thesis (Ph.D.)–University of Warwick.
Peter B. Gothen and André G. Oliveira, Rank two quadratic pairs and surface group representations, Geom. Dedicata 161 (2012), 335–375. MR 2994046, DOI 10.1007/s10711-012-9709-1
O. Guichard and A. Wienhard, Positivity and higher Teichmüller theory, Proc. of the 7th ECM (to appear).
Olivier Guichard and Anna Wienhard, Topological invariants of Anosov representations, J. Topol. 3 (2010), no. 3, 578–642. MR 2684514, DOI 10.1112/jtopol/jtq018
Jochen Heinloth, Uniformization of $\scr G$-bundles, Math. Ann. 347 (2010), no. 3, 499–528. MR 2640041, DOI 10.1007/s00208-009-0443-4
N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. MR 887284, DOI 10.1112/plms/s3-55.1.59
Nigel Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987), no. 1, 91–114. MR 885778, DOI 10.1215/S0012-7094-87-05408-1
N. J. Hitchin, Lie groups and Teichmüller space, Topology 31 (1992), no. 3, 449–473. MR 1174252, DOI 10.1016/0040-9383(92)90044-I
Nigel Hitchin, Langlands duality and $G_2$ spectral curves, Q. J. Math. 58 (2007), no. 3, 319–344. MR 2354922, DOI 10.1093/qmath/ham016
Nigel Hitchin and Laura P. Schaposnik, Nonabelianization of Higgs bundles, J. Differential Geom. 97 (2014), no. 1, 79–89. MR 3229050
Nigel Hitchin, Higgs bundles and characteristic classes, Arbeitstagung Bonn 2013, Progr. Math., vol. 319, Birkhäuser/Springer, Cham, 2016, pp. 247–264. MR 3618052, DOI 10.1007/978-3-319-43648-7_{8}
Anton Kapustin and Edward Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), no. 1, 1–236. MR 2306566, DOI 10.4310/CNTP.2007.v1.n1.a1
Jun Li, The space of surface group representations, Manuscripta Math. 78 (1993), no. 3, 223–243. MR 1206154, DOI 10.1007/BF02599310
David Nadler, Perverse sheaves on real loop Grassmannians, Invent. Math. 159 (2005), no. 1, 1–73. MR 2142332, DOI 10.1007/s00222-004-0382-3
M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567. MR 184252, DOI 10.2307/1970710
A. Peón-Nieto, Higgs bundles, real forms and the Hitchin fibration, 2013. Thesis (Ph.D.)–Autonomous University of Madrid.
A. Peón-Nieto, Cameral data for $SU(p+ 1, p)$-Higgs bundles, arXiv:1506.01318 (2015).
Laura P. Schaposnik, Spectral data for G-Higgs bundles, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (D.Phil.)–University of Oxford (United Kingdom). MR 3389247
Laura P. Schaposnik, Spectral data for $U(m,m)$-Higgs bundles, Int. Math. Res. Not. IMRN 11 (2015), 3486–3498. MR 3373057
L. P. Schaposnik, A geometric approach to orthogonal Higgs bundles, Eur. J. Math. (2017), DOI 10.1007/s40879-017-0206-9. arXiv:1608.00300.
Carlos T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95. MR 1179076
Carlos 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. MR 1307297
Carlos 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 (1995). MR 1320603
Eugene Z. Xia, The moduli of flat $\textrm {U}(p,1)$ structures on Riemann surfaces, Geom. Dedicata 97 (2003), 33–43. Special volume dedicated to the memory of Hanna Miriam Sandler (1960–1999). MR 2003688, DOI 10.1023/A:1023675007667