## Monodromy of rank 2 twisted Hitchin systems and real character varieties

HTML articles powered by AMS MathViewer

- by David Baraglia and Laura P. Schaposnik PDF
- Trans. Amer. Math. Soc.
**370**(2018), 5491-5534 Request permission

## Abstract:

We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of $L$-twisted $G$-Higgs bundles for the groups $G = GL(2,\mathbb {C})$, $SL(2,\mathbb {C})$, and $PSL(2,\mathbb {C})$. We also determine the Tate-Shafarevich class of the abelian torsor defined by the regular locus, which obstructs the existence of a section of the moduli space of $L$-twisted Higgs bundles of rank $2$ and degree $\deg (L)+1$. By counting orbits of the monodromy action with $\mathbb {Z}_2$-coefficients, we obtain in a unified manner the number of components of the character varieties for the real groups $G = GL(2,\mathbb {R})$, $SL(2,\mathbb {R})$, $PGL(2,\mathbb {R})$, $PSL(2,\mathbb {R})$, as well as the number of components of the $Sp(4,\mathbb {R})$ and $SO_0(2,3)$-character varieties with maximal Toledo invariant. We also use our results for $GL(2,\mathbb {R})$ to compute the monodromy of the $SO(2,2)$ Hitchin map and determine the components of the $SO(2,2)$ character variety.## References

- V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko,
*Singularities of differentiable maps. Vol. II*, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. Monodromy and asymptotics of integrals; Translated from the Russian by Hugh Porteous; Translation revised by the authors and James Montaldi. MR**966191**, DOI 10.1007/978-1-4612-3940-6 - Michael F. Atiyah,
*Riemann surfaces and spin structures*, Ann. Sci. École Norm. Sup. (4)**4**(1971), 47–62. MR**286136**, DOI 10.24033/asens.1205 - 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,
*Topological T-duality for general circle bundles*, Pure Appl. Math. Q.**10**(2014), no. 3, 367–438. MR**3282986**, DOI 10.4310/PAMQ.2014.v10.n3.a1 - David Baraglia,
*Topological $T$-duality for torus bundles with monodromy*, Rev. Math. Phys.**27**(2015), no. 3, 1550008, 55. MR**3342758**, DOI 10.1142/S0129055X15500087 - 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 - Christina Birkenhake and Herbert Lange,
*Complex abelian varieties*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 2004. MR**2062673**, DOI 10.1007/978-3-662-06307-1 - Joan S. Birman,
*Braids, links, and mapping class groups*, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR**0375281** - Steven B. Bradlow, Oscar García-Prada, and Ignasi Mundet i Riera,
*Relative Hitchin-Kobayashi correspondences for principal pairs*, Q. J. Math.**54**(2003), no. 2, 171–208. MR**1989871**, DOI 10.1093/qjmath/54.2.171 - 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 - Steven B. Bradlow and Laura P. Schaposnik,
*Higgs bundles and exceptional isogenies*, Res. Math. Sci.**3**(2016), Paper No. 14, 28. MR**3531373**, DOI 10.1186/s40687-016-0062-0 - Mark Andrea A. de Cataldo, Tamás Hausel, and Luca Migliorini,
*Topology of Hitchin systems and Hodge theory of character varieties: the case $A_1$*, Ann. of Math. (2)**175**(2012), no. 3, 1329–1407. MR**2912707**, DOI 10.4007/annals.2012.175.3.7 - D. J. Copeland, A special subgroup of the surface braid group, arXiv:0409461 (2004).
- D. Jeremy Copeland,
*Monodromy of the Hitchin map over hyperelliptic curves*, Int. Math. Res. Not.**29**(2005), 1743–1785. MR**2172340**, DOI 10.1155/IMRN.2005.1743 - Kevin Corlette,
*Flat $G$-bundles with canonical metrics*, J. Differential Geom.**28**(1988), no. 3, 361–382. MR**965220** - Igor Dolgachev and Anatoly Libgober,
*On the fundamental group of the complement to a discriminant variety*, Algebraic geometry (Chicago, Ill., 1980) Lecture Notes in Math., vol. 862, Springer, Berlin-New York, 1981, pp. 1–25. MR**644816** - Antun Domic and Domingo Toledo,
*The Gromov norm of the Kaehler class of symmetric domains*, Math. Ann.**276**(1987), no. 3, 425–432. MR**875338**, DOI 10.1007/BF01450839 - R. Y. Donagi and D. Gaitsgory,
*The gerbe of Higgs bundles*, Transform. Groups**7**(2002), no. 2, 109–153. MR**1903115**, DOI 10.1007/s00031-002-0008-z - S. K. Donaldson,
*Twisted harmonic maps and the self-duality equations*, Proc. London Math. Soc. (3)**55**(1987), no. 1, 127–131. MR**887285**, DOI 10.1112/plms/s3-55.1.127 - Gerd Faltings,
*Stable $G$-bundles and projective connections*, J. Algebraic Geom.**2**(1993), no. 3, 507–568. MR**1211997** - O. García-Prada, P. B. Gothen, and I. Mundet i Riera, The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations, arXiv:0909.4487v3 (2012).
- 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 - William M. Goldman,
*The symplectic nature of fundamental groups of surfaces*, Adv. in Math.**54**(1984), no. 2, 200–225. MR**762512**, DOI 10.1016/0001-8708(84)90040-9 - William M. Goldman,
*Topological components of spaces of representations*, Invent. Math.**93**(1988), no. 3, 557–607. MR**952283**, DOI 10.1007/BF01410200 - Peter B. Gothen,
*Components of spaces of representations and stable triples*, Topology**40**(2001), no. 4, 823–850. MR**1851565**, DOI 10.1016/S0040-9383(99)00086-5 - 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 - 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} - Eduard Looijenga,
*Cohomology and intersection homology of algebraic varieties*, Complex algebraic geometry (Park City, UT, 1993) IAS/Park City Math. Ser., vol. 3, Amer. Math. Soc., Providence, RI, 1997, pp. 221–263. MR**1442524**, DOI 10.1090/pcms/003/04 - Nitin Nitsure,
*Moduli space of semistable pairs on a curve*, Proc. London Math. Soc. (3)**62**(1991), no. 2, 275–300. MR**1085642**, DOI 10.1112/plms/s3-62.2.275 - R. W. Richardson,
*Conjugacy classes of $n$-tuples in Lie algebras and algebraic groups*, Duke Math. J.**57**(1988), no. 1, 1–35. MR**952224**, DOI 10.1215/S0012-7094-88-05701-8 - 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,
*Monodromy of the $\textrm {SL}_2$ Hitchin fibration*, Internat. J. Math.**24**(2013), no. 2, 1350013, 21. MR**3045345**, DOI 10.1142/S0129167X13500134 - Laura P. Schaposnik,
*Spectral data for $U(m,m)$-Higgs bundles*, Int. Math. Res. Not. IMRN**11**(2015), 3486–3498. MR**3373057** - Carlos T. Simpson,
*Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization*, J. Amer. Math. Soc.**1**(1988), no. 4, 867–918. MR**944577**, DOI 10.1090/S0894-0347-1988-0944577-9 - Carlos T. Simpson,
*Higgs bundles and local systems*, Inst. Hautes Études Sci. Publ. Math.**75**(1992), 5–95. MR**1179076**, DOI 10.1007/BF02699491 - V. G. Turaev,
*A cocycle of the symplectic first Chern class and Maslov indices*, Funktsional. Anal. i Prilozhen.**18**(1984), no. 1, 43–48 (Russian). MR**739088** - Katharine C. Walker,
*Quotient groups of the fundamental groups of certain strata of the moduli space of quadratic differentials*, Geom. Topol.**14**(2010), no. 2, 1129–1164. MR**2651550**, DOI 10.2140/gt.2010.14.1129 - Eugene Z. Xia,
*Components of $\textrm {Hom}(\pi _1,\textrm {PGL}(2,\mathbf R))$*, Topology**36**(1997), no. 2, 481–499. MR**1415600**, DOI 10.1016/0040-9383(96)00008-0 - Eugene Z. Xia,
*The moduli of flat $\textrm {PGL}(2,\textbf {R})$ connections on Riemann surfaces*, Comm. Math. Phys.**203**(1999), no. 3, 531–549. MR**1700170**, DOI 10.1007/s002200050624

## Additional Information

**David Baraglia**- Affiliation: School of Mathematical Sciences, The University of Adelaide, Adelaide SA 5005, Australia
- MR Author ID: 912405
- Email: david.baraglia@adelaide.edu.au
**Laura P. Schaposnik**- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
- Address at time of publication: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607; and Department of Mathematics, Freie Universität Berlin, 14195 Berlin, Germany
- MR Author ID: 1013453
- ORCID: 0000-0003-1417-2201
- Email: schapos@uic.edu
- Received by editor(s): January 21, 2016
- Received by editor(s) in revised form: November 28, 2016
- Published electronically: February 28, 2018
- Additional Notes: The work of the first author was supported by the Australian Research Council Discovery Project DP110103745.

The work of the second author was supported by the Simons Foundation through an AMS-Simons Travel Grant. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 5491-5534 - MSC (2010): Primary 14H60, 53C07; Secondary 14H70
- DOI: https://doi.org/10.1090/tran/7144
- MathSciNet review: 3812111