Communications in Mathematical Physics

, Volume 331, Issue 3, pp 1271–1300 | Cite as

Higgs Bundles and (A, B, A)-Branes



Through the action of anti-holomorphic involutions on a compact Riemann surface Σ we construct families of (A, B, A)-branes \({\mathcal{L}_{G_{c}}}\) in the moduli spaces \({\mathcal{M}_{G_{c}}}\) of G c -Higgs bundles on Σ. We study the geometry of these (A, B, A)-branes in terms of spectral data and show they have the structure of real integrable systems.


Modulus Space Riemann Surface Line Bundle Double Cover Spectral Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Ati71.
    Atiyah M.: Riemann surfaces and spin structures. Ann. Sci. École Norm. Sup. 4, 47–62 (1971)MATHMathSciNetGoogle Scholar
  2. AB82.
    Atiyah M.F., Bott R.: Yang-Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. A 308, 523–615 (1982)ADSMathSciNetCrossRefGoogle Scholar
  3. BaSch.
    Baraglia, D., Schaposnik, L.P.: Real structures on moduli spaces of Higgs bundles. Adv. Theor. Math. Phys. (2014, to appear). arXiv:1309.1195
  4. BiGo08.
    Biswas I., Gómez T.L.: Connections and Higgs fields on a principal bundle. Ann. Glob. Anal. Geom. 33(1), 19–46 (2008)MATHCrossRefGoogle Scholar
  5. BBCGG.
    Broughton S.A., Bujalance E., Costa A.F., Gamboa J.M., Gromadzki G.: Symmetries of Accola-Maclachlan and Kulkarni surfaces. Proc. Am. Math. Soc. 127, 637–646 (1999)MATHMathSciNetCrossRefGoogle Scholar
  6. BCGG.
    Bujalance E., Cirre J., Gamboa J.M., Gromadzki G.: Symmetry types of hyperelliptic Riemann surfaces. Memoires de la Societe Mathematique de France 86, 1–122 (2001)Google Scholar
  7. BNR89.
    Beauville A., Narasimhan M.S., Ramanan S.: Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398, 169–179 (1989)MATHMathSciNetGoogle Scholar
  8. Cor88.
    Corlette K.: Flat G-bundles with canonical metrics. J. Differ. Geom. 28, 361–382 (1988)MATHMathSciNetGoogle Scholar
  9. D87.
    Donaldson S.K.: Twisted harmonic maps and the self-duality equations. Proc. Lond. Math. Soc. (3) 55(1), 127–131 (1987)MATHMathSciNetCrossRefGoogle Scholar
  10. GH81.
    Gross B.H., Harris J.: Real algebraic curves. Ann. Sci. École Norm. Sup. (4) 14(2), 157–182 (1981)MATHMathSciNetGoogle Scholar
  11. Go84.
    Goldman W.M.: The symplectic nature of fundamental groups of surfaces. Adv. Math. 54, 200–225 (1984)MATHCrossRefGoogle Scholar
  12. Gu07.
    Gukov, S.: Surface operators and knot homologies. In: New Trends in Mathematical Physics, pp. 313–343 (2009)Google Scholar
  13. HT03.
    Hausel T., Thaddeus M.: Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153, 197–229 (2003)ADSMATHMathSciNetCrossRefGoogle Scholar
  14. Hit87.
    Hitchin N.J.: The self-duality equations on a Riemann surface. Proc. LMS 55(3), 59–126 (1987)MATHMathSciNetGoogle Scholar
  15. Hit87a.
    Hitchin N.J.: Stable bundles and integrable systems. Duke Math. J. 54(1), 91–114 (1987)MATHMathSciNetCrossRefGoogle Scholar
  16. Hit92.
    Hitchin N.J.: Lie groups and Teichmüller space. Topology 31(3), 449–473 (1992)MATHMathSciNetCrossRefGoogle Scholar
  17. Hit07.
    Hitchin N.J.: Langlands duality and G2 spectral curves. Q. J. Math. 58, 319–344 (2007)MATHMathSciNetCrossRefGoogle Scholar
  18. Huy.
    Huybrechts D.: Fourier-Mukai transforms in algebraic geometry. Oxford University Press, Oxford (2006)MATHCrossRefGoogle Scholar
  19. KMcC96.
    Kalliongis J., McCullough D.: Orientation-reversing involutions on handlebodies. Trans. AMS 348(5), 1739–1755 (1996)MATHMathSciNetCrossRefGoogle Scholar
  20. KW07.
    Kapustin A., Witten E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1, 1–236 (2007)MATHMathSciNetCrossRefGoogle Scholar
  21. MM03.
    Moerdijk I., Mrčun J.: Introduction to foliations and Lie groupoids. Cambridge University Press, Cambridge (2003)MATHCrossRefGoogle Scholar
  22. Ra75.
    Ramanathan A.: Stable principal bundles on a compact Riemann surface. Math. Ann. 213, 129–152 (1975)MATHMathSciNetCrossRefGoogle Scholar
  23. Sch11.
    Schaposnik, L.P.: Monodromy of the SL2 Hitchin fibration. Int. J. Math. 24 (2013)Google Scholar
  24. Sch.
    Schaposnik L.P.: Spectral data for G-Higgs bundles. University of Oxford, Oxford (2013)Google Scholar
  25. Se91.
    Seppälä M.: Moduli spaces of stable real algebraic curves. Ann. Sci. École Norm. Sup. (4) 24(5), 519–544 (1991)MATHGoogle Scholar
  26. S88.
    Simpson C.T.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformisation. J. AMS 1, 867–918 (1988)MATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesThe University of AdelaideAdelaideAustralia
  2. 2.Mathematisches InstitutRuprecht-Karls-Universität HeidelbergHeidelbergGermany
  3. 3.Department of MathematicsUniversity of Illinois at Urbana ChampaignUrbanaUSA

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