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Branes and moduli spaces of Higgs bundles on smooth projective varieties

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Abstract

Given a smooth complex projective variety M and a smooth closed curve \(X\, \subset \, M\) such that the homomorphism of fundamental groups \(\pi _1(X)\, \longrightarrow \, \pi _1(M)\) is surjective, we study the restriction map of Higgs bundles, namely from the Higgs bundles on M to those on X. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on M and X. We also consider the setup where a finite group is acting on M via holomorphic automorphisms or anti-holomorphic involutions, and the curve X is preserved by this action. Branes are studied in this context.

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References

  1. Baraglia, D., Schaposnik, L.P.: Higgs bundles and \((A, B, A)\)-branes. Commun. Math. Phys. 331, 1271–1300 (2014)

    Article  MathSciNet  Google Scholar 

  2. Baraglia, D., Schaposnik, L.P.: Real structures on moduli spaces of Higgs bundles. Adv. Theor. Math. Phys. 20, 525–551 (2016)

    Article  MathSciNet  Google Scholar 

  3. Biswas, I.: Parabolic principal Higgs bundles. J. Ramanujan Math. Soc. 23, 311–325 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Biswas, I., Majumder, S., Wong, M.L.: Parabolic Higgs bundles and \(\Gamma \)-Higgs bundles. J. Aust. Math. Soc. 95, 315–328 (2013)

    Article  MathSciNet  Google Scholar 

  5. Chen, T.-H., Ngô, B.C.: On the Hitchin morphism for higher dimensional varieties. Duke Math. J. 169, 1971–2004 (2020)

    MathSciNet  MATH  Google Scholar 

  6. Corlette, K.: Flat G-bundles with canonical metrics. J. Differ. Geom. 28, 361–382 (1988)

    Article  MathSciNet  Google Scholar 

  7. Donaldson, S.K.: Twisted harmonic maps and the self-duality equations. Proc. Lond. Math. Soc. 55, 127–131 (1987)

    Article  MathSciNet  Google Scholar 

  8. Heller, S., Schaposnik, L.P.: Branes through finite group actions. J. Geom. Phys. 129, 279–293 (2018)

    Article  MathSciNet  Google Scholar 

  9. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55, 59–126 (1987)

    Article  MathSciNet  Google Scholar 

  10. Hitchin, N.J.: Stable bundles and integrable systems. Duke Math. J. 54, 91–114 (1987)

    Article  MathSciNet  Google Scholar 

  11. Hitchin, N.J., Karlhede, A., Lindström, U., Rocek, M.: Hyperkähler metrics and supersymmetry. Commun. Math. Phys. 108, 535–589 (1987)

    Article  Google Scholar 

  12. Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1, 1–236 (2007)

    Article  MathSciNet  Google Scholar 

  13. Kobayashi, S.: Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15. Kanô Memorial Lectures, 5. Princeton University Press, Princeton (1987)

  14. Lamotke, K.: The topology of complex projective varieties after S. Lefschetz. Topology 20, 15–51 (1981)

    Article  MathSciNet  Google Scholar 

  15. Simpson, C.T.: Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization. J. Am. Math. Soc. 1, 867–918 (1988)

    Article  MathSciNet  Google Scholar 

  16. Simpson, C.T.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 7(5), 5–95 (1992)

    Article  MathSciNet  Google Scholar 

  17. Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety. I. Inst. Hautes Études Sci. Publ. Math. 79, 47–129 (1994)

  18. Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety. II, Inst. Hautes Études Sci. Publ. Math. 80, 5–79 (1994)

  19. Simpson, C.T.: The Hodge filtration on nonabelian cohomology, Algebraic geometry, Santa Cruz 1995, 217–281, Proc. Sympos. Pure Math. 62, Part 2, American Mathematical Society, Providence, RI (1997)

  20. Wentworth, R.A.: Higgs bundles and local systems on Riemann surfaces, Geometry and quantization of moduli spaces, 165–219. Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham (2016)

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Acknowledgements

We thank the two referees for going through the paper very carefully. IB is supported by a J. C. Bose Fellowship. LPS is partially supported by NSF CAREER Award DMS-1749013. On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Authors and Affiliations

  1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005, India

    Indranil Biswas

  2. Institute of Differential Geometry, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Sebastian Heller

  3. Department of Mathematics Statistics, and Computer Science, University of Illinois at Chicago, 851 S Morgan St, Chicago, IL, 60607, USA

    Laura P. Schaposnik

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  1. Indranil Biswas
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  2. Sebastian Heller
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  3. Laura P. Schaposnik
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Correspondence to Indranil Biswas.

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Biswas, I., Heller, S. & Schaposnik, L.P. Branes and moduli spaces of Higgs bundles on smooth projective varieties. Res Math Sci 8, 52 (2021). https://doi.org/10.1007/s40687-021-00286-z

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