Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 14 (2018), 111, 22 pages      arXiv:1710.03977      https://doi.org/10.3842/SIGMA.2018.111

The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations

Arata Komyo
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

Received January 23, 2018, in final form October 03, 2018; Published online October 13, 2018

Abstract
In this paper, we study the moduli spaces of parabolic connections with a quadratic differential. We endow these moduli spaces with symplectic structures by using the fundamental 2-forms on the moduli spaces of parabolic connections (which are phase spaces of isomonodromic deformation systems). Moreover, we see that the moduli spaces of parabolic connections with a quadratic differential are equipped with structures of twisted cotangent bundles.

Key words: parabolic connection; quadratic differential; isomonodromic deformation; twisted cotangent bundle.

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References

  1. Atiyah M.F., Bott R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523-615.
  2. Beilinson A., Bernstein J., A proof of Jantzen conjectures, in I.M. Gel'fand Seminar, Adv. Soviet Math., Vol. 16, Amer. Math. Soc., Providence, RI, 1993, 1-50.
  3. Beilinson A., Kazhdan D., Flat projective connection, unpublished.
  4. Beilinson A.A., Schechtman V.V., Determinant bundles and Virasoro algebras, Comm. Math. Phys. 118 (1988), 651-701.
  5. Ben-Zvi D., Biswas I., Theta functions and Szegő kernels, Int. Math. Res. Not. 2003 (2003), 1305-1340, math.AG/0211441.
  6. Ben-Zvi D., Biswas I., Opers and theta functions, Adv. Math. 181 (2004), 368-395, math.AG/0204301.
  7. Ben-Zvi D., Frenkel E., Geometric realization of the Segal-Sugawara construction, in Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., Vol. 308, Cambridge University Press, Cambridge, 2004, 46-97, math.AG/0301206.
  8. Bloch S., Esnault H., Relative algebraic differential characters, math.AG/9912015.
  9. Boalch P., Symplectic manifolds and isomonodromic deformations, Adv. Math. 163 (2001), 137-205.
  10. Faltings G., Stable $G$-bundles and projective connections, J. Algebraic Geom. 2 (1993), 507-568.
  11. Goldman W.M., The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200-225.
  12. Hitchin N., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
  13. Hitchin N., Geometrical aspects of Schlesinger's equation, J. Geom. Phys. 23 (1997), 287-300.
  14. Hurtubise J., On the geometry of isomonodromic deformations, J. Geom. Phys. 58 (2008), 1394-1406, arXiv:0804.0249.
  15. Inaba M.-A., Moduli of parabolic connections on curves and the Riemann-Hilbert correspondence, J. Algebraic Geom. 22 (2013), 407-480, math.AG/0602004.
  16. Inaba M.-A., Iwasaki K., Saito M.-H., Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part I, Publ. Res. Inst. Math. Sci. 42 (2006), 987-1089, math.AG/0309342.
  17. Iwasaki K., Moduli and deformation for Fuchsian projective connections on a Riemann surface, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38 (1991), 431-531.
  18. Komyo A., Hamiltonian structures of isomonodromic deformations on moduli spaces of parabolic connections, arXiv:1611.03601.
  19. Krichever I., Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations, Mosc. Math. J. 2 (2002), 717-752, hep-th/0112096.
  20. Narasimhan M.S., Seshadri C.S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540-567.
  21. Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math. (1994), 47-129.
  22. Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. II, Inst. Hautes Études Sci. Publ. Math. (1994), 5-79.

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