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  • Appendix C - Line bundles on Riemann surfaces

  • from Part III - Appendix
  • Eric D'Hoker, University of California, Los Angeles, Justin Kaidi, Kyushu University
  • Book:
    Modular Forms and String Theory
    Published online:
    28 November 2024
    Print publication:
    12 December 2024, pp 378-390

  • Summary

    In this appendix, we shall define and study complex line bundles over an arbitrary compact Riemann surface, provide their topological classification in terms divisors, and give the Riemann–Roch theorem. We shall prove various dimension formulas, including for the dimension of the moduli space of complex or conformal structures on a Riemann surface. We then discuss sections of line bundles from a more physics-oriented point of view in terms of spaces of vector fields, differential forms, and spinor fields.