Abstract
The purpose of this paper is to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control to a large extent the orientability of certain moduli spaces in Real gauge theory and Real algebraic geometry.
Similar content being viewed by others
References
Arbarello, E., Cornalba, M., Griffiths, Ph., Harris, J.: Geometry of Algebraic Curves. Vol. 1, Berlin- Heidelberg-New York: Springer, 1985
Atiyah, M.: K-theory. New York: W.A. Benjamin, 1967
Atiyah M.: Riemann surfaces and spin structures. Ann. Sci. l’É.N.S., 4e série, tome 4(1), 47–62 (1971)
Bertram A., Daskalopoulos G., Wentworth R.: Gromov invariantsfor holomorphic maps from Riemann surfaces to Grassmannians. J.Amer. Math. Soc. 9(2), 529–571 (1996)
Birkenhake, Ch., Lange, H.: Complex Abelian Varieties. Grundlehren, Berlin-Heidelberg-New York: Springer, 2004
Bismut J.-M., Gillet H., Soulé C.: Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion. Commun. Math. Phys. 115(1), 49–78 (1988)
Bismut J.-M., Gillet H., Soulé C.: Analytic torsion and holomorphic determinant bundles. II. Direct images and Bott-Chern forms. Commun. Math. Phys. 115(1), 79–126 (1988)
Bismut J.-M., Gillet H., Soulé C.: Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants. Commun. Math. Phys. 115(2), 301–351 (1988)
Costa A., Natanzon S.: Poincaré’s theorem for the modular group of real Riemann surface. Diff. Geom. Appl. 27(5), 680–690 (2009)
Gross B., Harris J.: Real algebraic curves. Ann. sci. ENS., 4e série 14, 157–182 (1981)
Grothendieck A.: Sur quelques points d’algébrehomologique, II. Tohoku Math. J. (2) 9(3), 119–122 (1957)
Johnson D.: Spin structures and quadratic forms on surfaces. J. London Math. Soc. (2) 22(2), 365–373 (1980)
Knudsen F., Mumford D.: The projectivity of the moduli spaceof stable curves I. Preliminaries on “det” and “div”. Math. Scand. 39, 19–55 (1976)
Libgober A.: Theta characteristics on singular curves, spin structures and Rohlin theorem. Ann. Sci. ENS, Sér. 4 21(4), 623–635 (1988)
Lübke, M., Teleman, A.: The Kobayashi-Hitchin correspondence. Singapore: World Scientific Pub Co., 1995
Natanzon, S.: Moduli of Riemann Surfaces, Real Algebraic Curves, and Their Superanalogs. Translations Of Mathematical Monographs, Volume 225, Providence, RI: Amer. Math. Soc., 2004
Okonek Ch., Teleman A.: Master spaces and the coupling principle: from geometric invariant theory to gauge theory. Commun.Math. Phys. 205, 437–58 (1999)
Okonek Ch., Teleman A.: Gauge theoretical equivariant Gromov-Witten invariants and the full Seiberg-Witten invariants ofruled surfaces. Commun. Math. Phys. 227(3), 551–585 (2002)
Quillen D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funk. Anal. iprilozen 19, 37–41 (1985)
Silhol, R.: Real Algebraic Surfaces. Lecture Notes in Math. 1392, Berlin: Springer-Verlag, 1989
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N. A. Nekrasov
The second author has been partially supported by the ANR project MNGNK, decision Nr. ANR-10- BLAN-0118.
Rights and permissions
About this article
Cite this article
Okonek, C., Teleman, A. Abelian Yang-Mills Theory on Real Tori and Theta Divisors of Klein Surfaces. Commun. Math. Phys. 323, 813–858 (2013). https://doi.org/10.1007/s00220-013-1793-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-013-1793-z