Communications in Mathematical Physics

, Volume 248, Issue 1, pp 1–31 | Cite as

A New Cohomology Theory of Orbifold

  • Weimin Chen
  • Yongbin Ruan
Article

Abstract

Based on the orbifold string theory model in physics, we construct a new cohomology ring for any almost complex orbifold. The key theorem is the associativity of this new ring. Some examples are computed.

Keywords

String Theory Theory Model Cohomology Ring Cohomology Theory String Theory Model 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baily, Jr., W.: The decomposition theorem for V-manifolds. Am. J. Math. 78, 862–888 (1956)MATHGoogle Scholar
  2. 2.
    Borcea, C.: K3-surfaces with involution and mirror pairs of Calabi-Yau manifolds. In: Mirror Symmetry II. Geene, B., Yau, S.-T.(eds)., Providence, RI: Am. Math. Soc. 2001, pp. 717–743Google Scholar
  3. 3.
    Batyrev, V.V., Dais, D.: Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry. Topology 35, 901–929 (1996)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Bott, R., Tu, L. W.: Differential Forms in Algebraic Topology. GTM 82, 1982Google Scholar
  5. 5.
    Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis, A symposium in honour of Bochner, Princeton, N.J.: Princeton University Press, 1970, pp. 195–199Google Scholar
  6. 6.
    Chen, W., Ruan, Y.: Orbifold Gromov-Witten theory. Cont. Math. 310, 25–86 (2002)MATHGoogle Scholar
  7. 7.
    Dixon, L., Harvey, J., Vafa, C., Witten, E.: Strings on orbifolds. Nucl.Phys. B261, 651 (1985)Google Scholar
  8. 8.
    Goresky, M., MacPherson, R.: Intersection homology theory. Topology 19, 135–162 (1980)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Kawasaki, T.: The signature theorem for V-manifolds. Topology 17, 75–83 (1978)CrossRefMATHGoogle Scholar
  10. 10.
    Kawasaki, T.: The Riemann-Roch theorem for complex V-manifolds. Osaka J. Math. 16, 151–159 (1979)MathSciNetMATHGoogle Scholar
  11. 11.
    Li, An-Min., Ruan, Y.: Symplectic surgery and GW-invariants of Calabi-Yau 3-folds. Invent. Math. 145(1), 151–218 (2001)CrossRefGoogle Scholar
  12. 12.
    Reid, M.: McKay correspondence. Seminarire BOurbaki, Vol. 1999/2000. Asterisque No. 176, 53–72 (2002)Google Scholar
  13. 13.
    Roan, S.: Orbifold Euler characteristic. Mirror symmetry, II, AMS/IP Stud. Adv. Math. 1, Providence, RI: Am. Math. Soc., , 1997, pp. 129–140Google Scholar
  14. 14.
    Ruan, Y.: Surgery, quantum cohomology and birational geometry. Am. Math.Soc.Trans (2), 196, 183–198 (1999)Google Scholar
  15. 15.
    Satake, I.: The Gauss-Bonnet theorem for V-manifolds. J. Math. Soc. Japan 9, 464–492 (1957)MATHGoogle Scholar
  16. 16.
    Scott, P.: The geometries of 3-manifolds. Bull. London. Math. Soc. 15, 401–487 (1983)MathSciNetMATHGoogle Scholar
  17. 17.
    Thurston, W.: The Geometry and Topology of Three-Manifolds. Princeton Lecture Notes, 1979Google Scholar
  18. 18.
    Voisin, C.: Miroirs et involutions sur les surfaces K3. In: Journées de géométrie algébrique d’Orsay, juillet 92, édité par A. Beauville, O. Debarre, Y. Laszlo, Astérisque 218, 273–323 (1993)Google Scholar
  19. 19.
    Zaslow, E.: Topological orbifold models and quantum cohomology rings. Commun. Math. Phys. 156(2), 301–331 (1993)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Weimin Chen
    • 1
    • 2
  • Yongbin Ruan
    • 1
  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.Mathematics DepartmentTulane UniversityNew OrleansUSA

Personalised recommendations