Israel Journal of Mathematics

, Volume 117, Issue 1, pp 335–352 | Cite as

The quantum euler class and the quantum cohomology of the Grassmannians



The Poincaré duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical “characteristic element;” in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the “quantum Euler class.” We prove that the characteristic element of a Frobenius algebraA is a unit if and only ifA is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmannians, and to the quantum cohomology of hypersurfaces. In addition we show that, in the case of the Grassmannians, the [quantum] Euler class equals, as [quantum] cohomology element and up to sign, the determinant of the Hessian of the [quantum] Landau-Ginzbug potential.


Complete Intersection Classical Cohomology Characteristic Element Dual Basis Euler Class 
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  1. [1]
    L. Abrams,Two-dimensional topological quantum field theories and Frobenius algebras, Journal of Knot Theory and its Ramifications5 (1996), 569–587.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    L. Abrams,Frobenius Algebra Structures in Topological Quantum Field Theory and Quantum Cohomology, Doctoral dissertation, The Johns Hopkins University, May, 1997.Google Scholar
  3. [3]
    F. W. Anderson and K. R. Fuller,Rings and Categories of Modules, second edition, Springer-Verlag, New York, 1992.MATHGoogle Scholar
  4. [4]
    A. Beauville,Quantum cohomology of complete intersections, Preprint alg-geom/9501008, 1995.Google Scholar
  5. [5]
    A. Bertram,Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian, International Journal of Mathematics5 (1994), 811–825.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    R. Bott and L. W. Tu,Differential Forms in Algebraic Topology, Springer-Verlag, New York, 1982.MATHGoogle Scholar
  7. [7]
    B. Crauder and R. Miranda,Quantum cohomology of rational surfaces, inThe Moduli Space of Curves (Texel Island, 1994), Progress in Mathematics 129, Birkhäuser, Boston, 1995, pp. 33–80.Google Scholar
  8. [8]
    C. Curtis and I. Reiner,Representation Theory of Finite Groups and Associative Algebras, Interscience Publishers, New York, 1962.MATHGoogle Scholar
  9. [9]
    B. Dubrovin,Geometry of 2d topological field theories, inIntegrable Systems and Quantum Groups, Lecture Notes in Mathematics1620, Springer-Verlag, Berlin, 1996, pp. 120–348.CrossRefGoogle Scholar
  10. [10]
    E. Getzler, Intersection theory on\(\bar {\mathcal{M}}_{1,4} \) and elliptic Gromov-Witten invariants, Journal of the American Mathematical Society10 (1997), 973–998.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    M. Kontsevich and Yu. Manin,Gromov-Witten classes, quantum cohomology, and enumerative geometry, Communications in Mathematical Physics164 (1994), 525–562.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    E. Kunz,Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser, Boston, 1985.MATHGoogle Scholar
  13. [13]
    D. McDuff and D. Salamon,J-holomorphic Curves and Quantum Cohomology, American Mathematical Society, Providence, 1994.MATHGoogle Scholar
  14. [14]
    J. W. Milnor,Morse Theory, Princeton University Press, Princeton, 1963.MATHGoogle Scholar
  15. [15]
    J. W. Milnor and J. D. Stasheff,Characteristic Classes, Princeton University Press, Princeton, 1974.MATHGoogle Scholar
  16. [16]
    Y. Ruan and G. Tian,A mathematical theory of quantum cohomology, Journal of Differential Geometry42 (1995), 259–367.MATHMathSciNetGoogle Scholar
  17. [17]
    S. Sawin,Direct sum decomposition and indecomposable tqft’s, Journal of Mathematical Physics36 (1995), 6673–6680.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    G. Scheja and U. Storch,Uber Spurfunktionen bei vollstandigen Durchschnitten, Journal für die Reine und Angewandte Mathematik278/279 (1975), 174–189.MathSciNetGoogle Scholar
  19. [19]
    B. Siebert and G. Tian,On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Preprint alg-geom/9403010, 1994.Google Scholar
  20. [20]
    G. Tian and G. Xu,On the semisimplicity of the quantum cohomology algebras of complete intersections, Preprint alg-geom/9611035, 1996.Google Scholar
  21. [21]
    W. V. Vasoconcelos,Arithmetic of Blowup Algebras, Cambridge University Press, New York, 1994.Google Scholar
  22. [22]
    E. Witten,Supersymmetry and Morse theory, Journal of Differential Geometry17 (1982), 661–692.MATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University 2000

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA

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