324 research outputs found

    Orbifold Quantum Riemann-Roch, Lefschetz and Serre

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    Given a vector bundle FF on a smooth Deligne-Mumford stack \X and an invertible multiplicative characteristic class \bc, we define the orbifold Gromov-Witten invariants of \X twisted by FF and \bc. We prove a "quantum Riemann-Roch theorem" which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A Quantum Lefschetz Hyperplane Theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus-0 orbifold Gromov-Witten invariants of \X and that of a complete intersection. This provides a way to verify mirror symmetry predictions for complete intersection orbifolds.Comment: major revision: numerous changes made, mistakes corrected, some new materials adde

    A_{n-1} singularities and nKdV hierarchies

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    According to a conjecture of E. Witten proved by M. Kontsevich, a certain generating function for intersection indices on the Deligne -- Mumford moduli spaces of Riemann surfaces coincides with a certain tau-function of the KdV hierarchy. The generating function is naturally generalized under the name the {\em total descendent potential} in the theory of Gromov -- Witten invariants of symplectic manifolds. The papers arXiv: math.AG/0108100 and arXive: math.DG/0108160 contain two equivalent constructions, motivated by some results in Gromov -- Witten theory, which associate a total descendent potential to any semisimple Frobenius structure. In this paper, we prove that in the case of K.Saito's Frobenius structure on the miniversal deformation of the An1A_{n-1}-singularity, the total descendent potential is a tau-function of the nnKdV hierarchy. We derive this result from a more general construction for solutions of the nnKdV hierarchy from n1n-1 solutions of the KdV hierarchy.Comment: 29 pages, to appear in Moscow Mathematical Journa