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On the quantum cohomology of a symmetric product of an algebraic curve | | Aaron Bertram
; Michael Thaddeus
; | | Date: |
9 Mar 1998 | | Subject: | Algebraic Geometry; Differential Geometry; Symplectic Geometry MSC-class: 14H99 | math.AG math.DG math.SG | | Abstract: | The dth symmetric product of a curve of genus g is a smooth projective variety. This paper is concerned with the little quantum cohomology ring of this variety, that is, the ring having its 3-point Gromov-Witten invariants as structure constants. This is of considerable interest, for example as the base ring of the quantum category in Seiberg-Witten theory. The main results give an explicit, general formula for the quantum product in this ring unless d is in the narrow interval [3/4 g, g-1). Otherwise, they still give a formula modulo third order terms. Explicit generators and relations are also given unless d is in [4/5 g - 3/5, g-1). The virtual class on the space of stable maps plays a significant role. But the central ideas ultimately come from Brill-Noether theory: specifically a formula of Harris-Tu for the Chern numbers of determinantal varieties. The case d = g-1 is especially interesting: it resembles that of a Calabi-Yau 3-fold, and the Aspinwall-Morrison formula enters the calculations. A detailed analogy with Givental’s work is also explained. | | Source: | arXiv, math.AG/9803026 | | Services: | Forum | Review | PDF | Favorites | |
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