Duke Mathematical Journal

On the quantum cohomology of a symmetric product of an algebraic curve

Aaron Bertram and Michael Thaddeus

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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 J. Harris and L. Tu for the Chern numbers of determinantal varieties. The case of 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 A. Givental's work is also explained.

Article information

Source
Duke Math. J., Volume 108, Number 2 (2001), 329-362.

Dates
First available in Project Euclid: 5 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1091737159

Digital Object Identifier
doi:10.1215/S0012-7094-01-10825-9

Mathematical Reviews number (MathSciNet)
MR1833394

Zentralblatt MATH identifier
1050.14052

Subjects
Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14H51: Special divisors (gonality, Brill-Noether theory)

Citation

Bertram, Aaron; Thaddeus, Michael. On the quantum cohomology of a symmetric product of an algebraic curve. Duke Math. J. 108 (2001), no. 2, 329--362. doi:10.1215/S0012-7094-01-10825-9. https://projecteuclid.org/euclid.dmj/1091737159


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