Noether-Lefschetz theory and the Yau-Zaslow conjecture
Abstract
The Yau-Zaslow conjecture determines the reduced genus 0 Gromov-Witten invariants of K3 surfaces in terms of the Dedekind eta function. Classical intersections of curves in the moduli of K3 surfaces with Noether-Lefschetz divisors are related to 3-fold Gromov-Witten theory via the K3 invariants. Results by Borcherds and Kudla-Millson determine the classical intersections in terms of vector-valued modular forms. Proven mirror transformations can often be used to calculate the 3-fold invariants which arise. Via a detailed study of the STU model (determining special curves in the moduli of K3 surfaces), we prove the Yau-Zaslow conjecture for all curve classes on K3 surfaces. Two modular form identities are required. The first, the Klemm-Lerche-Mayr identity relating hypergeometric series to modular forms after mirror transformation, is proven here. The second, the Harvey-Moore identity, is proven by D. Zagier and presented in the paper.
Cite
@article{arxiv.0807.2477,
title = {Noether-Lefschetz theory and the Yau-Zaslow conjecture},
author = {A. Klemm and D. Maulik and R. Pandharipande and E. Scheidegger},
journal= {arXiv preprint arXiv:0807.2477},
year = {2008}
}
Comments
40 pages