Noether-Lefschetz theory and the Yau-Zaslow conjecture

  • 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.

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Metadaten
Author:Albrecht Klemm, Davesh Maulik, Rahul Pandharipande, Emanuel ScheideggerGND
URN:urn:nbn:de:bvb:384-opus4-9739
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/1124
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2008-27)
Type:Preprint
Language:English
Publishing Institution:Universität Augsburg
Release Date:2008/07/21
GND-Keyword:Algebraische Geometrie; Superstringtheorie; Elliptische Modulform; Gromov-Witten-Invariante; K 3-Fläche; Noether-Lefschetz-Satz
Source:http://arxiv.org/abs/0807.2477
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik