Noether-Lefschetz theory and the Yau-Zaslow conjecture

Abstract

The Yau-Zaslow conjecture predicts the genus 0 curve counts of K 3 K3 surfaces in terms of the Dedekind η \eta function. The classical intersection theory of curves in the moduli of K 3 K3 surfaces with Noether-Lefschetz divisors is related to 3-fold Gromov-Witten invariants via the K 3 K3 curve counts. Results by Borcherds and Kudla-Millson determine these 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 K 3 K3 surfaces), we prove the Yau-Zaslow conjecture for all curve classes on K 3 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.

A modern fareytail

“…This partition function is earlier computed from this perspective in [50]. Recently, the interpretation of η(τ ) −24 as a generating function for D2-branes wrapping cycles in K3 has been put on a firmer mathematical basis [51]. It provides the (reduced) Gromov-Witten invariants of K3.…”

Maps, Sheaves and K3 Surfaces

“…Theorem 1 is proven in [32] by studying Theorem 2 applied to the STU model. There are four basic steps:…”

“…(ii) Theorem 2 is used to show the 3-fold BPS counts n Equation (13) was conjectured in [23] and proven by D. Zagier -the proof is presented in Section 4 of [32]. The strategy of the proof of the Yau-Zaslow conjectures is special to genus 0.…”

Counting Real Rational Curves onK3 Surfaces

“…We refer the interested reader to [20] for references to antecedent proofs of the conjecture for various intermediate cases. Let us here mention only the paper by Bryan and Leung [5], where they also, as Beauville, treat only the case of primitive effective divisors, but use a different method (via twistor families and their Gromov-Witten invariants) and extend the result to counting curves of arbitrary genus (with an appropriate number of nodes).…”