The Katz-Klemm-Vafa conjecture for K3 surfaces

TL;DR

This paper proves the KKV conjecture relating Gromov-Witten invariants of K3 surfaces to modular forms, using a novel approach involving stable pairs and degeneration techniques, applicable in all genera and classes.

Contribution

It provides a complete proof of the KKV conjecture for all genera and classes, linking Gromov-Witten invariants of K3 surfaces to modular forms through new methods.

Findings

Proof of the KKV conjecture in all cases

New Gromov-Witten multiple cover formulas

A new proof of the Yau-Zaslow formula

Abstract

We prove the KKV conjecture expressing Gromov-Witten invariants of K3 surfaces in terms of modular forms. Our results apply in every genus and for every curve class. The proof uses the Gromov-Witten/Pairs correspondence for K3-fibered hypersurfaces of dimension 3 to reduce the KKV conjecture to statements about stable pairs on (thickenings of) K3 surfaces. Using degeneration arguments and new multiple cover results for stable pairs, we reduce the KKV conjecture further to the known primitive cases. Our results yield a new proof of the full Yau-Zaslow formula, establish new Gromov-Witten multiple cover formulas, and express the fiberwise Gromov-Witten partition functions of K3-fibered 3-folds in terms of explicit modular forms.