Log BPS numbers of log Calabi-Yau surfaces

Abstract

Let ( S , E ) (S,E) be a log Calabi-Yau surface pair with E E a smooth divisor. We define new conjecturally integer-valued counts of A 1 \mathbb {A}^1 -curves in ( S , E ) (S,E) . These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along E E via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence.