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Log BPS numbers of log Calabi-Yau surfaces. (English) Zbl 07288869
Summary: Let $$(S,E)$$ be a log Calabi-Yau surface pair with $$E$$ a smooth divisor. We define new conjecturally integer-valued counts of $$\mathbb{A}^1$$-curves in $$(S,E)$$. These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along $$E$$ 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.
##### MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14J33 Mirror symmetry (algebro-geometric aspects)
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