We prove three fundamental properties of counting holomorphic cylinders in log
Calabi–Yau surfaces: positivity, integrality and the gluing formula. Positivity and
integrality assert that the numbers of cylinders, defined via virtual techniques, are in
fact nonnegative integers. The gluing formula roughly says that cylinders
can be glued together to form longer cylinders, and the number of longer
cylinders equals the product of the numbers of shorter cylinders. Our approach
uses Berkovich geometry, tropical geometry, deformation theory and the
ideas in the proof of associativity relations of Gromov–Witten invariants by
Maxim Kontsevich. These three properties provide evidence for a conjectural
relation between counting cylinders and the broken lines of Gross, Hacking and
Keel.
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