Enumeration of holomorphic cylinders in log Calabi–Yau surfaces, II : Positivity, integrality and the gluing formula

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