  We define the counting of holomorphic cylinders in log Calabi-Yau surfaces.
Although we start with a complex log Calabi-Yau surface, the counting is
achieved by applying methods from non-archimedean geometry. This gives rise to
new geometric invariants. Moreover, we prove that the counting satisfies a
property of symmetry. Explicit calculations are given for a del Pezzo surface
in detail, which verify the conjectured wall-crossing formula for the
focus-focus singularity. Our holomorphic cylinders are expected to give a
geometric understanding of the combinatorial notion of broken line by Gross,
Hacking, Keel and Siebert. Our tools include Berkovich spaces, tropical
geometry, Gromov-Witten theory and the GAGA theorem for non-archimedean
analytic stacks.

Algebraic Geometry;Symplectic Geometry