From wall structures to closed mirror symmetry. The case of $K_{{\mathbb{P}}^2}$: Renormalized periods over the positive real locus, closed Gromov-Witten invariants from wall functions, and tropical enumeration

Abstract

We recover closed Gromov-Witten invariants and renormalized mirror periods for $K_{{\mathbb{P}}^2}$ by the same operation on a wall function. This gives a direct passage from the Gross-Siebert construction of intrinsic mirror pairs to classical enumerative mirror symmetry. The link is a polynomiality theorem for punctured invariants. Assuming the expected identification with the normalized slab function, we also obtain a finite tree sum for closed Gromov-Witten invariants in terms of types of plane tropical curves. The mechanism is expected to extend to more general Calabi-Yau mirror pairs.

arXiv comment42 pages, 4 figures