Gross-Siebert intrinsic mirror ring for smooth log Calabi-Yau pairs

In this paper, we exhibit a formula relating punctured Gromov-Witten invariants used by Gross and Siebert to 2-point relative/logarithmic Gromov-Witten invariants with one point-constraint for any smooth log Calabi-Yau pair $(W,D)$. Denote by $N_{a,b}$ the number of rational curves in $W$ meeting $D$ in two points, one with contact order $a$ and one with contact order $b$ with a point constraint. (Such numbers are defined within relative or logarithmic Gromov-Witten theory). We then apply a modified version of deformation to the normal cone technique and the degeneration formula developed by Kim, Lho, Ruddat and Abramovich, Chen, Gross, Siebert to give a full understanding of $N_{e-1,1}$ with $D$ nef where $e$ is the intersection number of $D$ and a chosen curve class. Later, by means of punctured invariants as auxiliary invariants, we prove, for the projective plane with an elliptic curve $(\mathbb{P}^2, D)$, that all standard 2-pointed, degree $d$, relative invariants with a point condition, for each $d$, can be determined by exactly one of these degree $d$ invariants, namely $N_{3d-1,1}$, plus those lower degree invariants. In the last section, we give full calculations of 2-pointed, degree 2, one-point-constrained relative Gromov-Witten invariants for $(\mathbb{P}^2, D)$.