Refined floor diagrams relative to a conic and Caporaso-Harris type formula

Abstract

We prove a $q$-refined correspondence theorem between higher genus relative Gromov-Witten invariants with a Lambda class ${\lambda }_{g-g^{\prime }}$ insertion in the blow-up of ${\mathbb{P}}^2$ at $k$ points on a conic and the refined counts of genus $g^{\prime }$ floor diagrams relative to a conic, after the change of variables $q=e^{iu}$. We provide a Caporaso-Harris type recursive formula for the refined counts of higher genus floor diagrams. As an application of the correspondence theorem, we propose a higher genus version of the BPS polynomials of del Pezzo surfaces of degree $\ge 3$ and Hirzebruch surfaces, which generalize the higher genus Block-Göttsche polynomials.

43 pages, 2 figures, references are updated, comments welcome