Open-closed duality in higher genus and winding

Abstract

Let $X$ be a toric Fano surface. Let $\pi :\hat{X}\to X$ be a toric blow up at a point. We use the Topological Vertex [AKMV] to prove a higher genus, higher winding, open-closed duality between the toric Calabi-Yau 3-folds $K_X,K_{\hat{X}}$. For $g\ge 0,w\ge 1$, we show the equality $n_g(K_{\hat{X}},{\pi }^{\ast }\beta -wC)=(-1{)}^g{N}_{g,(w)}^{LMOV}(K_X/L,\beta )$, where $n_g(K_{\hat{X}},{\pi }^{\ast }\beta -wC)$ is the genus-$g$, Gopakumar-Vafa invariant of $K_{\hat{X}}$ in curve class ${\pi }^{\ast }\beta -wC$, where $\beta \in H_2(X,\mathbb{Z})$ and $C$ is the exceptional curve, and $N_{g,(w)}^{LMOV}(K_X/L,\beta )$ is the genus-$g$, LMOV invariant of an outer Aganagic-Vafa brane $L\subset K_X$ in class $\beta$ and representation $(w)$, or the Young Tableau of a single row with $w$ boxes.

arXiv comment24 pages, comments welcome