Holomorphic anomaly equation for $(\mathbb{P}^2,E)$ and the Nekrasov-Shatashvili limit of local $\mathbb{P}^2$

Abstract

We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with $X$ a smooth projective variety and $D$ a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $\lambda_g$-insertion is related to Gromov-Witten theory of the total space of $\mathcal{O}_X(-D)$ and local Gromov-Witten theory of $D$. Specializing to $(X,D)=(S,E)$ for $S$ a del Pezzo surface or a rational elliptic surface and $E$ a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold $\mathcal{O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve $E$. Specializing further to $S=\mathbb{P}^2$, we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $(\mathbb{P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local $\mathbb{P}^2$ and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of $(\mathbb{P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on $\mathbb{P}^2$, we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local $\mathbb{P}^2$ in the Nekrasov-Shatashvili limit.