Holomorphic anomaly equation for and the Nekrasov-Shatashvili limit of local

Abstract

AbstractWe prove a higher genus version of the genus$0$local-relative correspondence of van Garrel-Graber-Ruddat: for$(X,D)$a pair withXa smooth projective variety andDa 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 ofD.Specializing to$(X,D)=(S,E)$forSa del Pezzo surface or a rational elliptic surface andEa 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 curveE.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.