Holomorphic anomaly equation for (P^2,E) and the Nekrasov-Shatashvili limit of local P^2

International audienceWe 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 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 O_S(−E) and the stationary Gromov-Witten theory of the elliptic curve E. Specializing further to S= P^2 , we prove that higher genus generating series of maximal contact Gromov-Witten invariants of (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 P^2 and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of (P^2 ,E) and Betti numbers of moduli spaces of semistable one-dimensional sheaves on 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 P^2 in the Nekrasov-Shatashvili limit