Geodesic
Holomorphic anomaly equation for $({\mathbb P}^2,E)$ and the Nekrasov-Shatashvili limit of local ${\mathbb P}^2$
Forum of Mathematics, Pi, Tome 9 (2021)

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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. 
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     author = {Pierrick Bousseau and Honglu Fan and Shuai Guo and Longting Wu},
     title = {Holomorphic anomaly equation for $({\mathbb P}^2,E)$ and the {Nekrasov-Shatashvili} limit of local ${\mathbb P}^2$},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {9},
     year = {2021},
     doi = {10.1017/fmp.2021.3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2021.3/}
}
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Pierrick Bousseau; Honglu Fan; Shuai Guo; Longting Wu. Holomorphic anomaly equation for $({\mathbb P}^2,E)$ and the Nekrasov-Shatashvili limit of local ${\mathbb P}^2$. Forum of Mathematics, Pi, Tome 9 (2021). doi: 10.1017/fmp.2021.3

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