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

Published online by Cambridge University Press:  03 May 2021

Pierrick Bousseau
Laboratoire de mathématiques d’Orsay, Université Paris-Saclay, CNRS, 91405, Orsay, France; E-mail:
Honglu Fan
Department of Mathematics, University of Geneva, 24, rue du Général Dufour, Case postale 64, 1211 Geneva 4, Switzerland; E-mail:
Shuai Guo
School of Mathematical Sciences, Peking University, No.5 Yiheyuan Road Haidian District, 100871, Beijing, China; E-mail:
Longting Wu
Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092 Zurich, Switzerland; E-mail:


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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.

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