Abstract
In this paper, we study the higher genus equivariant Gromov—Witten invariants of \({K_{{\mathbb{P}^1} \times {\mathbb{P}^1}}}\) via Givental formalism. In particular, we prove the generating function satisfies finite generation property and holomorphic anomaly equation.
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Coates T., Corti A., Iritani H., Tseng H.-H., Computing genus-zero twisted Gromov-Witten invariants. Duke Math. J., 2009, 147(3): 377–438
Coates T., Givental A., Quantum Riemann—Roch, Lefschetz and Serre. Ann. of Math., 2007, 165(1): 15–53
Coates T., Iritani H., Gromov—Witten invariants of local ℙ2 and modular forms. Kyoto J. Math., 2021, 61(3): 543–706
Etingof P., Mathematical ideas and notions of quantum field theory. Available at http://www-math.mit.edu/etingof/lect.ps, 2002
Fang B., Ruan Y., Zhang Y., Zhou J., Open Gromov—Witten theory of \({K_{{\mathbb{P}^2}}}\), \({K_{{\mathbb{P}^1} \times {\mathbb{P}^1}}}\), KWℙ[1,1,2], \({K_{{\mathbb{F}_1}}}\) and Jacobi forms. Comm. Math. Phys., 2019, 369(2): 675–719
Garvan F., Cubic Modular Identities of Ramanujan, Hypergeometric Functions and Analogues of the Arithmetic-geometric Mean Iteration, Providence, RI: Amer. Math. Soc., 1994
Givental A.B., Gromov—Witten invariants and quantization of quadratic Hamiltonians. Mosc. Math. J., 2001, 1(4): 551–568
Givental A.B., Semisimple Frobenius structures at higher genus. Int. Math. Res. Not. IMRN, 2001, 2001(23): 1265–1286
Givental A.B., Symplectic geometry of Frobenius structures. In: Frobenius Manifolds, Wiesbaden: Friedr. Vieweg, 2004, 91–112
Guo S., Janda F., Ruan Y., Structure of higher genus Gromov—Witten invariants of quintic 3-folds. 2018, arXiv:1812.11908
Lho H., Gromov—Witten invariants of Calabi-Yau fibrations. 2019, arXiv:1904.10315
Lho H., Gromov—Witten invariants of Calabi—Yau manifolds with two Kähler parameters. Int. Math. Res. Not. IMRN, 2021, 2021(10): 7552–7596
Lho H., Pandharipande R., Stable quotients and the holomorphic anomaly equation. Adv. Math., 2018, 332: 349–402
Li J., Tian G., Virtual moduli cycles and Gromov—Witten invariants of algebraic varieties. J. Amer. Math. Soc., 1998, 11(1): 119–174
Pandharipande R., Pixton A., Zvonkine D., Relations on \({\overline M _{g,n}}\) via 3-spin structures. J. Amer. Math. Soc., 2015, 28(1): 279–309
Teleman C., The structure of 2D semi-simple field theories. Invent. Math., 2012, 188(3): 525–588
Wang X., Quasi-modularity and holomorphic anomaly equation for the twisted Gromov—Witten theory: \({\cal O}\left( 3 \right)\) over ℙ2. Acta Math. Sin. Engl. Ser., 2019, 35(12): 1945–1962
Acknowledgements
The author would like to specially thank professor Shuai Guo and Felix Janda for numerous discussing Givental theory and Calabi—Yau geometry. He would also like to thank the anonymous referee for careful reading of the manuscript and for the many helpful suggestions and comments. This work is supported by National Science Foundation of China (Nos. 11601279, 12071255) and Shandong Provincial Natural Science Foundation (No. ZR2021MA101).
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Wang, X. Finite Generation and Holomorphic Anomaly Equation for Equivariant Gromov—Witten Invariants of \({K_{{\mathbb{P}^1} \times {\mathbb{P}^1}}}\). Front. Math 18, 17–46 (2023). https://doi.org/10.1007/s11464-021-0225-1
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DOI: https://doi.org/10.1007/s11464-021-0225-1