Abstract
We define a formal Gromov–Witten theory of the quintic threefold via localization on \({\mathbb {P}}^4\). Our main result is a direct geometric proof of holomorphic anomaly equations for the formal quintic in precisely the same form as predicted by B-model physics for the true Gromov–Witten theory of the quintic threefold. The results suggest that the formal quintic and the true quintic theories should be related by transformations which respect the holomorphic anomaly equations. Such a relationship has been recently found by Q. Chen, S. Guo, F. Janda, and Y. Ruan via the geometry of new moduli spaces.
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Notes
For stability, marked points are required in genus 0 and positive degree is required in genus 1.
A second proof (in most cases) can be found in [10].
The negative exponent denotes the dual: \(\mathsf {S}\) is a line bundle and \(\mathsf {S}^{-5}=(\mathsf {S}^\star )^{\otimes 5}\).
Since the formal quintic theory is homogeneous of degree 0, the specialization \(\lambda _i = \zeta ^i\lambda _0\) could also be taken as in [22]. However, for other specializations, we expect the finite generation of Theorem 1 and the holomorphic anomaly equation of Theorem 2 to take different forms. A study of the dependence on specialization will appear in [21]. For local \(\mathbb {P}^2\) considered in [22] and \(\mathbb {C}^3/\mathbb {Z}_3\) considered in [23], the theories are independent of specialization.
Our functions \(K_2\) and \(A_{2k}\) are normalized differently with respect to \(C_0\) and \(C_1\). The dictionary to exactly match the notation of [1, (2.52)] is to multiply our \(K_2\) by \((C_0C_1)^2\) and our \(A_{2k}\) by \((C_0C_1)^{2k}\).
See Remark 27.
Corresponding to a stratum of the moduli space of stable curves \(\overline{M}_{g,n}\).
Self-edges correspond to loops of \({\mathsf {T}}\)-invariant rational curves.
The moduli spaces \(\overline{Q}_{0,0}({\mathbb {P}}^m,d)\) and \(\overline{Q}_{0,1}({\mathbb {P}}^m,d)\) are empty by the definition of a stable quotient.
The associated weights on \(H^0({\mathbb {P}}^4,\mathcal {O}_{{\mathbb {P}}^4}(1))\) are \(\lambda _0,\lambda _1,\lambda _2,\lambda _3,\lambda _4\) and so match the conventions of Sect. 1.2.
See Sections 2 and 5 of [6].
In Gromov–Witten theory, a parallel relation is obtained immediately from the WDDV equation and the string equation. Since the map forgetting a point is not always well-defined for quasimaps, a different argument is needed here [8].
We follow here the notation of [26] for \(B_k\).
We use the variables \(x_1\) and \(x_2\) here instead of x and y.
We follow here the notation of Sect. 2.
Flags are either half-edges or markings.
In case e is self-edge, \(v_1=v_2\).
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Acknowledgements
We thank I. Ciocan-Fontanine, E. Clader, Y. Cooper, B. Kim, A. Klemm, Y.-P. Lee, A. Marian, M. Mariño, D. Maulik, D. Oprea, E. Scheidegger, Y. Toda, and A. Zinger for discussions over the years about the moduli space of stable quotients and the invariants of Calabi–Yau geometries. The work of Q. Chen, F. Janda, S. Guo, and Y. Ruan as presented at the Workshop on higher genus is crucial for the wider study of the formal (and true) quintic. We are very grateful to them for sharing their ideas with us. R. P. was partially supported by SNF-200020182181, ERC-2012-AdG-320368-MCSK, ERC-2017-AdG-786580-MACI, SwissMAP, and the Einstein Stiftung. H. L. was supported by the Grants ERC-2012-AdG-320368-MCSK and ERC-2017-AdG-786580-MACI. This project has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 786580).
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Lho, H., Pandharipande, R. Holomorphic Anomaly Equations for the Formal Quintic. Peking Math J 2, 1–40 (2019). https://doi.org/10.1007/s42543-018-0008-0
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DOI: https://doi.org/10.1007/s42543-018-0008-0