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The logarithmic gauged linear sigma model

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Abstract

We introduce the notion of log R-maps, and develop a proper moduli stack of stable log R-maps in the case of a hybrid gauged linear sigma model. Two virtual cycles (canonical and reduced) are constructed for these moduli stacks. The main results are two comparison theorems relating the reduced virtual cycle to the cosection localized virtual cycle, as well as the reduced virtual cycle to the canonical virtual cycle. This sets the foundation of a new technique for computing higher genus Gromov–Witten invariants of complete intersections.

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Notes

  1. A different approach to higher genus Gromov–Witten invariants of quintic threefolds has been developed by Chang–Guo–Li–Li–Liu [12,13,14, 18, 19].

  2. This is different from the “twisted superpotential” used in the physics literature [52, (2.27)].

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Acknowledgements

The first author would like to thank Dan Abramovich, Mark Gross and Bernd Siebert for the collaborations on foundations of stable log maps which influenced the development of this project. Last two authors wish to thank Shuai Guo for the collaborations which inspired the current work. The authors would like to thank Adrien Sauvaget, Rachel Webb and Dimitri Zvonkine for discussions related to the current work. The authors thank Huai-Liang Chang, Young-Hoon Kiem, Jun Li and Wei-Ping Li for their inspiring works on cosection localization needed in our construction. Part of this research was carried out during a visit of the Institute for Advanced Studies in Mathematics at Zhejiang University. Three of us would like to thank the Institute for the support. The first author was partially supported by NSF grant DMS-1700682 and DMS-2001089. The second author was partially supported by an AMS Simons Travel Grant and NSF grants DMS-1901748 and DMS-1638352. The last author was partially supported by Institute for Advanced Study in Mathematics of Zhejiang University, NSF grant DMS-1405245 and NSF FRG grant DMS-1159265 .

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Authors and Affiliations

  1. Department of Mathematics, Boston College, Chestnut Hill, MA, 02467, USA

    Qile Chen

  2. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46656, USA

    Felix Janda

  3. Institute for Advanced Study in Mathematics, Zhejiang University, Hangzhou, China

    Yongbin Ruan

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  1. Qile Chen
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Correspondence to Qile Chen.

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Chen, Q., Janda, F. & Ruan, Y. The logarithmic gauged linear sigma model. Invent. math. 225, 1077–1154 (2021). https://doi.org/10.1007/s00222-021-01044-2

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