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Towards logarithmic GLSM:
the $r$–spin case
Qile Chen, Felix Janda, Yongbin Ruan and Adrien Sauvaget |
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Geometry & Topology 26 (2022) 2855–2939
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Abstract |
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We establish the logarithmic foundation for compactifying the moduli stacks of the gauged linear sigma model using stable log maps. We then illustrate our method via the key example of Witten’s –spin class to construct a proper moduli stack with a reduced perfect obstruction theory whose virtual cycle recovers the –spin virtual cycle of Chang, Li and Li. Indeed, our construction of the reduced virtual cycle is built upon their work by appropriately extending and modifying the Kiem–Li cosection along certain logarithmic boundary. In a follow-up article, we push the technique to a general situation. One motivation of our construction is to fit the gauged linear sigma model in the broader setting of Gromov–Witten theory so that powerful tools such as virtual localization can be applied. A project along this line is currently in progress, leading to applications including computing loci of holomorphic differentials, and calculating higher-genus Gromov–Witten invariants of quintic threefolds. |
Keywords
$r$–spin, stable logarithmic maps, virtual cycles
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Mathematical Subject Classification 2010
Primary: 14D23, 14N35
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References |
Publication
Received: 25 December 2018
Revised: 7 March 2021
Accepted: 18 August 2021
Published: 23 January 2023
Proposed: Jim Bryan
Seconded: Mark Gross, Marc Levine
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Authors |
| Qile Chen | |
| Department of Mathematics Boston College Chestnut Hill, MA United States |
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| Felix Janda | |
| Department of Mathematics University of Notre Dame Notre Dame, IN United States |
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| Yongbin Ruan | |
| Institute for Advanced Study in
Mathematics Zhejiang University Hangzhou China |
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| Adrien Sauvaget | |
| Institut de Mathématiques de
Jussieu Université Pierre et Marie Curie Paris France |
CNRS, Université de
Cergy–Pontoise Laboratoire de Mathématiques AGM, UMR 8088 Cergy-Pontoise France |
