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Towards logarithmic GLSM: the $r$–spin case

Qile Chen, Felix Janda, Yongbin Ruan and Adrien Sauvaget

Geometry & Topology 26 (2022) 2855–2939

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 r–spin class to construct a proper moduli stack with a reduced perfect obstruction theory whose virtual cycle recovers the r–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.

$r$–spin, stable logarithmic maps, virtual cycles
Mathematical Subject Classification 2010
Primary: 14D23, 14N35
Received: 25 December 2018
Revised: 7 March 2021
Accepted: 18 August 2021
Published: 23 January 2023
Proposed: Jim Bryan
Seconded: Mark Gross, Marc Levine
Qile Chen
Department of Mathematics
Boston College
Chestnut Hill, MA
United States
Felix Janda
Department of Mathematics
University of Notre Dame
Notre Dame, IN
United States
Yongbin Ruan
Institute for Advanced Study in Mathematics
Zhejiang University
Adrien Sauvaget
Institut de Mathématiques de Jussieu
Université Pierre et Marie Curie
CNRS, Université de Cergy–Pontoise
Laboratoire de Mathématiques AGM, UMR 8088