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.