Towards Logarithmic GLSM: The r-spin case

TL;DR

This paper develops a logarithmic framework for the moduli stacks of the gauged linear sigma model, exemplified by the r-spin case, enabling the application of Gromov-Witten theory tools to compute invariants.

Contribution

It establishes a logarithmic foundation for the GLSM moduli stacks and constructs a proper moduli stack with a reduced perfect obstruction theory for the r-spin case.

Findings

Constructed a proper moduli stack with a reduced perfect obstruction theory.

Recovered the r-spin virtual cycle via the new construction.

Extended cosection techniques along logarithmic boundaries.

Abstract

In this article, we establish the logarithmic foundation for compactifying the moduli stacks of the gauged linear sigma model using stable log maps of Abramovich-Chen-Gross-Siebert. 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-Li. Indeed, our construction of the reduced virtual cycle is built upon the work of Chang-Li-Li by appropriately extending and modifying the Kiem-Li cosection along certain logarithmic boundary. In the subsequent 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.