The logarithmic gauged linear sigma model

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 for a new technique for computing higher genus Gromov-Witten invariants of compact Calabi-Yau manifolds. Contents 1. Introduction 1 2. Logarithmic R-maps 7 3. A tale of two virtual cycles 16 4. Examples 28 5. Properties of the stack of stable logarithmic R-maps 30 6. Reducing perfect obstruction theories along boundary 42 References 47

Virtual cycles on projective completions and quantum Lefschetz formula

“…We prove Theorems 1 and 2 using the projective completion N :" PpN ' O M q. In [CJR,Theorem 1.11=Theorem 3.21], Chen-Janda-Ruan proved a similar comparison result to Theorem 2 using a different compactification of N . Recall the p-field space N is the moduli space of stable objects of pC, L, u, pq, where…”

Punctured logarithmic maps

“…The third motivation is from work of the second author on the logarithmic gauged linear sigma model. In the up-coming paper [CJR2], punctured maps provide the key tool to compute the invariants of the logarithmic gauged linear sigma model of [CJR1]. This leads to a powerful geometric method for computing higher genus invariants of complete intersections.…”

Virtual cycles of stable (quasi)-maps with fields

“…Beginning with the work of Givental [28] and Lian-Liu-Yau [41], there has been significant progress in establishing these conjectures in genus 0, and Zinger (in collaboration with Li, Vakil and Zagier) [52] and Kim-Lho [38] have proved mirror theorems in genus 1. By contrast, progress on the higher-genus problem is limited and also comparatively recent: three separate approaches are due to Chang-Guo-Li-Li-Liu in [15,16,12,11,10], to Oberdieck-Pixton in [43], and to Guo, Ruan, and the first and second author in [19,18,29,30]. A key ingredient in both the first and last approach is the moduli of stable maps with p-fields, and this moduli space is what we study in the current paper.…”

“…The main application of this paper is to the logarithmic gauged linear sigma model (log GLSM) of [20,19], which provides a well-behaved compactification of the moduli of stable maps with p-fields, and via a localization formula [18] a way to compute invariants. In fact, the log GLSM has been used in [29,30] to investigate the Gromov-Witten invariants of quintic threefolds.…”