We introduce the notion of log R-maps generalizing the notion of stable maps with p-fields, 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 Kiem-Li cosection localized virtual cycle, as well as the reduced virtual cycle to the canonical virtual cycle. This sets the foundation of a new technique for computing higher genus Gromov--Witten invariants of compact Calabi--Yau manifolds.

This is a joint work with Felix Janda and Yongbin Ruan.