Abstract:  We introduce the notion of log R-maps generalizing 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. This moduli stack carries two virtual fundamental classes -- the canonical virtual cycle and the reduced virtual cycle. The main results are two comparison theorems: 

(1) We identify the reduced virtual cycle with the Kiem-Li cosection localized virtual cycle which was shown to recover Gromov-Witten theory of certain critical locus. 

(2) We relate the reduced virtual cycle to the canonical virtual cycle which can have larger symmetry in many interesting examples. 

This is part of a project aiming at the foundation of a new technique for computing higher genus Gromov--Witten invariants of complete intersections.

The talk consists of joint work with Felix Janda, Yongbin Ruan, and Adrien Sauvaget.