Stability conditions in the mathematical Gauged Linear Sigma Model

The theory of Mixed-Spin-P (MSP) fields was introduced by Chang-Li-Li-Liu for the quintic threefold, aiming at studying its higher-genus Gromov-Witten invariants. Chang-Guo-Li has successfully applied it to prove conjectures including the BCOV Feynman rule, Yamaguchi-Yau's polynomiality conjecture and the Holomorphic Anomaly Equation. Meanwhile, Fan-Jarvis-Ruan introduced a mathematical theory of Gauged Linear Sigma Model (GLSM), associating a counting theory to a GIT quotient with a super-potential, under suitable assumptions. This paper provides a common generalization of both works, by introducing new stability conditions in the mathematical GLSM. We show that our stability condition guarantees the separatedness and properness of the cosection degeneracy locus in the moduli. It generalizes the MSP fields construction to more general GIT quotients, including Calabi-Yau global complete intersections in toric varieties. This hopefully provides a geometric platform to effectively compute their higher-genus Gromov-Witten invariants.