Torus localization and wall crossing for cosection localized virtual cycles

The theory of virtual fundamental class defines important invariants such as the Gromov-Witten and the Donaldson-Thomas invariants. It has been generalized to the cosection localized virtual cycle which has applications in Seiberg-Witten, Fan-Jarvis-Ruan-Witten and other invariants. In this paper, we prove the formulas of virtual pullback, torus localization and wall crossing for cosection localized virtual cycles.