Virtual cycles of gauged Witten equation

In this paper, we construct virtual cycles on moduli spaces of solutions to the perturbed gauged Witten equation over a fixed smooth r-spin curve, under the framework of [G. Tian and G. Xu, Analysis of gauged Witten equation, J. reine angew. Math. 740 (2018), 187–274]. Together with the wall-crossing formula proved in the companion paper [G. Tian and G. Xu, A wall-crossing formula for the correlation function of gauged linear σ-model, preprint], this paper completes the construction of the correlation function for the gauged linear σ-model announced in [G. Tian and G. Xu, Correlation functions in gauged linear σ-model, Sci. China Math. 59 (2016), 823–838] as well as the proof of its invariance.