This is a joint work with Gang Tian. I will talk about the analytical properties of the classical equation of motion in gauged linear $\sigma$-model, which we call the gauged Witten equation. This is a generalization of the Witten equation in Landau-Ginzburg A-model (Fan-Jarvis-Ruan, Witten) and the symplectic vortex equation (Mundet, Cieliebak-Gaio-Salamon). We will also discuss a mathematical definition of the correlation function, using the moduli space of gauged Witten equation, when the curve is fixed.