Floer theory of Landau–Ginzburg model

In this report, I will give the definition of the Floer homology of Laudau–Ginzburg model with a superpotential function $W$. This is originated from the study of the moduli problem of Witten equation with Lefschetz boundary condition. By mild assumption of $W$, one can obtain the $C^0$ estimate of the solutions. By perturbing the Kaehler potential associated to the Kaehler class, we can get the transversality of the Lefschetz thimbles of W, which implies the Fredholm property of the Witten map. As in Lagrangian intersection homology theory of Cauchy–Riemann operator, the Maslov index is used to give a grading of the Floer LG homology. The orientation is simple here and then it is routine to get the defintion of this homology theory. This Floer LG homology and the corresponding Category theory may have important application in recent work of open string theory of LG model. This is a joint work with Wenfeng Jiang.