Equivariant Homological Mirror Symmetry for $\mathbb{C}$ and $\mathbb{C}{P}^1$

Abstract

In this paper we define an equivariant Floer $A_{\infty }$ algebra for $\mathbb{C}$ and $\mathbb{C}{P}^1$ by using Cartan model. We then prove an equivariant homological mirror symmetry, i.e. an equivalence between an $A_{\infty }$ category of equivariant Lagrangian branes and the category of matrix factorizations of Givental's equivariant Landau-Ginzburg potential function.