  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.

Symplectic Geometry;Algebraic Geometry