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

TL;DR

This paper establishes an equivariant homological mirror symmetry for the complex plane and projective line by constructing an equivariant Floer algebra and proving an equivalence with matrix factorizations of an equivariant Landau-Ginzburg potential.

Contribution

It introduces an equivariant Floer $A_ abla$ algebra for $C$ and $CP^1$ and proves an equivariant homological mirror symmetry relating Lagrangian branes to matrix factorizations.

Findings

Construction of equivariant Floer $A_ abla$ algebra for $C$ and $CP^1$

Proof of an equivalence between equivariant Lagrangian branes and matrix factorizations

Validation of equivariant homological mirror symmetry in this setting

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.