Mirror Symmetry of the Affine Toda Systems

TL;DR

This paper establishes a homological mirror symmetry between the affine Toda system's wrapped Fukaya category and coherent sheaves on the dual group's centralizer, linking geometric Langlands duality.

Contribution

It proves a new homological mirror symmetry for affine Toda systems related to complex reductive groups and their Langlands duals, advancing geometric Langlands theory.

Findings

Homological mirror symmetry between Fukaya category and coherent sheaves.

Interpretation as geometric Langlands equivalence with wild ramification.

Connection between affine Toda systems and Langlands dual groups.

Abstract

For a complex reductive group $G$, we prove a homological mirror symmetry between the wrapped Fukaya category of the affine Toda system for $G$ and coherent sheaves on the regular centralizer group scheme for the Langlands dual group $G^\vee$. This can be interpreted as a geometric Langlands equivalence for $\mathbb{P}^1$ with mildest wild ramification at $0$ and $\infty$.