Homological Mirror Symmetry for the universal centralizers I: The adjoint group case

TL;DR

This paper establishes homological mirror symmetry for the universal centralizer associated with complex semisimple Lie groups, linking Fukaya categories and coherent sheaves in the adjoint group case, as a foundational step for the general case.

Contribution

It proves the homological mirror symmetry conjecture for the universal centralizer in the adjoint group case, connecting symplectic and algebraic geometry in this setting.

Findings

Homological mirror symmetry is established for the universal centralizer in the adjoint case.

The A-side involves a partially wrapped Fukaya category on the Toda space.

The B-side is the category of coherent sheaves on a quotient of a dual maximal torus.

Abstract

We prove homological mirror symmetry for the universal centralizer $J_G$ (a.k.a Toda space), associated to any complex semisimple Lie group $G$. The A-side is a partially wrapped Fukaya category on $J_G$, and the B-side is the category of coherent sheaves on the categorical quotient of a dual maximal torus by the Weyl group action (with some modification if $G$ has a nontrivial center). This is the first and the main part of a two-part series, dealing with $G$ of adjoint type. The general case will be proved in the forthcoming second part [Jin2].