Homological mirror symmetry for Milnor fibers of simple singularities

Abstract

We prove homological mirror symmetry for Milnor fibers of simple singularities in dimensions greater than one, which are among the log Fano cases of Conjecture 1.5 in arXiv:1806.04345. The proof is based on a relation between matrix factorizations and Calabi--Yau completions. As an application, we give an explicit computation of the Hochschild cohomology group of the derived $n$-preprojective algebra of a Dynkin quiver for any $n\ge 1$, and the symplectic cohomology group of the Milnor fiber of any simple singularity in any dimension greater than one.

23 pages, revised following the suggestions of the referees

References in corpus (4)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Matrix Factorizations and Representations of Quivers I
• Holomorphic curves for Legendrian surgery
• Homological mirror symmetry for invertible polynomials in two variables

Cited by in corpus (4)

• Ginzburg algebras of triangulated surfaces and perverse schobers
• Mirror symmetry for Berglund-Hübsch Milnor fibers
• Exotic families of symplectic manifolds with Milnor fibers of $ADE$-type
• Nonexistence of exact Lagrangian tori in affine conic bundles over ${\mathbb{C}}^n$