---
tag: B5N9WJH5F0
kind: conjecture
name: Homological mirror symmetry
msc: [53D37, 14J33, 18G80]
uses: [4HS60JABC5]
about: [4HS60JABC5]
sources:
  - url: https://en.wikipedia.org/wiki/Homological_mirror_symmetry
same_as:
  wikidata: Q11338004
  wikipedia: Homological mirror symmetry
history:
  posed_by: "Maxim Kontsevich"
  first_stated: "1994"
---

## Statement

Let $X$ be a compact Calabi–Yau manifold, viewed as a symplectic manifold.
Let $\check{X}$ be a Calabi–Yau manifold that is mirror to $X$ in the sense of mirror symmetry.
Then the derived category of coherent sheaves $D^b\mathrm{Coh}(\check{X})$ is equivalent, as a triangulated category, to the split-closed derived [[Fukaya category|4HS60JABC5]] $D^\pi \mathcal{F}(X)$.
The same holds with the roles of $X$ and $\check{X}$ exchanged.

## Remarks

Kontsevich proposed the conjecture in his address to the International Congress of Mathematicians in Zürich in 1994, as a categorical explanation of mirror symmetry.
The conjecture is not a single precise statement, because "mirror" is not defined in general.
It is proved in many cases where a mirror pair is given by an explicit construction.
Polishchuk and Zaslow proved it for elliptic curves, Seidel for the quartic surface, Abouzaid and Smith for the four-dimensional torus, and Sheridan for Calabi–Yau hypersurfaces in projective space.
Versions for Fano manifolds, whose mirrors are Landau–Ginzburg models, replace the derived category on one side by a category of matrix factorizations or a Fukaya–Seidel category.

## Explanations

### L1 human

Mirror symmetry is a duality from string theory that pairs two very different spaces.
On one side one looks at the symplectic geometry of a space, through its Lagrangian submanifolds.
On the other side one looks at the algebraic geometry of the mirror space, through its coherent sheaves.
Homological mirror symmetry predicts that these two worlds have the same algebraic structure, packaged as a category.
For an elliptic curve, this means that closed curves on a torus correspond to vector bundles and sheaves on the mirror elliptic curve.