---
tag: YK3DMN2NSK
kind: concept
name: Mirror symmetry
msc: [14J33, 53D37, 81T30]
sources:
  - url: https://en.wikipedia.org/wiki/Mirror_symmetry_(string_theory)
same_as:
  wikidata: Q5914418
  wikipedia: Mirror symmetry (string theory)
---

## Summary

Mirror symmetry is a correspondence, first predicted by string theory, between pairs of [[Calabi–Yau manifolds|STYAB4PFDW]] $X$ and $\check{X}$.
The symplectic geometry of $X$, encoded by its [[Gromov–Witten invariants|XVJA0JD79P]], matches the complex geometry of $\check{X}$, encoded by the variation of its [[Hodge structures|ZSM083M97C]].
For Calabi–Yau threefolds the Hodge numbers are exchanged: $h^{1,1}(X) = h^{2,1}(\check{X})$ and $h^{2,1}(X) = h^{1,1}(\check{X})$.
It is not a single theorem but a program with several precise forms, such as [[homological mirror symmetry|B5N9WJH5F0]].

## Explanations

### L1 human

Two spaces that look unrelated can carry the same hidden information, read in two different languages.
On one side one counts curves inside a space, which is a hard enumerative problem.
On the mirror side one computes certain integrals of differential forms, called periods, which is much easier.
In 1991 Candelas, de la Ossa, Green and Parkes used this to predict the number of rational curves of each degree on a quintic threefold in $\mathbb{P}^4$.
The predicted numbers include $2875$ lines and $609250$ conics.
These predictions were later proved by mathematicians, and mirror symmetry became a major field of geometry.