---
created: 2022-02-23T18:45
updated: 2024-01-01T23:13
title: Mirror Symmetry
aliases:
  - Mirror Symmetry
---


---

- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Fukaya category](Unsorted/Fukaya%20category.md)
	- [derived category of coherent sheaves](derived%20category%20of%20coherent%20sheaves)
	- [Hodge theory MOC](Unsorted/Hodge%20theory%20MOC.md)
	- [Calabi-Yau](Unsorted/Calabi-Yau.md), [kahler](Unsorted/Kahler.md),
	- [Gromov-Witten invariants](Gromov-Witten%20invariants)
	- [symplectic geometry](MOCs/symplectic%20geometry.md)
	- [SYZ](Unsorted/SYZ%20torus%20fibration.md)
	- [triangulated category](triangulated%20category)
	- [Gorenstein](Gorenstein)
	- [del Pezzo](del%20Pezzo)
	- [Log geometry](Unsorted/Log%20geometry.md)
	- [exceptional collection](exceptional%20collection)
	- [McKay correspondence](McKay%20correspondence.md)
	- [Fano](Unsorted/Fano%20variety.md)
	- [Landau Ginzberg](Unsorted/Landau%20Ginzberg.md)
	- [vanishing cycles](Unsorted/vanishing%20cycles.md)
	- [del Pezzo](Unsorted/del%20Pezzo.md)
	- [exceptional collection](exceptional%20collection)
	- [[invertible polynomial]]
	- [dual complex](dual%20complex.md)
	- [[holomorphic symplectic variety]]

---



# Mirror Symmetry

Gromov witten = closed string part
Fukaya categories = open string part.

- $\PP^1 \mapstofrom (\cstar, W = x_1 + x_2\inv)$.
- $\PP^2\mapstofrom ((\cstar)\cartpower{2}, W = x_1 + x_2 + x_1\inv x_2\inv)$.

![](2023-06-07-1.png)

![](2023-06-15.png)

![](2023-06-07-4.png)

- **A Side: Symplectic**: $\D\Fuk(\hat{X})$ for $\hat{X}$ symplectic, 
	- Boundary conditions: Lagrangian submanifolds with brane structures
	- Numbers: Counts of rational curves
- **B Side: Complex**: $\dbcoh(X)$ for $X$ smooth projective.
	- Boundary conditions: holomorphic subvarieties equipped with holomorphic vector bundles.
	- Numbers: Taylor coefficients of periods of Hodge structures on $\hat{X}$, regarded as functionals on the moduli of complex structures on $\hat{X}$.

Main approaches:

- [SYZ](SYZ%20torus%20fibration.md)
	- Related to [Hitchin systems](Hitchin%20fibration) and [Langlands duality](root%20system.md).
- [[Gross-Siebert]]
- [[Kontsevich-Soibelman]].
- [HMS](homological%20mirror%20symmetry.md)
- [Gromov-Witten](Gromov-Witten%20invariants.md)
- [Specific predictions for quintic threefolds](https://arxiv.org/pdf/alg-geom/9411018.pdf#page=4&zoom=160,-136,725)

![](2023-04-02-12.png)

Idea: degenerate $X$ a CY to $\mcx_0$ an SNC CY with [[coregularity]] 0 ([maximally unipotent](unipotent.md) degeneration).

![](attachments/2023-03-31-71.png)

![](attachments/2023-03-31-72.png)

![](attachments/2023-03-31-73.png)

 Slogan: the symplectic geometry of a Calabi-Yau should have "the same" enmerative invariants as those in the complex-analytic geometry of its mirror.
The homological mirror symmetry conjecture predicts a correspondence between the derived category of coherent sheaves of a variety and the symplectic data (packaged in the Fukaya category) of its mirror object.

![](attachments/2023-03-31-74.png)


# Lagrangian submanifolds

For $X$ a [Calabi-Yau](Unsorted/Calabi-Yau.md)
- A-side: the Fukaya category of $X$ corresponds to $A\dash$branes on $X$, so roughly [Lagrangian](Lagrangian) submanifolds equipped with a [flat bundle](Unsorted/flat%20bundle.md).
- B-side: [DCoh](DCoh) of $X$ corresponds to $B\dash$branes on $X$.
  
![](attachments/2023-03-31-75.png)

Other field theories replace $X$ with a [Fano](Unsorted/Fano%20variety.md).

# For Fanos

![](attachments/2023-03-31-76.png)
![](attachments/2023-03-31-77.png)

Importance of [toric](Unsorted/toric.md) structure:
![](attachments/2023-03-31-78.png)


# Examples

Showing an congruence of [Weil zeta functions](Weil%20zeta%20function) for a [K3](K3%20surfaces.md)
![](attachments/2023-03-31-79.png)