---
created: 2022-04-05T23:42
updated: 2023-12-28T18:12
aliases:
  - SYZ
  - SYZ mirror symmetry
  - mirror symmetry
  - SYZ mirror
  - SYZ torus fibration
  - SYZ fibration
---
# SYZ mirror symmetry

Conjecturally, the general fiber $\mcx_t$ of a punctured family of CYs $\mcx^\circ \to \Delta^\circ$ can be given the structure of a special Lagrangian torus fibration $\mcx_t \to B$, one can "dualize" the fibration to obtain a mirror CY $\hat{\mcx_t} \to B$ over the same base. The common base $B$ of these two fibrations is conjecturally of the form $\Gamma(\mcx_0)$, the dual complex of a degeneration $\mcx\to \Delta$ extending $\mcx^\circ$.
# SYZ torus fibration

Existence proved in the non-archimedean case in 2019, still an open problem in the archimedean setting.

![](attachments/2023-01-19-syz-conj.png)
![](attachments/2023-01-19-dbfuk.png)
![](attachments/2023-01-19-degen.png)
![](attachments/2023-01-19-dual-vb.png)
![](attachments/2023-01-19troptree.png)

# Geometric Langlands

![](attachments/2023-03-12geom.png)