Mirror Symmetry¶
Job: duality relating complex geometry of a Calabi–Yau \(X\) to symplectic geometry of a mirror \(\check X\) — Hodge number swap, enumerative predictions, and categorical formulations (HMS, SYZ).
Core¶
Physically motivated (string theory, \(N=2\) SCFT) and mathematically refined: mirror symmetry equates A-model data (symplectic / Gromov–Witten / Fukaya) of \(X\) with B-model data (complex / variation of Hodge structure / coherent sheaves) of \(\check X\), and vice versa.
| Layer | Content |
|---|---|
| Classical | \(h^{p,q}(X)=h^{n-p,q}(\check X)\) (CY \(n\)-folds); Yukawa / GW predictions |
| HMS | \(D^\pi\operatorname{Fuk}(X)\simeq D^b\operatorname{Coh}(\check X)\) (Kontsevich) |
| SYZ | dual special Lagrangian \(T^n\)-fibrations |
| Examples | toric mirrors, quintic threefold family |
Reading map¶
| Page | Focus |
|---|---|
| Foundations | Hodge swap, HMS pointer, SYZ, toric/quintic one-liners |
| Gromov–Witten & Quantum Cohomology | moduli of stable maps; VFC; WDVV; quantum cohomology; mirror theorem; quintic \(N_d\) |
| Fukaya Category & HMS | \(A_\infty\)-Fukaya; bounding cochains; wrapped Fukaya; HMS proven cases; elliptic curve PZ |
| SYZ & Gross–Siebert | SYZ special Lagrangian fibration; toric degenerations; scattering / cluster; log CY |
| Hitchin & Geometric Langlands | Hitchin fibration; \(G\leftrightarrow\check G\) mirror; BAA/BBB branes; GL via mirror |
Dependency order¶
- Complex / Kähler geometry of Calabi–Yau — complex geometry, Kähler.
- Symplectic / Lagrangian Floer — symplectic geometry, Floer.
- Algebraic geometry of coherent sheaves / derived categories — AG, homological algebra.
- Optional: tropical large-complex-structure limits — tropical geometry.
Interfaces¶
| Area | Contact |
|---|---|
| Complex geometry | Hodge structures, CY metrics, period maps |
| Symplectic geometry | Fukaya category, GW invariants, Lagrangian tori |
| Algebraic geometry | derived categories of coherent sheaves; toric varieties |
| Tropical geometry | dual intersection complexes; combinatorial mirrors |
| Mathematical physics | SCFT, topological strings, D-branes |
| Homological algebra | triangulated / dg / \(A_\infty\) categories |
Notation (paper-standard)¶
- \(X\leftrightarrow\check X\); \(h^{p,q}\); complexified Kähler moduli \(\mathcal{M}_K\) vs complex moduli \(\mathcal{M}_{\mathrm{cx}}\)
- \(\operatorname{Fuk}(X)\), \(D^b\operatorname{Coh}(X)\); SYZ fibration \(f:X\to B\)
See it¶
Quintic threefold. Smooth quintic \(X\subset\mathbb{P}^4\) has \(h^{1,1}=1\), \(h^{2,1}=101\). Mirror \(\check X\) (Greene–Plesser / Batyrev orbifold construction) has \(h^{1,1}=101\), \(h^{2,1}=1\). Period integrals of \(\check X\) predict genus-0 GW invariants of \(X\) (Candelas–de la Ossa–Green–Parkes) — verified mathematically by Givental, Lian–Liu–Yau.
Check¶
Why is mirror symmetry not merely a Hodge diamond reflection for arbitrary complex manifolds?
The duality pairs moduli: complex structure deformations of \(X\) (\(\sim H^{n-1,1}\) for CY) with Kähler/symplectic deformations of \(\check X\) (\(\sim H^{1,1}\)), plus matching quantum-corrected products (GW vs Yukawa). Hodge swap is the numerical shadow of that identification, not the full statement; non-CY mirrors and Fano/Landau–Ginzburg pairs need modified diamonds / categories.
What is the A-side object corresponding to a holomorphic vector bundle on the B-side under HMS?
An object of the Fukaya category: roughly a Lagrangian submanifold with local system / bounding cochain (Floer data). Structure sheaf \(\mathcal{O}_p\) at a point often mirrors a Lagrangian torus fiber with a flat \(U(1)\) connection (SYZ heuristic).
Pitfalls¶
- Mirror is not unique as a variety — families, birational models, and derived-equivalent spaces; categorical mirror is often the robust object.
- Hodge numbers for orbifolds / singular mirrors need stringy Hodge numbers (Batyrev).
- HMS is a theorem in many cases (toric, Seidel, Sheridan quintic, Abouzaid, …) but remains a program in full generality.
Refs¶
- Aspinwall et al., Dirichlet Branes and Mirror Symmetry (Clay)
- Cox–Katz, Mirror Symmetry and Algebraic Geometry
- Hori et al., Mirror Symmetry (Clay)
- Kontsevich, “Homological algebra of mirror symmetry” (ICM 1994)
Next¶
Foundations · Gromov–Witten & Quantum Cohomology · Fukaya Category & HMS · SYZ & Gross–Siebert · Hitchin & Geometric Langlands · Complex geometry · Symplectic geometry · Tropical geometry · Algebraic geometry · Mathematical physics