Mirror Symmetry: Foundations¶
Job: classical Hodge-number duality, Kontsevich’s homological mirror symmetry (HMS) as a categorical statement, the SYZ torus-fibration heuristic, and the flagship examples (toric mirrors, quintic).
Core¶
Calabi–Yau package (minimal)¶
A compact Kähler manifold \(X\) of complex dimension \(n\) is Calabi–Yau in the sense used here if \(K_X\cong\mathcal{O}_X\) (trivial canonical bundle) and usually \(H^{0,q}(X)=0\) for \(0<q<n\) (strict CY). Yau’s theorem supplies a Ricci-flat Kähler metric in each Kähler class (Kähler geometry).
Moduli (rough).
| Sector | Tangent space (CY) | Geometry |
|---|---|---|
| Complex structures | \(H^1(X,T_X)\cong H^{n-1,1}\) | period map / VHS |
| Complexified Kähler | \(H^{1,1}(X,\mathbb{C})\) (formal) | symplectic / A-model |
Mirror symmetry identifies the complex moduli of \(X\) with the Kähler moduli of \(\check X\) (and vice versa), including flat structures and quantum corrections.
Hodge numbers swap¶
Classical numerical mirror symmetry. For mirror Calabi–Yau \(n\)-folds \(X,\check X\), $$ h^{p,q}(X)=h^{n-p,q}(\check X). $$ In particular for threefolds: \(h^{1,1}(X)=h^{2,1}(\check X)\), \(h^{2,1}(X)=h^{1,1}(\check X)\). Hodge diamond reflected through the diagonal / mid-plane.
Enumerative upgrade. Genus-0 Gromov–Witten invariants of \(X\) are extracted from period integrals and Yukawa couplings on the complex-structure moduli of \(\check X\) (B-model). Higher genus: topological recursion / BCOV, still partly conjectural in places.
Homological mirror symmetry (Kontsevich)¶
Conjecture (HMS). For mirror pairs (in a suitable sense), there is an equivalence of triangulated categories $$ D^\pi\operatorname{Fuk}(X)\simeq D^b\operatorname{Coh}(\check X) $$ (or a Landau–Ginzburg / matrix factorization variant when mirrors are not compact CY). Here:
- \(D^b\operatorname{Coh}(\check X)\): bounded derived category of coherent sheaves (B-side; algebraic / complex).
- \(D^\pi\operatorname{Fuk}(X)\): split-closed derived Fukaya category — \(A_\infty\)-category of Lagrangians with Floer morphisms, then twisted complexes / idempotent completion (A-side; symplectic).
Philosophy. Complex geometry \(\leftrightarrow\) symplectic geometry at the level of noncommutative “spaces” (categories), not merely of Hodge numbers. Deformations of categories match moduli.
Status (pointer). Proved for many toric Fano / CY hypersurface situations, Seidel’s genus-2 work, Sheridan’s quintic, Abouzaid’s HMS for tori and some SYZ setups, etc. Formulations vary (wrapped Fukaya, partially wrapped, relative HMS).
SYZ conjecture¶
Strominger–Yau–Zaslow: near large complex structure limits, a CY \(X\) admits a fibration \(f:X\to B\) by special Lagrangian tori \(T^n\) (singular fibers allowed), and the mirror \(\check X\) is the dual torus fibration (dualizing nonsingular \(T^n\)-fibers; correcting singularities via instanton / wall-crossing data).
| Ingredient | Role |
|---|---|
| Special Lagrangian | $\omega |
| Affine base \(B\) | integral affine structure from action coordinates |
| Dual fibers | \(\operatorname{Hom}(\pi_1(F),U(1))\) / dual \(T^n\) |
| Quantum corrections | holomorphic disks bound Lagrangians; scatter diagrams / walls |
SYZ links mirror symmetry to tropical geometry: the base \(B\) with discriminant locus is a tropical skeleton of the degeneration.
Examples (one-liners)¶
Toric / Batyrev. Reflexive polytope \(\Delta\subset M_{\mathbb{R}}\) defines Fano toric \(P_\Delta\); anticanonical hypersurfaces are CY. Mirror from dual polytope \(\Delta^\circ\) (Batyrev). Combinatorial Hodge numbers from face data; HMS largely accessible via toric Fukaya / coherent-constructible correspondence (Bondal, Fang–Liu–Treumann–Zaslow, …).
Quintic threefold. \(X=\{F_5=0\}\subset\mathbb{P}^4\): \(h^{1,1}=1\), \(h^{2,1}=101\). Greene–Plesser mirror: quotient of Fermat quintic by \((\mathbb{Z}/5)^3\) with crepant resolution \(\check X\), \(h^{1,1}=101\), \(h^{2,1}=1\). Periods of the Picard–Fuchs equation predict GW invariants of \(X\) (CdOGP curve counts).
Elliptic curves. Mirror of \(E_\tau\) is a symplectic torus with complexified area \(\rho\); HMS is due to Polishchuk–Zaslow / later refinements — theta functions \(\leftrightarrow\) Floer products of circles.
Fano / LG. Mirror of Fano \(X\) is often a Landau–Ginzburg model \((Y,W)\) with matrix factorizations on the B-side; Hodge diamond statements become Hodge–Tate / vanishing cycle packages.
See it¶
Yukawa coupling vs quantum cohomology. On the B-side, the cubic Yukawa \(Y_{ijk}=\int_{\check X}\Omega\wedge\partial_i\partial_j\partial_k\Omega\) (periods) defines a product on \(H^{n-1,1}\). Under mirror symmetry it matches the genus-0 quantum product on \(H^{1,1}(X)\) deformed by GW invariants. The famous CdOGP computation equates a hypergeometric period series on the mirror quintic with the generating function \(\sum N_d q^d\) of rational curve counts on the quintic — a concrete numerical A=B match.
Check¶
Why do special Lagrangians appear in SYZ rather than arbitrary Lagrangians?
Ricci-flat metrics calibrate special Lagrangians (Harvey–Lawson): they are volume-minimizing in their homology class and match the \(B\)-model supersymmetric cycle condition. Dualizing the flat \(T^n\) with respect to the induced affine structure is well-defined; arbitrary Lagrangians lack the metric calibration tying A- and B-moduli.
State the expected HMS partners of (i) structure sheaf \(\mathcal{O}_{\check X}\) and (ii) skyscraper \(\mathcal{O}_p\).
Heuristically: (i) a Lagrangian section (or Clifford-type torus with suitable brane data) of the SYZ fibration; (ii) a fiber torus \(F_b\) equipped with a flat line bundle / \(U(1)\) local system corresponding to \(p\) in the dual fiber. Precise statements depend on the HMS setup (compact vs wrapped).
Pitfalls¶
- Stringy Hodge numbers for orbifold mirrors: use age-shifted Chen–Ruan / Batyrev stringy \(h^{p,q}\), not naive resolution Betti numbers alone.
- Kähler moduli are complexified: \(B+i\omega\in H^2(X,\mathbb{C})/H^2(X,\mathbb{Z})\) — the \(B\)-field is essential for matching complex moduli dimensions.
- Fukaya category technicalities (bounding cochains, Novikov coefficients, perturbations) are load-bearing, not optional, for HMS statements.
- Multiple mirrors related by birational maps / Fourier–Mukai partners can share derived categories — “the” mirror is often a family or a category.
Refs¶
- Kontsevich, ICM 1994 address on HMS
- Cox–Katz, Mirror Symmetry and Algebraic Geometry
- Aspinwall et al., Dirichlet Branes and Mirror Symmetry
- Gross–Siebert, tropical / logarithmic mirror symmetry program papers
- Candelas–de la Ossa–Green–Parkes, Nucl. Phys. B 359 (1991)
Next¶
Mirror index · Gromov–Witten & Quantum Cohomology · Fukaya Category & HMS · SYZ & Gross–Siebert · Hitchin & Geometric Langlands · Complex geometry · Kähler geometry · Symplectic foundations · Floer · Tropical geometry · AG cohomology · Mathematical physics