SYZ Mirror Symmetry for Toric Calabi-Yau: Knot Invariants via D-Brane Duality — E8 Intelligence Research

FINDING: SYZ mirror symmetry for toric Calabi-Yau varieties establishes a duality between the A-model (symplectic, counting holomorphic disks) and B-model (complex, algebraic geometry) via Lagrangian torus fibrations, with knot invariants emerging from the mirror's D-brane categories. | MATH: SYZ conjecture: mirror pair (X, X^∨) are dual torus fibrations over a common base B, with fibers T^n and T^n∨; for toric CY, base B is a polytope with integer vertices (reflexive polytopes for compact case). Aganagic's homological knot invariants: derived category of coherent sheaves D^b(Coh(X^∨)) on the mirror CY X^∨, with knot complement realized as a Lagrangian submanifold L_K ⊂ X; knot invariant = Hom_{D^b}(O_{L_K}, O_{L_K}) or Ext groups. Key constants: Kähler moduli (complexified volumes) t = ∫_{S^2} ω + iB ∈ H^2(X, ℂ)/2πiH^2(X, ℤ); quantum parameter q = e^{2πit}. | CONNECTION: Toric CY varieties are built from cones over reflexive polytopes — these polytopes encode root systems of affine Li Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com