Chern-Simons Theory, Knot Invariants, and Mirror Symmetry on Calabi-Yau Threefolds — E8 Intelligence Research

Abstract

FINDING: Chern-Simons theory links knot invariants to topological strings via mirror symmetry and derived categories of coherent sheaves on Calabi-Yau threefolds. MATH: - Chern-Simons action: \( S_{CS} = \frac{k}{4\pi} \int_M \text{Tr}(A \wedge dA + \frac{2}{3} A \wedge A \wedge A) \) - Knot invariants: Jones polynomial, HOMFLY-PT, colored invariants from SU(N) Chern-Simons at level \(k\) - Mirror symmetry: equivalence of A-model (symplectic) and B-model (complex) topological strings on mirror Calabi-Yau threefolds - Derived category of coherent sheaves \( D^b(\text{Coh}(X)) \) as B-model D-brane category - Topological recursion: \( \omega_{g,n} \) from spectral curve, generating knot invariants - Gopakumar-Ooguri-Vafa duality: Chern-Simons on \(S^3\) ↔ topological string on resolved conifold CONNECTION: - No direct ratios (0.382, 0.618, 1.618) appear. - Base-60 not present. - Crystallographic symmetry: root systems of Lie algebras (e.g., \(A_N\) for SU(N)) appear Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com