Topological Recursion and Mirror Symmetry in Chern-Simons Wilson Loops — E8 Intelligence Research

Abstract

FINDING: Wilson loop expectation values in Chern-Simons theory are governed by topological recursion and mirror symmetry, linking gauge theory invariants to algebraic curve moduli. MATH: - Chern-Simons action: \( S_{CS} = \frac{k}{4\pi} \int_M \text{Tr}(A \wedge dA + \frac{2}{3} A \wedge A \wedge A) \) - Wilson loop expectation: \( \langle W_R(K) \rangle = \int [DA] \, e^{iS_{CS}} \, \text{Tr}_R \, \text{Pexp}(\oint_K A) \) - Topological recursion: \( \omega_{g,n} \) on spectral curve \( \Sigma \) (algebraic curve with modular properties) - Mirror symmetry: Gopakumar-Ooguri-Vafa duality maps CS invariants to Gromov-Witten invariants of resolved conifold, with \( t = \frac{2\pi i}{k+N} \) ('t Hooft coupling). - Seifert loop invariants: \( \langle W_R(\text{Seifert}) \rangle \) related to quantum \( \mathfrak{sl}_N \) invariants at \( q = e^{2\pi i/(k+N)} \). CONNECTION: - Modular forms appear via spectral curve \( \Sigma \) (e.g., \( y^2 = x^3 + ax + b \) with modular para Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com