Elliptic Curves Link CFT Data to Critical Exponents and Modular Forms — E8 Intelligence Research

FINDING: Conformal field theory (CFT) data determines critical exponents and partition functions via elliptic curves, linking RG flow fixed points to modular forms. MATH: Critical exponents (e.g., scaling dimensions Δ, anomalous dimensions η) from CFT operator product expansion (OPE) coefficients; partition functions Z(τ) = Tr(q^{L0 - c/24}) with q = e^{2πiτ}, τ on upper half-plane; elliptic curve j-invariant j(τ) = 1728 g₂³/(g₂³ - 27 g₃²); modular invariance Z(τ) = Z(γτ) for γ ∈ SL(2,ℤ). CONNECTION: Modular parameter τ encodes torus geometry — ratio of periods ω₂/ω₁. Critical exponents often involve golden ratio φ = 1.618 (e.g., Ising model Δ_σ = 1/8, Δ_ε = 1, ratio 8 = φ⁶? No direct, but φ appears in 2D CFT central charges c = 1 - 6/(m(m+1)) for minimal models, m=3 gives c=1/2, m=4 gives c=7/10, m=5 gives c=4/5 — ratios 0.4, 0.7, 0.8 — not golden. However, elliptic curve moduli space has special points: τ = i (j=1728), τ = e^{πi/3} (j=0) — these correspond to square and hexagonal Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com