Witten's Chern-Simons Path Integral Derives Jones Polynomial via SU(2) Wilson Loops — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Witten's Chern-Simons formulation derives the Jones polynomial as a Wilson loop expectation value in SU(2) gauge theory at level k, with the framing anomaly fixed by the ribbon structure of the knot. MATH: - Partition function: Z(M) = ∫ DA exp( (ik/4π) ∫_M Tr(A ∧ dA + (2/3)A³) ) - Wilson loop: W_R(K) = Tr_R P exp(∮_K A) - Jones polynomial: V_K(q) = ⟨W_R(K)⟩ with q = exp(2πi/(k+2)) - Level-shift: k → k + h∨ = k + 2 for SU(2) (dual Coxeter number h∨ = 2) - Framing anomaly: each self-linking contributes phase exp(2πi h∨ / (k+h∨)) = exp(2πi·c/24·...); conformal weight h_R = C₂(R)/(k+h∨) - For SU(2) fundamental: h = 3/(4(k+2)), so framing phase = exp(2πi·3/(4(k+2))) - Skein relation: q^{1/2} V(L₊) − q^{−1/2} V(L₋) = (q^{1/2} − q^{−1/2}) V(L₀) CONNECTION: - SU(2) root system = A₁, rank 1, dual Coxeter number h∨ = 2, Weyl group Z₂ - Level k+2 quantization: the shift by h∨ = 2 is the dual Coxeter number — a root-lattice invariant - Conformal weight h_R = C₂(R)/(k+h∨) ties directly t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225328351