Effective sigma model of an E₈ group field theory: kinetic uniqueness, the Skyrme sector, and topological terms on the Wolf space EIX

We construct the long-wavelength effective theory of the slow Goldstone field ψ(x) ∈ EIX = E₈ / (E₇ × SU(2)) of the adjoint E₈ group field theory developed in the companion paper [Bednařík, Foundations of an E₈ group field theory, 2026]. Wang–Ziller uniqueness and real irreducibility of the isotropy module (56, 2) fix the two-derivative Lagrangian up to a single positive coefficient κ₂, with Schur factor c_H = 1/4 on ad²_{V_A}|_𝔪. Two of the five primitive E₈-invariant quartics of the (4,4) sector vanish on the slow configuration by abelianness of the four-dimensional algebraic substrate; the surviving three reduce, conditional on a large-N_c single-trace dominance hypothesis, to one Skyrme term with tree-level squared Schur factor c_H' = 1/16. The pseudo-Goldstone potential collapses to a single non-negative parameter μ⁴ via tree-level Casimir constancy, E₈-equivariance of the leading Coleman–Weinberg fluctuation matrix, and Schur uniqueness. Integrality H⁴(EIX; ℤ) = ℤ, generated by the second Chern class of the Sp(1)-factor of the Wolf-space holonomy, supplies a circle-valued θ-term; the rational vanishing H⁵(EIX; ℚ) = 0 excludes a non-trivial Wess–Zumino–Witten five-form. The principal theorem assembles these pieces into a closed effective action with three continuous coefficients (κ₂, λ_sk, μ⁴) and one discrete topological angle θ ∈ [0, 2π), carrying the sign constraints κ₂ > 0, λ_sk > 0, μ⁴ ≥ 0 from stability, the Derrick virial identity, and potential boundedness. Verification scripts: https://github.com/lukasbednarik/E8-GFT