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 Goldstonefield ψ(x) ∈ EIX = E₈ / (E₇ × SU(2)) of the adjoint E₈ group fieldtheory developed in the companion paper [Bednařík, Foundations of anE₈ 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 positivecoefficient κ₂, with Schur factor c_H = 1/4 onad²_{V_A}|_𝔪. Two of the five primitive E₈-invariant quartics of the (4,4) sectorvanish on the slow configuration by abelianness of the four-dimensionalalgebraic substrate; the surviving three reduce, conditional on alarge-N_c single-trace dominance hypothesis, to one Skyrme term withtree-level squared Schur factor c_H' = 1/16. The pseudo-Goldstone potential collapses to a single non-negativeparameter μ⁴ via tree-level Casimir constancy, E₈-equivariance of theleading Coleman–Weinberg fluctuation matrix, and Schur uniqueness. Integrality H⁴(EIX; ℤ) = ℤ, generated by the second Chern class of theSp(1)-factor of the Wolf-space holonomy, supplies a circle-valuedθ-term; the rational vanishing H⁵(EIX; ℚ) = 0 excludes a non-trivialWess–Zumino–Witten five-form. The principal theorem assembles these pieces into a closed effectiveaction 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