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 of the companion paper [doi:10.5281/zenodo.19843240]. Wang–Ziller uniqueness and real irreducibility of the isotropy representation fix the two-derivative kinetic term up to a single positive coefficient κ₂. At four-derivative order, the five primitive E₈-invariant quartic couplings reduce, conditional on a large-Nc single-trace dominance hypothesis, to a single Skyrme term with coefficient λ_sk > 0 required by the Derrick virial identity. The pseudo-Goldstone potential collapses to a single non-negative mass parameter μ⁴ by E₈-equivariance of the Coleman–Weinberg effective potential. Integrality H⁴(EIX; ℤ) = ℤ, generated by the second Chern class of the holonomy Sp(1)-bundle, supplies a circle-valued θ-term; the rational vanishing H⁵(EIX; ℚ) = 0 excludes a Wess–Zumino–Witten term. 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 identity, and potential boundedness. Verification scripts are available at https://github.com/lukasbednarik/E8-GFT