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 λₛk > 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, λₛk, μ⁴) and one discrete topological angle θ ∈ [0, 2π), carrying the sign constraints κ₂ > 0, λₛk > 0, μ⁴ ≥ 0 from stability, the Derrick identity, and potential boundedness. Verification scripts are available at https: //github. com/lukasbednarik/E8-GFT