Effective sigma model of an E₈ group field theory on the Wolf space E₈/(E₇ × SU(2))

We construct the long-wavelength effective theory of the slow Goldstone field ψ(x) ∈ EIX = E₈/(E₇ × SU(2)) of an adjoint E₈ group field theory, on the emergent four-dimensional base of its condensate. Wang–Ziller uniqueness and the real irreducibility of the isotropy representation (56, 2) fix the two-derivative kinetic term up to a single positive coefficient κ₂, with a trace-arithmetic Schur factor c_H^(EIX) = 1/4. At four-derivative order the five primitive E₈-invariant quartic couplings reduce to three, by two structural vanishings that follow from the abelian closure of the emergent momenta. Conditional on a large-N_c single-trace dominance hypothesis, these collapse further to a single Skyrme term with tree-level Schur factor c_H'^(EIX) = 1/16. Its coefficient λ_sk is required positive by the Derrick virial identity, conditional on the existence of a nontrivial static soliton. The pseudo-Goldstone potential is orbit-flat at tree level and at one loop, by E₈-equivariance of the Coleman–Weinberg potential; any explicit-breaking mechanism that lifts it is fixed in form, by Schur uniqueness on the isotropy representation, to a single non-negative parameter μ⁴, whose value is not determined here. The integral cohomology H⁴(EIX; ℤ) = ℤ, generated by the second Chern class of the holonomy Sp(1)-bundle, supplies a circle-valued θ-term, while 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 five continuous coefficients (κ₂, κ₄', κ₄'', λ_sk, μ⁴) and one discrete topological angle θ ∈ [0, 2π): the algebraic structure of the Wolf space reduces the eleven group-field-theory Wilson coefficients to four continuous parameters and this angle, after a field rescaling, and the optional single-trace hypothesis sharpens the quartic sector to a single Skyrme coupling, giving two. Every result other than this final collapse, including the kinetic uniqueness, the Schur factors, the topological terms, and the operative sign λ_sk > 0, is independent of the hypothesis.