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Published April 28, 2026 | Version v1
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Foundations of an E₈ group field theory: action uniqueness, vacuum selection, and a four-dimensional algebraic substrate

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Description

We formulate a local quantum field theory for an adjoint-valued scalar field
Φ : E₈ → 𝔢₈ on the compact real form of the exceptional Lie group, from four
standard meta-principles of local Lagrangian QFT and four 𝔢₈-specific
postulates, and trace its consequences in four steps.
 
(i) Separate E₈^L × E₈^R × E₈^Ad invariance, the Casimir-degree spectrum of
𝔢₈, Ostrogradski exclusion, and stability collapse the IR action to two
leading and nine sub-leading Wilson coefficients with a bounded remainder.
 
(ii) On the open half-line c₂ < 0 the action admits a Bose–Einstein-type
symmetry-breaking vacuum on the round 247-sphere of 𝔢₈, stratified by Levi
sub-root-systems.
 
(iii) Two algebraic filters (cubic-anomaly safety; existence of an SU(2)
substrate for a Skyrme-type soliton) together with one structural-geometric
input (compact quaternion-Kähler structure) select the unique adjoint orbit
EIX = E₈ / (E₇ × SU(2)).
 
(iv) Suter's rank–antichain identity yields a four-dimensional abelian
sector 𝔞 ⊂ 𝔪_EIX with dim_ℝ 𝔞 = 4 and [P_μ, P_ν] = 0, and the lower
homotopy of EIX vanishes, ruling out Kibble-type defects from the phase
transition.
 
The promotion to a smooth Lorentzian four-manifold is recorded as a
hypothesis: dimension and abelian closure follow from (iv); Lorentzian
signature and the global Poincaré subgroup are closed at leading bosonic
Gaussian (and at a 𝒟_stab-interior point) by Osterwalder–Schrader
reconstruction; metric reconstruction is closed at leading + BV-BRST
sub-leading Sakharov order via a Camporesi–Higuchi spectral-zeta
computation on EIX, with structural coefficient
𝒱_ind^(EIX) = 432/3 = 144 to within a ≤ 3.7% finite-part bound.
 
We record explicitly the structural selections (compact real form;
single-copy adjoint-valued field; truncation (n,k) ≤ (4,4); BEC branch;
Cas₂ vacuum truncation; Wolf-space input F3) and the residual open
problems. The construction is parallel to the non-compact E₈ programs of
Lisi (2007), Manogue–Dray–Wilson (2022), Wilson (2025), and Furey (2018),
and is not addressed by the Distler–Garibaldi (2010) no-go theorem.
Standard-Model embedding, particle content, and the Wilsonian calibration
of Newton's constant are deferred.
 
Verification scripts: https://github.com/lukasbednarik/E8-GFT

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Software

Repository URL
https://github.com/lukasbednarik/E8-GFT
Programming language
Python
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Active

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