Localized cubic anomalies of the SU(4) irrep 60 and their Wilson-line invariance

For the six-dimensional SU(4) gauge-Higgs unification model of arXiv:2603.05857, with the colour-fundamental block (3, 60) (60 = (0, 2, 1), intrinsic parity η = (+, +), 6D chirality +), exact integer computation over the 40 weights of 60 gives the localized non-Abelian cubic anomalies at an f = 0, 1 fixed point: SU(3)_C³ = (1/4) · c3num = −3/2 and SU(3)_{123}³ = (−1/8) · s3num = −3/4, with the integer numerators certified here. The weight list is an input (from SU(4) character data).

The Higgs is a Wilson line along a U(1) Cartan direction of SU(4) and shifts every abelian charge q₁₅ ↦ q₁₅ + θ. The pure non-Abelian cubics carry no U(1) charge and are exactly θ-invariant, while a mixed C² · U(1) piece depends on θ: the anomaly obstruction decouples from electroweak symmetry breaking. The file uses no Mathlib; every check is decide or rfl.

Main results

• dim_60: the 40 weights carry dimension 60.
• c3_exact, s3_exact: c3num = −6, s3num = 6.
• c3_wilson_invariant, s3_wilson_invariant: both cubic numerators are independent of the Wilson line θ.
• mixed_depends: the mixed C² · U(1)₁₅ numerator is not.

References

• arXiv:2603.05857 (the six-dimensional SU(4) gauge-Higgs unification model).
• C. Marín, Anomaly- and Tadpole-Compatible Fermion Completion of Six-Dimensional SU(4) Gauge-Higgs Unification, Zenodo, doi:10.5281/zenodo.21432625.

@[reducible, inline]

abbrev GaugeHiggsUnification.R60LocalizedAnomaly.St : Type

One weight of the 60: (mult, p0, q15, h3), with p0 = ±1 the ℤ₂ parity, q15 the local U(1)₁₅ charge and h3 the SU(3)_{123} Cartan probe.

Equations

• GaugeHiggsUnification.R60LocalizedAnomaly.St = (Int × Int × Int × Int)

def GaugeHiggsUnification.R60LocalizedAnomaly.S : List St

The 40 weights of 60 = (0, 2, 1).

Equations

• One or more equations did not get rendered due to their size.

def GaugeHiggsUnification.R60LocalizedAnomaly.dim : Int

Total dimension carried by the 40 weights.

Equations

• One or more equations did not get rendered due to their size.

def GaugeHiggsUnification.R60LocalizedAnomaly.c3num (_theta : Int) : Int

SU(3)_C³ numerator Σ mult · p0. The pure colour cubic carries no U(1) charge, so it takes the Wilson line θ and ignores it.

Equations

• One or more equations did not get rendered due to their size.

def GaugeHiggsUnification.R60LocalizedAnomaly.s3num (_theta : Int) : Int

SU(3)_{123}³ numerator Σ mult · p0 · h3³; it carries no U(1) charge either.

Equations

• One or more equations did not get rendered due to their size.

def GaugeHiggsUnification.R60LocalizedAnomaly.mixnum (theta : Int) : Int

Mixed C² · U(1)₁₅ numerator Σ mult · p0 · (q15 + θ) · h3²; the Wilson line enters.

Equations

• One or more equations did not get rendered due to their size.

theorem GaugeHiggsUnification.R60LocalizedAnomaly.dim_60 : dim = 60

The 40 weights carry dimension 60.

theorem GaugeHiggsUnification.R60LocalizedAnomaly.c3_exact : c3num 0 = -6

The SU(3)_C³ numerator is −6, so the anomaly is −3/2.

theorem GaugeHiggsUnification.R60LocalizedAnomaly.s3_exact : s3num 0 = 6

The SU(3)_{123}³ numerator is 6, so the anomaly is −3/4.

theorem GaugeHiggsUnification.R60LocalizedAnomaly.c3_wilson_invariant (θ₁ θ₂ : Int) : c3num θ₁ = c3num θ₂

The SU(3)_C³ anomaly is independent of the Higgs Wilson line.

theorem GaugeHiggsUnification.R60LocalizedAnomaly.s3_wilson_invariant (θ₁ θ₂ : Int) : s3num θ₁ = s3num θ₂

The SU(3)_{123}³ anomaly is independent of the Higgs Wilson line.

theorem GaugeHiggsUnification.R60LocalizedAnomaly.mixed_depends : mixnum 0 ≠ mixnum 1

The mixed C² · U(1) piece is not Wilson-line invariant.