THE COMPLETE E8 × E8 LAGRANGIAN FROM 496 SYMMETRIES TO OUR UNIVERSE — IN ONE ACTION THE MASTER ACTION — 10D STRING FRAME 𝒮 = 1/2κ²₁₀ ∫d¹⁰x √(−g) e⁻²ᶠ [ℛ + 4(∂Φ)² − ½|H₃|² − α′/4(tr|F₁|² + tr|F₂|²)] + [ψ̄ₘΓᴹᴺᴾ𝒟ₙψₚ + λ̄Γᴹ𝒟ₘλ + α′/4 χ̄ᵃΓᴹ𝒟ₘχᵃ] + (Yukawa + 4-fermi) ■ Gravity ■ B-field ■ Visible E₈ ■ Hidden E₈ ■ Fermions 7 fields · 3 lines · entire universe §1 THE 7 FUNDAMENTAL FIELDS 4 bosons + 3 fermions = everything g MN 10D Metric Graviton · 35 DOF → 4D graviton + moduli scalars B MN Kalb-Ramond 2-Form Antisymmetric tensor · 28 DOF → Axion (solves Strong CP) Φ Dilaton String coupling · g_s = e^Φ · 1 DOF → Sets all coupling constants A M a E₈×E₈ Gauge Field 496 generators · 496×8 DOF → All gauge bosons (g, W, Z, γ) ψ M Gravitino M-W spinor-vector · 56 DOF → SUSY partner of graviton · LSP dark matter candidate λ Dilatino M-W spinor · 8 DOF → SUSY partner of dilaton χ a Gauginos Adjoint of E₈×E₈ · 496×8 DOF → quarks, leptons! (upon compactification) §2 BOSONIC SECTOR §3 FERMIONIC SECTOR EINSTEIN-HILBERT + DILATON 𝒮_grav = 1/2κ² ∫d¹⁰x √(−g) e⁻²ᶠ (ℛ + 4∂ₘΦ∂ᴹΦ) ℛ = Ricci scalar (gravity) · Φ = dilaton · e⁻²ᶠ = string tree-level prefactor KALB-RAMOND B-FIELD 𝒮_B = −1/4κ² ∫d¹⁰x √(−g) e⁻²ᶠ |H₃|² H₃ = dB₂ − α′/4(ω₃ʸᴹ − ω₃ᴸ) Modified field strength — mixes gauge + gravity → anomaly cancellation E₈ × E₈ YANG-MILLS 𝒮_YM = −α′/8κ² ∫ √(−g) e⁻²ᶠ (trF⁽¹⁾² + trF⁽²⁾²) F⁽¹⁾ = visible E₈ (→ SM forces) · F⁽²⁾ = hidden E₈ (→ shadow matter) 248 + 248 = 496 gauge bosons · sectors couple only through gravity GRAVITINO + DILATINO + GAUGINO KINETICS ψ̄ₘΓᴹᴺᴾ𝒟ₙψₚ gravitino + λ̄Γᴹ𝒟ₘλ dilatino + α′/4 χ̄ᵃΓᴹ𝒟ₘχᵃ gaugino KEY INTERACTION TERMS ½ ψ̄ₘΓᴹᴺᴾᑫᴿψₙ Hₚᑫᴿ gravitino-torsion α′/4 χ̄ᵃΓᴹΓᴺᴾψₘ Fₙₚ ← Yukawa origin! χ̄ΓψF → upon compactification → all quark/lepton masses SUSY TRANSFORMATIONS · 𝒩 = 1 · 16 SUPERCHARGES BOSONS → FERMIONS: δg_MN = ε̄Γ_(M ψ_N)   δΦ = ½ε̄λ   δA_M = ε̄Γ_M χ FERMIONS → BOSONS: δψ = 𝒟ε − ⅛HΓε   δλ = Γ∂Φε   δχ = −½FΓε §4 ANOMALY CANCELLATION — WHY E₈ × E₈ THE BIANCHI IDENTITY UNIQUELY SELECTS E₈ × E₈ dH₃ = α′/4 (tr ℛ∧ℛ − tr F∧F) Only E₈×E₈ or SO(32) are anomaly-free in 10D STANDARD EMBEDDING A_gauge = ω_spin ⟹ E₈ → E₆ × SU(3) 248 DECOMPOSITION 248 = (78,1) ⊕ (27,3) ⊕ (2̄7,3̄) ⊕ (1,8) (78, 1) E₆ gauge bosons → all forces (27,3)+(2̄7,3̄) 3 gen quarks + leptons (1, 8) Extra-dim curvature n_gen = ½|χ(CY)| → topology fixes 3 gen §5 4D EFFECTIVE THEORY KÄHLER POTENTIAL 𝒦(Φ, Φ̄) scalar kinetic terms + geometry SUPERPOTENTIAL 𝒲(Φ) particle masses + Yukawa couplings GAUGE KINETIC FUNCTION f_ab(Φ) gauge couplings + θ-angles THE SCALAR POTENTIAL — SOURCE OF ALL SYMMETRY BREAKING V = e^(𝒦/M²) (𝒦^iī D_i𝒲 D_ī𝒲̄ − 3/M² |𝒲|²) + ½(Re f)⁻¹ DᵃDᵇ F-terms → GUT breaking: E₆→SO(10)→SU(5)→SM · D-terms → gauge invariance · −3|𝒲|² → cosmological constant problem §6 COMPACTIFICATION & ROSETTA STONE ℳ¹⁰ = ℳ⁴ × 𝒳⁶ ℳ⁴ = our spacetime · 𝒳⁶ = Calabi-Yau (Ricci-flat, SU(3) holonomy) TOPOLOGY → PHYSICS |χ|/2 = 3 generations · h¹·¹ = moduli / inflatons · h²·¹ = Yukawa couplings · π₁ = Wilson lines → SM ROSETTA STONE — LAGRANGIAN ↔ SYMMETRY BREAKING POSTER LAGRANGIAN TERM WHAT IT PRODUCES ↔ SYMMETRY BREAKING POSTER tr F⁽¹⁾² visible Yang-Mills 78 gauge bosons → all SM forces E6 GUT → SO(10)→SM → STANDARD MODEL (27,3)⊕(2̄7,3̄) from 248 3 families of quarks + leptons 3 GENERATIONS UNOBS. MATTER V(𝒦,𝒲,f) scalar potential drives every symmetry breaking step SINGLET (S) VEV MECHANISM → every ↓ arrow χ̄ΓψF Yukawa coupling all particle masses upon compactification QUARK / LEPTON TABLE (poster bottom) tr F⁽²⁾² hidden Yang-Mills mirror E₈ → dark sector forces HIDDEN SECTOR SHADOW MATTER δε (SUSY) supercharges boson ↔ fermion pairing · LSP stability SUSY R-PARITY HIGGS DOUBLETS (1,8) + g_MN curvature + metric extra-dim geometry + 4D gravity 170 LEFTOVERS 66 EXOTIC FORCES Visualized by Prof. Andy Haas, NYU Physics, 2026