Clifford Residues and Charge Quantization

We derive the quantization of action, particle number, and electric charge in a Lagrangian spin bundle over M ≡ M#\∪DJ , Penrose’s conformal compactification of Minkowsky space, with the world tubes of massive particles removed. Our Lagrangian density, Lg, is the spinor factorization of the MaurerCartan 4-form Ω; it’s action, Sg, measures the covering number of the 4 internal u (1) × su (2) phases over external spacetime M. Under PTC symmetry, Lg reduces to the second Chern form TrKL ∧ KR for a left ⊕ right chirality spin bundle. We prove a residue theorem for gl (2,C)valued forms, which says that, when we “sew in” singular loci DJ over which the u (1)× su (2) phases of the matter fields have some extra twists compared to the 8 vacuum modes, the additional contributions to the action, electric charge, lepton and baryon numbers are all topologically quantized. Because left and right chirality 2-forms are chiral dual, forms are quantized over their dual cycles. Thus it is the interaction c2 (E), with a globally nontrivial magnetic field, that forces electric fields to be topologically quantized over spatial 2 cycles, ∫ S2 Kore θ ∧ e = 4πN .