Fermion Representations from the Metric Bundle: Clifford Algebras and the Pati-Salam Spinor

Paper 2 of 5 in the Metric Bundle Programme.

In a companion paper, we showed that the DeWitt supermetric on the space of Lorentzian metrics over four-dimensional spacetime has signature (6,4), yielding the Pati-Salam gauge group \SU(4) × \SU(2)_L × \SU(2)_R as the maximal compact subgroup of \SO(6,4). In this paper, we derive the fermion content of the theory from the Clifford algebra of the positive-norm sector of the normal bundle. The Clifford algebra \Cl(\R^6) \otimes \C ≅ M_8(\C) acts on an eight-dimensional spinor \C^8, which decomposes under the chirality operator as \C^8 = S^+ ⊕ S^- = \mathbf{4} ⊕ \bar{\mathbf{4}} of \SU(4) ≅ \Spin(6). We prove that the \SU(3) subalgebra defined as the centraliser of a complex structure J in \mathfrak{so}(6) coincides exactly with the \SU(3) from the Pati-Salam breaking \SU(4) → \SU(3) × \U(1)_{B-L}, with the \U(1) charge reproducing the baryon-minus-lepton number in the ratio +1/3 : -1 (triplet : singlet). Combined with the (\mathbf{2},\mathbf{1}) ⊕ (\mathbf{1},\mathbf{2}) spinor of \Spin(4) ≅ \SU(2)_L × \SU(2)_R, the full representation is the \mathbf{16} of \Spin(10) ⊃ \Spin(6) × \Spin(4): one complete generation of Standard Model fermions including a right-handed neutrino, with all hypercharges correctly reproduced by Y = (B{-}L)/2 + T_{3R}.

Part of a five-paper series deriving the Pati-Salam gauge group, fermion content, gauge dynamics, anomaly cancellation, and the three-generation structure from the geometry of the metric bundle Y¹⁴ = Met(X⁴).