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…