We present a self-contained computation showing that the Pati–Salam gaugegroup SU(4) × SU(2)L ×SU(2)R and the fermion content of one chiral generationof the Standard Model (including a right-handed neutrino) arise from the geometryof the bundle of pointwise Lorentzian metrics over a four-dimensional spacetimemanifold. The key step is the use of the trace-reversal involution, which modifies thenatural fiber metric from signature (7,3) to signature (6,4). The structure groupon the fibers is then SO(6,4), whose maximal compact subgroup is SO(6)×SO(4),and whose spin cover is SU(4) × SU(2) × SU(2). The Weyl spinor representationof Spin(6,4) is 16-dimensional and decomposes under this subgroup as (4,2,1) ⊕(¯ 4, 1, 2), which is precisely one chiral generation of Standard Model fermions in thePati–Salam scheme. No fields beyond the spacetime metric are introduced. Westate what this result does and does not establish regarding physical unification.
Lawrence McCurrach (Sat,) studied this question.