A hidden symmetry of complex spacetime and the emergence of the standard model algebraic structure

TL;DR

The paper analyzes how real spacetime, viewed as a Lorentzian fiber, sits inside a complex spacetime with symmetry $P_{\mathbb{C}}$, highlighting a mismatch with real fibers and the absence of linear spinor representations for $P_{\mathbb{C}}$. It shows that a Spin$^{h}$ extension on the relevant coset $SU(3)/SO(3)$ is needed, facilitated by an auxiliary $SU(2)$-like bundle to realize spin states. Through a bosonic $SU(3)$ realization and careful matching of transition functions, an exact complex $U(1)\oplus SU(3)$ symmetry emerges, reorganizing massive spinor states into a flavor-like doublet. This leads to a 5-dimensional color-space degeneracy parametrized by the coset $SU(3)/SO(3)$, suggesting a Standard-Model–like algebraic structure that could be gauged and has potential implications for how complex spacetime symmetries relate to particle quantum numbers.

Abstract

When spacetime is considered as a subspace of a wider complex spacetime manifold, there is a mismatch of the elementary linear representations of their symmetry groups, the real and complex Poincaré groups. In particular, no spinors are allowed for the complex case. When a spin$^{h}$ structure is implemented on principal bundles in complex spacetime, one is naturally led to an algebraic structure analogous to the one of the standard model.