Complex hidden symmetries in real spacetime and their algebraic structures

TL;DR

The paper investigates how real spacetime, viewed as a Lorentzian fiber inside a complex manifold with symmetry $P_{\\mathbb{C}}$, may inherit hidden algebraic structures akin to the Standard Model, despite a mismatch with the real Poincaré group $P_{\\mathbb{R}}$ that prevents elementary half-integer spins in the complex setting. By analyzing spin$^{h}$ structures on associated bundles and focusing on coset manifolds $SU(3)/SO(3)$ and $SO(3)/SO(2)$, the work reveals spinor degeneracies and additional quantum numbers arising from the larger symmetry, notably through an auxiliary $SU(2)$ bundle and emergent $SU(3)$ flavor-like organization for massive states, with a parallel flavor-type enrichment in the massless sector via $U(2)/SO(2)$. The results connect topological obstructions (Wu manifold) to concrete representation-theoretic constructions, and provide explicit algebraic frameworks (Appendices A and B) that support a dynamical interpretation where hidden complex symmetries could generate the observed particle-physics-like structure. Overall, the work suggests a geometric origin for extra quantum numbers and degeneracies, offering a route to relate higher spacetime symmetries to phenomenological features reminiscent of the Standard Model, while outlining several avenues for further exploration in higher coset spaces and related division-algebra extensions.

Abstract

Considering real spacetime as a Lorentzian fiber in a complex manifold, there is a mismatch of the elementary linear representations of their symmetry groups, the real and complex Poincaré groups. No spinors are allowed as linear irreducible representations for the complex case, but when a spin$^{h}$ structure is implemented on the associated principal bundles, one is naturally led to an algebraic structure similar to the one of the standard model. This last (dynamical) structure might therefore be inherited from the kinematical symmetries of a larger space.