Complex hidden symmetries in real spacetime and their algebraic structures

TL;DR

This paper explores the algebraic structures of symmetries in real spacetime viewed as a Lorentzian fiber within a complex manifold, revealing potential links to the standard model through spin$^{h}$ structures.

Contribution

It introduces a novel perspective on spacetime symmetries by analyzing their complex algebraic structures and proposes a connection to the standard model via spin$^{h}$ structures.

Findings

Mismatch between real and complex Poincaré group representations

Spin$^{h}$ structures enable algebraic frameworks similar to the standard model

Potential inheritance of dynamical structures from larger symmetry spaces

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 Poincar\'{e} 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.