Spin-Clifford Bundles and Spin Connections on the Hyperbolic Clifford Algebra - Part I

Abstract

We develop the theory of spinor fields associated with the hyperbolic Clifford algebra $\mathbb{R}^{H}=\mathcal{C}\ell(H_V)$, which is associated with hyperbolic space $H_V=V\oplus V^{*}$. Algebraically, we exhibit an explicit real primitive idempotent which, unlike the Lorentzian case, remains primitive after complexification and generates the ideal of Witten superfields. We also describe the chirality decomposition and the hyperbolic spin group. Geometrically, we construct a canonical hyperbolic spin structure on every oriented manifold, showing that no topological obstructions arise beyond orientability and we study the associated spinor bundles, spin-Clifford bundles and ideal spinor subbundles. Finally, we introduce the isotropic extension of an arbitrary linear connection to $TM\oplus T^*M$, which is always compatible with the hyperbolic metric, and the associated hyperbolic spin connection, whose well-definedness, compatibilities, curvature and explicit form are established, providing the geometric framework for Dirac operators on the hyperbolic Clifford algebra, which we will present in a forthcoming article.