Geometric interpretation of the Dirac-sector conserved quantity

Identify the geometric meaning of the conserved quantity r^2 e^{\alpha}(|\xi_+|^2 - |\eta_+|^2 - |\xi_-|^2 + |\eta_-|^2) for static spherically symmetric twisted chiral spinor solutions of the coupled Einstein–Yang–Mills–Higgs–Dirac–Yukawa system.

Background

The reduced static spherically symmetric system possesses a conserved quantity involving the radial coordinate r, the metric coefficient \alpha, and the amplitudes \xi_\pm and \eta_\pm of the twisted chiral spinor fields. The paper establishes conservation of this expression directly from the reduced Dirac equations, but does not identify the corresponding geometric object or symmetry. The author notes that an analogous conserved quantity occurs for Riemannian Dirac–Yang–Mills equations in spherical symmetry, suggesting that the unresolved issue concerns the geometric interpretation of a potentially general spinorial or Noether-type invariant.