Topologically nontrivial solution in Einstein-Dirac gravity on the Hopf bundle

TL;DR

This paper presents a novel topologically nontrivial solution in Einstein-Dirac gravity with a cosmological constant, where the spatial section is modeled as a Hopf bundle, linking topology to spinor current density.

Contribution

It introduces a new solution connecting topology and spinor fields in Einstein-Dirac gravity, including both gravitating and nongravitating cases with explicit quantum number characterization.

Findings

Hopf invariant relates to spinor current density.

Diagonal energy-momentum tensor achieved with two Dirac spinors.

Nongravitating Dirac solutions characterized by quantum numbers m, n.

Abstract

The topologically nontrivial solution in Einstein-Dirac gravity with cosmological constant is obtained. The spacetime has the Hopf bundle as a spatial section. It is shown that the Hopf invariant is related to the spinor current density. Two Dirac spinors are used for obtaining a diagonal energy-momentum tensor. The solutions for the nongravitating Dirac equation on the background of Lorentzian spacetime with the Hopf bundle as a spatial section are also obtained. Nongravitating solutions of the Dirac equation are defined by two quantum half-integer numbers $m, n$.