Euler and Pontryagin currents of the Dirac operator

TL;DR

The paper develops a framework to express Euler and Pontryagin topological currents on spin manifolds as divergences of real tensorial connections that concurrently act as sources in the Dirac equation. By performing a polar decomposition of spinors in dimensions $4$, $3$, and $2$, it shows that space-time and gauge tensorial connections serve as covariant potentials for curvature and Maxwell fields, enabling the topological currents to be written as divergences of real vectors and to enter the Dirac dynamics. The construction yields dimension-specific realizations of the currents ($G^{\mu}_{4}, K^{\mu}_{4}$ in 4D; $G^{\mu}_{2}, K^{\mu}_{2}$ in 2D; $G^{\mu}_{3}, K^{\mu}_{3}$ in 3D) and underscores their role as dynamical sources for spinor fields, providing a unifying tensorial view of topology–quantum matter interplay with ties to index theory. The results offer a concrete route to connect geometric invariants to spinor dynamics and potential topological effects in quantum matter via the Dirac operator.

Abstract

On differential manifolds with spinor structure, it is possible to express the Euler and Pontryagin currents in terms of tensors that also appear as source in the Dirac equation. It is hence possible to tie concepts rooted in geometry and topology to dynamical characters of quantum matter.