On the spectrum of the magnetic Dirac operator

TL;DR

This work develops the spectral theory of the magnetic Dirac operator on closed spin manifolds by deriving Friedrich- and Hijazi-type eigenvalue bounds that incorporate the magnetic potential via $D^{\eta}=D+i\eta\cdot$, the geometry through scalar curvature $S$ and the exterior derivative $d\eta$, and by analyzing equality cases that reveal Sasaki and η-Einstein structures. It also investigates diamagnetic behavior, showing that the first eigenvalue need not decrease under magnetic perturbation and providing obstructions in certain geometries, notably the round sphere. The paper then presents explicit spectrum computations for two canonical settings: the round $\mathbb{S}^3$ with Hopf fibration and the flat torus $\mathbb{T}^n$ under spin$^c$ twists, illustrating how topology, spin structure, and gauge data shape spectral data. Finally, it treats the Killing-magnetic-field case via a transversal Dirac framework, deriving refined bounds for basic eigenvalues in terms of the O'Neill tensor and transversal curvature, thereby linking spectrum to the geometry of Riemannian flows.

Abstract

The magnetic Dirac operator describes the relativistic motion of a charged particle in a magnetic field. Although this operator got a lot of attention in physics many of its fundamental mathematical properties remain unexplored and this article is a first step towards filling this gap. To this end we provide a number of eigenvalue estimates for the magnetic Dirac operator on closed Riemannian manifolds and explicitly compute its spectrum for specific choices of the magnetic field on the flat torus and on the three-dimensional round sphere.

Figures (1)

• Figure 1: Eigenvalues of $D^{t\eta}$ on $\mathbb{S}^3$ as functions of $t$: $\frac{3}{2}+k+t$, $\frac{3}{2}+k-t$, $\frac{1}{2}\pm\sqrt{f_0(k,p,t)}$

Theorems & Definitions (32)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof