Monopoles for Gravitation and for Higher Spin Fields

TL;DR

This work generalizes electric-magnetic duality to massless bosonic higher-spin fields, introducing Dirac-string constructions for magnetic sources and deriving a universal quantization condition that ties electric and magnetic charges to asymptotic charges via $\frac{1}{2\pi\hbar} Q_{\gamma_1 \cdots \gamma_{s-1}}(v) P^{\gamma_1 \cdots \gamma_{s-1}}(u) \in \mathbb{Z}$. For spin 2, the condition can be written as $\frac{4GP_\gamma Q^\gamma}{\hbar} \in \mathbb{Z}$ and reduces to expressions involving the electric energy and magnetic mass; the authors illustrate this with explicit linearized solutions (electric mass giving Schwarzschild and magnetic mass yielding Taub-NUT). Beyond the linear theory, they argue that duality-inspired constraints persist in full gravity, using the Taub-NUT solution to extract angular-momentum-based quantization and to show asymptotic flatness in the Regge-Teitelboim sense. The results illuminate how duality, conserved asymptotic charges, and magnetic poles might coexist in a gravitational setting, though a complete non-linear, interacting magnetic sector remains open. Overall, the paper provides a coherent framework for magnetic sources in higher-spin gauge theories and highlights the Taub-NUT geometry as a key non-linear example supporting duality and quantization ideas.

Abstract

We consider massless higher spin gauge theories with both electric and magnetic sources, with a special emphasis on the spin two case. We write the equations of motion at the linear level (with conserved external sources) and introduce Dirac strings so as to derive the equations from a variational principle. We then derive a quantization condition that generalizes the familiar Dirac quantization condition, and which involves the conserved charges associated with the asymptotic symmetries for higher spins. Next we discuss briefly how the result extends to the non linear theory. This is done in the context of gravitation, where the Taub-NUT solution provides the exact solution of the field equations with both types of sources. We rederive, in analogy with electromagnetism, the quantization condition from the quantization of the angular momentum. We also observe that the Taub-NUT metric is asymptotically flat at spatial infinity in the sense of Regge and Teitelboim (including their parity conditions). It follows, in particular, that one can consistently consider in the variational principle configurations with different electric and magnetic masses.