Dipole symmetries from the topology of the phase space and the constraints on the low-energy spectrum

TL;DR

This paper reveals the existence of dipole conservation laws in bosonic field theories, linking symplectic geometry to constraints on low-energy spectra and illustrating the concepts with axion electrodynamics.

Contribution

It establishes a general framework connecting symplectic forms to dipole symmetries and explores implications for low-energy excitations and momentum definitions.

Findings

Dipole conservation laws arise from the symplectic form.

Presence of non-exact symplectic forms can eliminate well-defined momentum.

Systems with these properties exhibit additional light modes.

Abstract

We demonstrate the general existence of a local dipole conservation law in bosonic field theory. The scalar charge density arises from the symplectic form of the system, whereas the tensor current descends from its stress tensor. The algebra of spatial translations becomes centrally extended in presence of field configurations with a finite nonzero charge. Furthermore, when the symplectic form is closed but not exact, the system may, surprisingly, lack a well-defined momentum density. This leads to a theorem for the presence of additional light modes in the system whenever the short-distance physics is governed by a translationally invariant local field theory. We also illustrate this mechanism for axion electrodynamics as an example of a system with Nambu--Goldstone modes of higher-form symmetries.