Fractons, dipole symmetries and curved spacetime

We study complex scalar theories with dipole symmetry and uncover a no-go theorem that governs the structure of such theories and which, in particular, reveals that a Gaussian theory with linearly realised dipole symmetry must be Carrollian. The gauging of the dipole symmetry via the Noether procedure gives rise to a scalar gauge field and a spatial symmetric tensor gauge field. We construct a worldline theory of mobile objects that couple gauge invariantly to these gauge fields. We systematically develop the canonical theory of a dynamical symmetric tensor gauge field and arrive at scalar charge gauge theories in both Hamiltonian and Lagrangian formalism. We compute the dispersion relation of the modes of this gauge theory, and we point out an analogy with partially massless gravitons. It is then shown that these fractonic theories couple to Aristotelian geometry, which is a non-Lorentzian geometry characterised by the absence of boost symmetries. We generalise previous results by coupling fracton theories to curved space and time. We demonstrate that complex scalar theories with dipole symmetry can be coupled to general Aristotelian geometries as long as the symmetric tensor gauge field remains a background field. The coupling of the scalar charge gauge theory requires a Lagrange multiplier that restricts the Aristotelian geometries.

Automated summary

We study complex scalar theories with dipole symmetry and reveal a theorem that dictates the structure of such theories, indicating that a Gaussian theory with linearly realized dipole symmetry must be Carrollian. We gauge the dipole symmetry through the Noether procedure, resulting in a scalar gauge field and a spatial symmetric tensor gauge field. We develop a worldline theory of mobile objects that couple gauge invariantly to these gauge fields. Additionally, we construct canonical theories of dynamical symmetric tensor gauge fields and obtain scalar charge gauge theories in both Hamiltonian and Lagrangian formalisms. Through calculations, we discover the dispersion relation of the modes of this gauge theory, revealing an analogy with partially massless gravitons. Furthermore, we demonstrate that these fractonic theories couple to Aristotelian geometry, a non-Lorentzian geometry characterized by the absence of boost symmetries. We extend previous results by coupling fracton theories to curved space and time, showing that complex scalar theories with dipole symmetry can be coupled to general Aristotelian geometries as long as the symmetric tensor gauge field remains a background field.