Fracton and Non-Lorentzian Particle Duality: Gauge Field Couplings and Geometric Implications

Fractons, characterized by restricted mobility and governed by higher-moment conservation laws, represent a novel phase of matter with deep connections to tensor gauge theories and emergent gravity. This work systematically explores the duality between fractons and non-Lorentzian particles-Carroll and Galilean-within electromagnetic (EM) fields. By constructing canonical actions for fractons in rank-2 gauge fields, we derive their equations of motion and demonstrate a new set of dualities between fractons and non-Lorentzian particles in gauge fields. The algebraic underpinnings of these dualities are clarified through symmetry analyses, revealing structural parallels between the fracton and Carroll/Galilean algebras. Furthermore, by gauging the fracton algebra, we develop a framework for coupling fracton gauge fields and background geometry, linking them to non-Lorentzian spacetimes and deriving the corresponding constraint on geometry. These results unify fracton dynamics with non-relativistic and ultra-relativistic limits of physics, offering insights into emergent gravity and exotic condensed matter systems.