Topological Elastic Spacetime: A Geometric Interpretation of Particles and Interactions Based on Fracton-Elasticity Duality

This paper explores a conceptual framework that models the physical vacuum as a (3+1)-dimensional elastic medium with geometric ordering. In this picture, fundamental fermions are identified as topological defects — dislocations and disclinations — in the vacuum lattice, while the four fundamental interactions are interpreted as different geometric response modes of this medium. The mathematical structure of this framework originates from the fracton-elasticity duality, which provides a natural language for the correspondence between elastic strain and higher-rank gauge fields. We explicitly demonstrate that the Burgers vector of dislocation lines can be mapped to conserved quantum numbers such as electric charge, with their discreteness guaranteed by a topological quantization condition. The weak-field limit of gravity is derived from the elastic equilibrium equations of the strain tensor, recovering the linearized Einstein field equations. The main contribution of this paper is to establish concrete correspondences between the mathematical objects of fracton-elasticity duality and particle physics concepts, while offering alternative geometric perspectives on the hierarchy problem and the cosmological constant. Although far from complete, this framework may provide a useful reference point for interdisciplinary research connecting condensed matter physics, high-energy physics, and gravitational theory.