Symmetric Tensor Gauge Theories on Curved Spaces

TL;DR

The paper addresses how spatial curvature affects symmetric tensor gauge theories and the mobility of fracton-like excitations. By analyzing four rank-two theories, it demonstrates that generic curvature induces curvature-driven hopping and potential loss of gauge invariance, with an exponential suppression of mobility in the weak-curvature limit, yielding asymptotic fracton behavior. Remarkably, traceless scalar charge theory remains gauge-invariant and preserves sharp mobility on Einstein manifolds, while traceful theories generally fail to do so; traceless vector theory exhibits robustness only on a restricted class of manifolds with constant curvature scalar. These results guide understanding of fracton phases in curved backgrounds and suggest avenues for exploring curved-space generalizations and dynamical consequences, including a summary table of manifold compatibilities.

Abstract

Fractons and other subdimensional particles are an exotic class of emergent quasi-particle excitations with severely restricted mobility. A wide class of models featuring these quasi-particles have a natural description in the language of symmetric tensor gauge theories, which feature conservation laws restricting the motion of particles to lower-dimensional sub-spaces, such as lines or points. In this work, we investigate the fate of symmetric tensor gauge theories in the presence of spatial curvature. We find that weak curvature can induce small (exponentially suppressed) violations on the mobility restrictions of charges, leaving a sense of asymptotic fractonic/sub-dimensional behavior on generic manifolds. Nevertheless, we show that certain symmetric tensor gauge theories maintain sharp mobility restrictions and gauge invariance on certain special curved spaces, such as Einstein manifolds or spaces of constant curvature.

Figures (2)

• Figure 1: The mobility restriction on fractons can be lifted via virtual processes involving the parallel transport of dipoles. First, a fracton moves in a specific direction and emits a dipole oriented in the opposite direction. That dipole can then rotate upon being parallel transported around a closed curve. The fracton can then reabsorb the rotated dipole, resulting in net motion of the fracton.
• Figure 2: On a lattice, the mobility restriction on fractons is violated by a (two-dimensional) virtual dipole propagating around a disclination defect, $i.e.$ a quantized unit of curvature (and torsion), which rotates the dipole and results in net motion of the fracton.