Worldline Formulations of Covariant Fracton Theories

TL;DR

The paper develops worldline, first-quantized BRST/BV formulations of covariant fracton gauge theories, introducing two core models (tensor and vector) plus a deformed vector model to capture a broad class of theories. By quantizing constraints on augmented phase spaces with bosonic oscillators and constructing nilpotent BRST charges, the authors reproduce the BV spectrum and BRST transformations of spacetime fracton theories, with explicit mappings such as $2\alpha-\beta=0$, $2\alpha+3\beta=0$ (for $D\neq4$), and a deformation that covers nearly all cases. Gauge-fixing is explored from both BV and worldline viewpoints, including BV-BRST gauge-fixing and Siegel gauge, yielding partially on-shell actions and wave-like equations in tensor form and more intricate structures in vector forms. The work demonstrates the viability of worldline methods to encode covariant fracton dynamics and suggests pathways to path-integral formulations, potential boundary analyses, and connections to condensed-matter contexts, while noting limitations related to trace constraints and traceless limits. Overall, the approach provides a consistent bridge between spacetime BV formulations and first-quantized worldline theories for a rich class of fracton gauge theories.

Abstract

We develop worldline formulations of covariant fracton gauge theories. These are a one-parameter family of gauge theories of a rank-two symmetric tensor field, invariant under a scalar gauge transformation involving a double derivative. These theories, which can be interpreted as linearized gravity theories invariant under longitudinal diffeomorphisms, provide a covariant framework for studying Lorentz-breaking fracton quasiparticles, which are excitations with restricted mobility due to dipole-moment conservation. We construct three worldline models. The first two are obtained by deducing their constraint structure directly from the spacetime gauge transformations. By applying BRST quantization, we show that these models reproduce the BV spectrum and the associated BRST transformations of two specific fracton theories. The third model is defined as a deformation of the second one: although free, it is analyzed by drawing inspiration from the standard treatment of interacting worldline systems, and is shown to capture almost the entire family of covariant fracton theories. Finally, we discuss the gauge-fixing, comparing the BV-BRST spacetime perspective with the worldline analogue of the ``Siegel gauge" employed in string field theory.