# Fractonic Chern-Simons and BF theories

## TL;DR

This paper develops effective field theories for 3D fracton order inspired by Chern-Simons and BF theories, capturing key features like restricted mobility, topological degeneracy, and fractional statistics.

## Contribution

It introduces a novel framework for fractonic gauge theories using higher-rank gauge fields, extending topological field theory concepts to 3D fracton phases.

## Key findings

- Reproduces features of 2D topological theories in 3D fracton context
- Realizes a gapped fracton phase with line-restricted mobility quasiparticles
- Shows ground state degeneracy depends on topology and geometry

## Abstract

Fracton order is an intriguing new type of order which shares many common features with topological order, such as topology-dependent ground state degeneracies, and excitations with mutual statistics. However, it also has several distinctive geometrical aspects, such as excitations with restricted mobility, which naturally lead to effective descriptions in terms of higher rank gauge fields. In this paper, we investigate possible effective field theories for 3D fracton order, by presenting a general philosophy whereby topological-like actions for such higher-rank gauge fields can be constructed. Our approach draws inspiration from Chern-Simons and BF theories in 2+1 dimensions, and imposes constraints binding higher-rank gauge charge to higher-rank gauge flux. We show that the resulting fractonic Chern-Simons and BF theories reproduce many of the interesting features of their familiar 2D cousins. We analyze one example of the resulting fractonic Chern-Simons theory in detail, and show that upon quantization it realizes a gapped fracton order with quasiparticle excitations that are mobile only along a sub-set of 1-dimensional lines, and display a form of fractional self-statistics. The ground state degeneracy of this theory is both topology- and geometry- dependent, scaling exponentially with the linear system size when the model is placed on a 3-dimensional torus. By studying the resulting quantum theory on the lattice, we show that it describes a $\mathbb{Z}_s$ generalization of the Chamon code.