Algebraic theory of abelian fractons and beyond
Abstract
We develop an algebraic theory of fusion, mobility, and statistics in abelian gapped fracton orders and topological phases. The framework generalizes the algebraic description of abelian anyons to systems with restricted-mobility and spatially extended excitations. We consider translation-invariant bosonic systems whose pointlike and extended excitations have finite fusion order. The basic objects are braided fusion complexes (BFCs): chain complexes encoding fusion and mobility, equipped with a braiding chain map encoding statistics. Their equivalence classes define braided fusion theories (BFTs). Our framework encompasses arbitrary translation-invariant Pauli codes, as well as all abelian anyon theories in two spatial dimensions, including chiral theories. We give evidence for the conjecture that every nondegenerate BFT can be realized as the excitation theory of a gapped phase and that the BFT describing such a phase is a phase invariant. We derive constraints on pointlike fusion and mobility, showing, for example, that neither the Pauli X nor Z sectors of the X-cube and anisotropic models can occur alone in a nondegenerate BFT. For nondegenerate theories with only pointlike excitations, we extract a simpler theory of mutual statistics, consisting of a fusion module and a map to its statistical dual. This theory of mutual statistics is an invariant of the BFT and, at odd fusion order, gives an equivalent description of the full braided fusion theory. We interpret remote detection as the generalization of mutual statistics to fracton orders. In the BFC setting, we give a precise formulation of the excitation-detector principle: every nontrivial excitation can be detected by an operator supported at spatial infinity, and every nontrivial detector detects some excitation. Assuming a unique infinite-space ground state, we prove that this principle is equivalent to nondegeneracy.
Related
- Finite abelian subgroups of algebraic groups
- A complete description of the algebraic structure of pseudocompact abelian groups
- Higher abelian gauge theory in BV formalism
- A multiplicative universal property for real algebraic K-theory
- Spectra and Support Theory for Wide Subcategories in n-Abelian Categories
BibTeX
@misc{shirley2026algebraic,
title = {{Algebraic theory of abelian fractons and beyond}},
author = {Wilbur Shirley and Evan Wickenden and Bowen Yang and Agnès Beaudry and Michael Hermele},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40596},
eprint = {2610.40596},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40596},
}
RIS
TY - UNPB
TI - Algebraic theory of abelian fractons and beyond
AU - Wilbur Shirley
AU - Evan Wickenden
AU - Bowen Yang
AU - Agnès Beaudry
AU - Michael Hermele
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40596
UR - https://arxiv.org/abs/2610.40596
AB - We develop an algebraic theory of fusion, mobility, and statistics in abelian gapped fracton orders and topological phases. The framework generalizes the algebraic description of abelian anyons to systems with restricted-mobility and spatially extended excitations. We consider translation-invariant bosonic systems whose pointlike and extended excitations have finite fusion order. The basic objects are braided fusion complexes (BFCs): chain complexes encoding fusion and mobility, equipped with a braiding chain map encoding statistics. Their equivalence classes define braided fusion theories (BFTs). Our framework encompasses arbitrary translation-invariant Pauli codes, as well as all abelian anyon theories in two spatial dimensions, including chiral theories. We give evidence for the conjecture that every nondegenerate BFT can be realized as the excitation theory of a gapped phase and that the BFT describing such a phase is a phase invariant. We derive constraints on pointlike fusion and mobility, showing, for example, that neither the Pauli X nor Z sectors of the X-cube and anisotropic models can occur alone in a nondegenerate BFT. For nondegenerate theories with only pointlike excitations, we extract a simpler theory of mutual statistics, consisting of a fusion module and a map to its statistical dual. This theory of mutual statistics is an invariant of the BFT and, at odd fusion order, gives an equivalent description of the full braided fusion theory. We interpret remote detection as the generalization of mutual statistics to fracton orders. In the BFC setting, we give a precise formulation of the excitation-detector principle: every nontrivial excitation can be detected by an operator supported at spatial infinity, and every nontrivial detector detects some excitation. Assuming a unique infinite-space ground state, we prove that this principle is equivalent to nondegeneracy.
ER -