Classification of connected \'etale algebras in pre-modular fusion categories up to rank three

TL;DR

This paper classifies connected étale algebras in pre-modular fusion categories of rank up to three, providing insights into their structure, Lagrangian algebras, and implications for ground state degeneracy and symmetry breaking in physics.

Contribution

It offers a complete classification of connected étale algebras in low-rank pre-modular fusion categories, including degenerate and non-(pseudo-)unitary cases, with applications to physics.

Findings

Classification of algebras up to rank three

Analysis of Lagrangian algebras and their properties

Implications for ground state degeneracy and symmetry breaking

Abstract

We classify connected \'etale algebras $A$'s in pre-modular fusion categories $\mathcal B$ with $\text{rank}(\mathcal B)\le3$ including degenerate and non-(pseudo-)unitary ones. We comment on Lagrangian algebras and physical applications to ground state degeneracy and proof of spontaneous $\mathcal B$-symmetry breaking.