An algebraic theory of planon-only fracton orders

In this talk, I will describe an algebraic theory for planon-only abelian fracton orders. These are three-dimensional gapped phases with the property that fractional excitations are abelian particles restricted to move in parallel planes. The fusion and statistics data can be identified with a finitely generated module over a Laurent polynomial ring together with a U(1)-valued quadratic form. These systems thus lend themselves to an elegant algebraic theory which we expect will lead to easily computable phase invariants and a classification. As a starting point, we establish a necessary condition for physical realizability, the excitation-detector principle, which I will explain. We conjecture that this criterion is also sufficient for realizability. I will also discuss preliminary classification results.
This talk is based on joint with Michael Hermele, Wilbur Shirley and Evan Wickenden.