= Lagrangian subgroup of Abelian anyons
For a <bosonic> Abelian <anyon> theory, a <Lagrangian subgroup of Abelian anyons> is a subgroup of types of <anyons> closed under <anyon fusion>, with trivial <topological spin> for every member and trivial mutual <braiding> among members. Every <anyon> outside the subgroup must braid nontrivially with at least one member; pairwise compatibility alone is insufficient.
These conditions describe a maximal condensable collection for an appropriate fully gapped boundary. In the <toric code>, the vacuum together with the electric <anyon> gives one choice, and the vacuum together with the magnetic <anyon> gives another. Electric and magnetic <anyons> cannot both belong to the same collection because their mutual <braiding> is nontrivial. Additional physical context, including a nonchiral bulk and any imposed <symmetry>, matters when using this data to construct a boundary.<Protected edge modes without symmetry by Michael Levin>[Introduction, conclusion and Appendix E: mutual-statistics maximality and the extra bosonic-spin condition]{ref}
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