A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation

TL;DR

This paper explores the mathematical structure of fermionic gapped boundaries in 2+1D topological order, revealing new insights into excitations, fusion rules, and their relation to conformal field theories.

Contribution

It introduces a systematic framework for understanding fermionic condensates at gapped boundaries using super Frobenius algebras and extends the defect Verlinde formula to a twisted version.

Findings

Classified excitations at fermionic gapped boundaries.

Derived fusion rules and endomorphisms including Majorana fermions.

Connected boundary phenomena with super modular invariant CFTs.

Abstract

We study gapped boundaries characterized by "fermionic condensates" in 2+1 d topological order. Mathematically, each of these condensates can be described by a super commutative Frobenius algebra. We systematically obtain the species of excitations at the gapped boundary/ junctions, and study their endomorphisms (ability to trap a Majorana fermion) and fusion rules, and generalized the defect Verlinde formula to a twisted version. We illustrate these results with explicit examples. We also connect these results with topological defects in super modular invariant CFTs. To render our discussion self-contained, we provide a pedagogical review of relevant mathematical results, so that physicists without prior experience in tensor category should be able to pick them up and apply them readily