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

TL;DR

This work develops a comprehensive, self-contained framework for gapped boundaries and defects in 2+1d topological orders arising from fermion condensation. By promoting bosonic Frobenius algebras to super-commutative Frobenius algebras, the authors systematically construct boundary excitations as modules and junction excitations as bimodules, and derive both the (twisted) defect Verlinde formula and half-linking numbers that govern fusion. The framework is illustrated with explicit analyses of the Toric code and D(S3), and is connected to super modular invariants and spin-CFT defects, including Majorana-mode accounting at junctions. This work provides practical computational tools for physicists and lays groundwork for further explorations of fermionic condensates, spin-structured boundaries, and categorical symmetries in higher dimensions.

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

Figures (37)

• Figure 1: Pentagon equation w.r.t. the fusion space $(V^{abcd}_e)^*$.
• Figure 2: Hexagon equation w.r.t. the fusion space $(V^{abc}_d)^*$.
• Figure 3: A defect wrapping a cycle on the torus, while generating either periodic or anti-periodic boundary conditions for free fermions in the other cycle, determining the spin-structure on the torus.
• Figure 4: Fusion of $ef$ and $fm$ junctions.
• Figure 5: Both the Kitaev model kitaev_fault-tolerant_2003 and the Wen Plaquette model Wen:2003yv realize the toric code topological order. They are illustrated in the same picture here. The black lines denote the lattice of Kitaev's toric code model and the blue lines denote the lattice of the Wen plaquette model Wen:2003yv. Note that in the former, the spin-degrees of freedom lives on the links, whereas in the latter, they live on the vertices. Therefore, where the black lines intersect the blue lattice lives a spin 1/2 degree of freedom. The plaquettes are divided into two sets, the $Z$ and $X$ plaquettes. The Hamiltonian acts in a way depending on this division, as reviewed briefly in (\ref{['eq:Wen_H']}).