Loops in 4+1d Topological Phases

Understand

2+1d topological phases are well characterized by the fusion rules and braiding/exchange statistics of fractional point excitations.

• In 4+1d, some topological phases contain only fractional loop excitations.
• What kind of loop statistics exist? We study the 4+1d gauge theory with 2-form $\mathbb{Z}_2$ gauge field (the loop only toric code) and find that while braiding statistics between two different types of loops can be nontrivial, the self `exchange' statistics are all trivial.
• In particular, we show that the electric, magnetic, and dyonic loop excitations in the 4+1d toric code are not distinguished by their self-statistics.