Year: 2026
Type: preprint
Abstract: Ising anyons and $SU(2)_2$ Chern--Simons theory share the fusion rules and $F$-symbols, and their $R$-symbols agree up to braiding orientation and an overall phase. The quantum-group description, $U_q(\mathfrak{sl}_2)$ at $q = e^{iπ/4}$, is nevertheless not semisimple: tensor powers of the fundamental representation contain an additional spin-$3/2$ representation and reducible indecomposable representations, all of zero quantum dimension. We construct explicit highest-weight bases for up to four strands, compute the Racah and braiding matrices, and show that for four strands changes of fusion basis and braiding mix the physical sector with an indecomposable summand. We prove that this mixing affects neither the braid-closure invariant nor the plat-closure amplitude for any number of strands, so that each braiding matrix can be replaced by its physical block. The full representation theory thus reproduces the Ising-anyon data for all observables relevant to topological quantum computation.
Source: arXiv (Cornell University)
Authors: Belov, Artem, Morozov, Andrey
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7161553694