Non-Abelian Fusion, Shrinking and Quantum Dimensions of Abelian Gauge Fluxes

TL;DR

This paper develops a field-theoretical framework to understand fusion, shrinking, and quantum dimensions of Abelian gauge flux loops in 3D topological orders, revealing non-Abelian fusion behaviors despite Abelian fluxes.

Contribution

It introduces a novel field-theoretical approach to analyze 3D loop fusion and shrinking rules, connecting them to topological invariants and quantum dimensions.

Findings

Loops can fuse into non-Abelian types despite carrying Abelian fluxes.

Loop shrinking rules are algebraically consistent with fusion rules.

Fusing loops and anti-loops can produce multiple vacua, unlike 2D anyons.

Abstract

Braiding and fusion rules of topological excitations are indispensable topological invariants in topological quantum computation and topological orders. While excitations in 2D are always particle-like anyons, those in 3D incorporate not only particles but also loops -- spatially nonlocal objects -- making it novel and challenging to study topological invariants in higher dimensions. While 2D fusion rules have been well understood from bulk Chern-Simons field theory and edge conformal field theory, it is yet to be thoroughly explored for 3D fusion rules from higher dimensional bulk topological field theory. Here, we perform a field-theoretical study on (i) how loops that carry Abelian gauge fluxes fuse and (ii) how loops are shrunk into particles in the path integral, which generates fusion rules, loop-shrinking rules, and descendent invariants, e.g., quantum dimensions. We first assign a gauge-invariant Wilson operator to each excitation and determine the number of distinct excitations through equivalence classes of Wilson operators. Then, we adiabatically shift two Wilson operators together to observe how they fuse and are split in the path integral; despite the Abelian nature of the gauge fluxes carried by loops, their fusions may be of non-Abelian nature. Meanwhile, we adiabatically deform world-sheets of unknotted loops into world-lines and examine the shrinking outcomes; we find that the resulting loop-shrinking rules are algebraically consistent to fusion rules. Interestingly, fusing a pair of loop and anti-loop may generate multiple vacua, but fusing a pair of anyon and anti-anyon in 2D has one vacuum only. By establishing a field-theoretical ground for fusion and shrinking in 3D, this work leaves intriguing directions, e.g., symmetry enrichment, quantum gates, and physics of braided monoidal 2-category of 2-group.