Abstract
String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group and a 4-cocycle twist of 's cohomology group in three-dimensional space and one-dimensional time . We establish the topological spin and the spin-statistics relation for the closed strings and their multistring braiding statistics. The twisted gauge theory can be characterized by a representation of a modular transformation group, . We express the generators and in terms of the gauge group and the 4-cocycle . As we compactify one of the spatial directions into a compact circle with a gauge flux inserted, we can use the generators and of an subgroup to study the dimensional reduction of the 3D topological order to a direct sum of degenerate states of 2D topological orders in different flux sectors: . The 2D topological orders are described by 2D gauge theories of the group twisted by the 3-cocycle , dimensionally reduced from the 4-cocycle . We show that the generators, and , fully encode a particular type of three-string braiding statistics with a pattern that is the connected sum of two Hopf links. With certain 4-cocycle twists, we discover that, by threading a third string through two-string unlink into a three-string Hopf-link configuration, Abelian two-string braiding statistics is promoted to non-Abelian three-string braiding statistics.
14 More- Received 31 July 2014
- Revised 4 December 2014
DOI:https://doi.org/10.1103/PhysRevB.91.035134
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