Non-Abelian string and particle braiding in topological order: Modular SL(3,Z) representation and (3 + 1)- dimensional twisted gauge theory

Abelian string and particle braiding in topological order: Modular SL(3,Z) representation and (3 + 1)-dimensional twisted gauge theory." Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group G and a 4-cocycle twist ω 4 of G's cohomology group H 4 (G,R/Z) in three-dimensional space and one-dimensional time (3 + 1D). We establish the topological spin and the spin-statistics relation for the closed strings and their multistring braiding statistics. The 3 + 1D twisted gauge theory can be characterized by a representation of a modular transformation group, SL(3,Z). We express the SL(3,Z) generators S xyz and T xy in terms of the gauge group G and the 4-cocycle ω 4. As we compactify one of the spatial directions z into a compact circle with a gauge flux b inserted, we can use the generators S xy and T xy of an SL(2,Z) subgroup to study the dimensional reduction of the 3D topological order C 3D to a direct sum of degenerate states of 2D topological orders C 2D b in different flux b sectors: C 3D = ⊕ b C 2D b. The 2D topological orders C 2D b are described by 2D gauge theories of the group G twisted by the 3-cocycle ω 3(b) , dimensionally reduced from the 4-cocycle ω 4. We show that the SL(2,Z) generators, S xy and T xy , 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.