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

Abstract

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 ω4 of G's cohomology group H4(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 Sxyz and Txy in terms of the gauge group G and the 4-cocycle ω4. As we compactify one of the spatial directions z into a compact circle with a gauge flux b inserted, we can use the generators Sxy and Txy of an SL(2,Z) subgroup to study the dimensional reduction of the 3D topological order C3D to a direct sum of degenerate states of 2D topological orders Cb2D in different flux b sectors: C3D=⊕bCb2D. The 2D topological orders Cb2D are described by 2D gauge theories of the group G twisted by the 3-cocycle ω3(b), dimensionally reduced from the 4-cocycle ω4. We show that the SL(2,Z) generators, Sxy and Txy, 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.