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 ω[subscript 4] of G's cohomology group H[superscript 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[superscript xyz] and T[superscript xy] in terms of the gauge group G and the 4-cocycle ω[subscript 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[superscript xy] and T[superscript xy] of an SL(2,Z) subgroup to study the dimensional reduction of the 3D topological order C[superscript 3D] to a direct sum of degenerate states of 2D topological orders C[2D over b] in different flux b sectors: C[superscript 3D] = ⊕[subscript b]C[2D over b]. The 2D topological orders C[2D over b] are described by 2D gauge theories of the group G twisted by the 3-cocycle ω[subscript 3(b)], dimensionally reduced from the 4-cocycle ω[subscript 4]. We show that the SL(2,Z) generators, S[superscript xy] and T[superscript 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.