PIRSA:14050015

Non-Abelian String and Particle Braiding in 3+1D Topological Order

APA

Wang, J. (2014). Non-Abelian String and Particle Braiding in 3+1D Topological Order. Perimeter Institute. https://pirsa.org/14050015

MLA

Wang, Juven. Non-Abelian String and Particle Braiding in 3+1D Topological Order. Perimeter Institute, May. 01, 2014, https://pirsa.org/14050015

BibTex

          @misc{ pirsa_14050015,
            doi = {},
            url = {https://pirsa.org/14050015},
            author = {Wang, Juven},
            keywords = {},
            language = {en},
            title = {Non-Abelian String and Particle Braiding in 3+1D Topological Order},
            publisher = {Perimeter Institute},
            year = {2014},
            month = {may},
            note = {PIRSA:14050015 see, \url{https://pirsa.org}}
          }
          

Abstract

String and particle excitations are examined in a class of 3+1D topological order described by a discrete gauge theory with a gauge group G and a 4-cocycle twist ω4∈H4(G,R/Z) of G's cohomology group. We demonstrate the topological spin and the spin-statistics relation for the closed strings, and their multi-string braiding. The 3+1D twisted gauge theory can be characterized by a representation of SL(3,Z) modular transformation, which we find its generators Sxyz and Txy in terms of the gauge group G and the 4-cocycle ω4. As we compactify one of the 3D's direction z into a compact circle inserted with a gauge flux b, we can use the generators of SL(2,Z) subgroup of SL(3,Z), Sxy and Txy, to study the dimension reduction of the 3D topological order C3D to a direct sum of degenerate states of 2D topological orders C2Db in different fluxb sectors: C3D=⊕bC2Db. The 2D topological orders C2Db are described by 2D gauge theories of the group G twisted by the 3-cocycles ω3(b)dimensionally reduced from the 4-cocycle ω4. We show that the SL(2,Z) generators, Sxy and Txy, fully encodes a particular type of three-string braiding statistics for the connected sum of two Hopf links 221#221 configuration. With certain 4-cocycle twist, we find that, by threading a third string through two-string unlink 021 into three-string Hopf links 221#221 configuration, Abelian two-string statistics is promoted to non-Abelian three-string statistics. (Work done in arxiv.org/abs/1404.7854)