22,964 research outputs found
Canonical singularities of orders over surfaces
We classify the possible ramification data and etale local structure of
orders over surfaces with canonical singularities.Comment: This contains major revisions, primarily to help introduce the reader
to the minimal model program for orders on surface
Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
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 3 dimensional space and 1
dimensional time (3+1D). We establish the topological spin and the
spin-statistics relation for the closed strings, and their multi-string
braiding statistics. The 3+1D twisted gauge theory can be characterized by a
representation of a modular transformation group SL. We express
the SL 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 SL 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-cocycles ,
dimensionally reduced from the 4-cocycle . We show that the
SL 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
three-string Hopf-link configuration, Abelian two-string braiding statistics is
promoted to non-Abelian three-string braiding statistics.Comment: 36 pages, many figures, 17 tables. v3: Accepted by Phys. Rev. B. Add
acknowledgements to Louis H. Kauffma
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