# Dehn twists and topological order

I am trying to understand notion of a "Dehn twist" and how it relates to topological order.

In particular refering to http://arxiv.org/abs/1208.4834 it is stated that Xiao Gang Wen's paper on "Topological Order in Rigid States" (http://dao.mit.edu/~wen/pub/topo.pdf) is supposed to provide an introduction to "non-abelian adiabatic Berry phases associated with Dehn twists for a U(1) Chern Simons Theory". Skimming through the respective paper however I could not find the notion of a "Dehn twist" appearing at all? Maybe it appears under a different name or it is not given a name at all?

I would be very happy for any support.

Best.

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## 2 Answers

In my paper, the Dehn twist is referred as modular transformation. See section V of http://arxiv.org/abs/1212.5121 which is available in arXiv. The unitary transformation generated by the Dehn twist is called the non-Abelian geometric phase.

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Any oriented closed surface is a torus with g holes (for an actual torus g=1, for a sphere g=0, etc.), where g is called the genus. Associated to these surfaces is the mapping class group, which is the group of equivalence classes of homeomorphisms (topological isomorphisms) of the surface to itself, where two such mappings are considered equivalent when they can be continuously deformed into each other. Dehn first proved that for an orientable genus g surface, this group is generated by what are now called Dehn twists. A Dehn twist is easy to understand: take a surface, cut open a tube, twist it around a full turn, and glue all points back to their original positions. This defines a map from the surface to itself mapping a point on the surface to the corresponding point on the twisted surface.

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