Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

(propose a free ad)

In the context of anyon braiding, we have $S$ and $T$ matrices which describe the mutual and self statistics of anyons. In the context of conformal field theory on a torus, we have modular transformations $S$ and $T$. ($T:\tau\rightarrow\tau+1$, $S:\tau\rightarrow -\frac{1}{\tau}$ with $\tau=\omega_2/\omega_1$ the modular parameter and $\omega_i$'s the periods of the lattice on a torus.)

What's the relationship between those two? I think the question could be related to the Dehn twists but don't know how.

This post imported from StackExchange Physics at 2015-09-30 17:14 (UTC), posted by SE-user Zhuxi Luo

You get a three sphere by gluing two solid tori along their boundary by the mapping class group element S. You can get a Hopf link by filling each core of the two solid tori with a line operator. Thus, the (full) braiding phase between quasiparticles a and b is computed as

$\langle T^2, a | S | T^2, b \rangle$,

where the state $|T^2, a\rangle$ is the state on the torus obtained by performing the open path integral over the solid torus with the a line in its core.

user contributions licensed under cc by-sa 3.0 with attribution required