Relationship between modular transformations and anyon braiding

Question

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

Answer 1

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.