# Relationship between modular transformations and anyon braiding

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.

Update 20151021: Thanks to the references recommended by Prof. Wen in his answer, I can now reformulate my original confusion and clarify it.

Originally I don't understand why we can identify the unitary matrices $S$ and $T$ in the context of non-abelian geometric phases with the $S$ and $T$ matrices in the context of anyon statistics: they seem to describe two different things. In the former case, they are "geometrical" modular transformations, while in the later case, they describe how quasiparticles "interact" with others (or themselves).

The Verlinde formula says $N_k^{ij}=\sum_l \frac{S_{li} S_{lj} (S_{lk})^*}{S_{l1}}$, which I originally understood as "The fusion of anyons ($N_k^{ij}$) is determined by the mutual statistics ($S$) of anyons." In this way, I couldn't see how modular transformations could enter the game and affect anyon statistics after all.

However, I figured out later that Verlinde's original work was saying "The modular transformation $S$ diagonalizes the fusion rules." Namely, the $S$ in the formula should be understood as "modular transformation" instead of "mutual statistics of anyon". This way, it seems clear to me that the modular transformation determines the internal degrees of freedom of anyons and thereby bridges the seemingly "two different things".

$S$ and $T$ matrices describe modular transformations. Fusion coefficient and particle spins $(N^{ij}_k,s_i)$ describe the mutual and self statistics of anyons. Their relation is well known in category theory (Our papers http://arxiv.org/abs/1506.05768 and http://arxiv.org/abs/1507.04673 used those relations).