My question from reading this paper:
Michael G. G. Laidlaw and Cécile Morette DeWitt, Feynman Functional Integrals for Systems of Indistinguishable Particles. Phys. Rev. D 3, 1375 (971).
Definition: the configuration space of $n$ indistinguishable particles in $d-$dim Eucildean space $\mathbb{R}^d$ is $M_n=(\mathbb{R}^{dn}-D)/S_n$ with $D=\{(\mathbf{r}_1,\cdots,\mathbb{r}_n) | \mathbf{r}_i=\mathbf{r}_j \text{ for some}\ i\neq j\}$, $S_n$ the permutation group.
In this paper they proved the following theorem:
In $d-$dimensional space, the statistics of $n$ indistinguishable particles is classified by different $1$-dimensional irreducible representation of $\pi_1(M_n)$, the fundamental group of $M_n$.
For $d\ge3$, $\pi_1(M_n)=S_n$, there are only two types of $1$-dim rep. of $S_n$, i.e. Boson and Fermion. This theorem excludes the possibility of Parastatistics, i.e. higher dimensional rep. of permutation group.
For $d=2$, $\pi_1(M_n)=B_n$ the Braid group. So $1$-dim rep. of $B_n$ is Abelian anyon, see Yong-Shi Wu, General Theory for Quantum Statistics in Two Dimensions.
My questions:
- From this theorem, it seems that only Abelian anyon can occur in $2$-dim, because it requires $1$-dim rep. of fundamental group. So why can there exist non-Abelian anyon, i.e. the higher dim. rep. of $B_n$ that violates this theorem?
- Or is there some loophole or assumption in this theorem that allows the non-Abelian anyon to break?