# Anyons without fractional spin?

Is it possible to have particles obeying anyonic statistics but not having fractional spin?

I am wondering, because while spin in quantum physics arises from the geometry/topology of spacetime, statistics is connected to the geometry/topology of configuration space.

What would particles like this be called?

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Bosons and fermions are special case of anyons. Bosons and fermions are representation of the symmetric group $S_n$. Anyons are representation of the Braid group $B_n$. And you have a surjective homomorphism from $B_n$ to $S_n$. So $B_n$ is a much richer group than $S_n$. See an Anyon primer for more information. – Trimok Jul 3 '13 at 10:03
Thank you @Trimok. I know the braid group well and how it arises in the topological analysis of the configuration space of $n$ indistinguishable particles. In $2d$ spin is fractional. But my question would be, are there structures, i.e. particle-like objects, obeying anyonic statistics but do not have fractional spin (maybe none). Are there physical systems like this and how are these objects called? – Hamurabi Jul 3 '13 at 22:52
It depends by what you mean by "anyonic statistics". For me, "anyonic statistics" means a representation of $B_n$. So I just wanted to point, that $S_n$ is a subgroup of $B_n$. Or, said differently, bosons, which have integer spin, for instance, are only a very special case of anyonic statistics. Now, if we exclude bosons, I think there is no integer spin particle obeying anyonic statistics. If we exchange 2 anyons, this corresponds to a phase change $e^{2i\pi s}$, where $s$ is the spin. $s$ integer means bosons, and $s$ semi-integer means fermions. Other particles have not a integer spin. – Trimok Jul 4 '13 at 2:32
What if we do not have a spin structure attached to the fields leading to an $n$-particle config. space? – Hamurabi Jul 4 '13 at 13:46
The phase change is related to the spin (even non integer and non semi-integer). – Trimok Jul 4 '13 at 14:38

## 1 Answer

Due to spin-statistics theorem, anyons always have fractional spin.

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Ok. But when one evaluates the Hilbert space dimension of anyons dwelling on a surface using the Verlinde formula and the fusion multiplicities $N_{ij}^k$, one sums over the labels. Are these labels spins? – Hamurabi Jul 30 '13 at 10:17
No. these labels are not spins. – Xiao-Gang Wen Jul 30 '13 at 10:35
So what are they? Don't they come from the liealgebra-valued connection one-forms $A$, the colouring of the punctures/anyons/particles? – Hamurabi Jul 30 '13 at 12:11
The labels, ie the indices of $N^k_{ij}$, label the topological type of quasiparticles. Each topological type of quasiparticle has a fractional spin (ie a fractional angular momentum). – Xiao-Gang Wen Oct 5 '13 at 11:18
thank you. maybe then it is better to ask for the physical motivation of the connection field $A$ and what physically its colouring by $su(2)$ means? – Hamurabi Oct 8 '13 at 16:51