Fermions, Bosons, Anyons on a 2-manifold My understanding of fermions, bosons, and anyons is that anyons are disallowed in 3+1 dimensions (or $n+1 | n\geq3$) because of the topology of spacetime. The paths of swapping two particles twice is contractible in 3+1. In 2+1, this isn't the case and so anyons can exist.
However, what is the case for other 2-manifolds than $\boldsymbol{\mathrm{R}}^2$? Can anyons exist on an $\boldsymbol{\mathrm{S}}^2$ manifold for example?
More importantly, what is the appropriate formalism for approaching this problem?
 A: A general formalism to probe whether or not you have anyons/generalised statistics, in $d$ spatial dimensions and arbitrary topology is the following.
Consider $n$-identical particles moving in a $d$-dimensional spatial manifold, $\Sigma_d$. Since the particles are indistinguishable the configuration space for the $n$-particles is the orbifold:
$$\mathrm{Sym}^n(\Sigma_d) := \raise{0.2em}{\underset{n\ \text{times}}{\underbrace{\Sigma_d\times\cdots\times\Sigma_d}}}\Big/\raise{-0.2em}{\mathrm{S}_n},$$
where $\mathrm{S}_n$ is the permutation group of $n$ letters. The important space, from which the rest follow is $\mathrm{Sym}^2(\Sigma_d)=\raise{0.13em}{\Sigma_d\times\Sigma_d}\Big/\raise{-0.15em}{\mathbb{Z}_2}$, since this tells you what happens when you exchange two particles.
Quantum mechanically, on a multiply connected space, $\mathrm{Sym}^2(\Sigma_d)$, all observables involving two such particles decompose as
$$\mathcal{O}(x,y) = \sum_{\alpha \in \pi_1(\mathrm{Sym}^2(\Sigma_d))} \chi(\alpha)\; \mathcal{O}_\alpha(x,y),$$
where $\mathcal{O}$ stands for observable, and the sum runs over all connected components of $\mathrm{Sym}^2(\Sigma_d)$ (here $\pi_1(\mathrm{Sym}^2(\Sigma_d))$ is the fundamental group of the configuration space). We have further allowed the freedom of a weight $\chi(\alpha)$ for every connected component.
Now, demanding that the observables satisfy the following two physical requirements:

*

*Physical observables cannot depend on the mesh used to calculate the homotopy classes

*Observables need to satisfy the standard convolution property $\mathcal{O}(x,y) = \displaystyle\int \mathrm{d}{z}\ \mathcal{O}(x,z)\,\mathcal{O}(z,y),$
restricts the weights $\chi(\alpha)$ significantly. The weights must satisfy the constraints

*

*$\left|\chi(\alpha)\right|^2=1$

*$\chi(\alpha)\chi(\beta) = \chi(\alpha\;\beta)$.

Therefore the weights provide a one-dimensional unitary representation of $\pi_1(\mathrm{Sym}^2(\Sigma_d))$. The different choices of representations correspond to the different possible statistics. The general strategy is, then:

*

*Choose $\Sigma_d$

*Calculate $\mathrm{Sym}^2(\Sigma_d)$

*Calculate $\pi_1\!\left(\mathrm{Sym}^2(\Sigma_d)\right)$

*Look at how many one-dimensional unitary representations it admits.

For example it is easy to check that $\pi_1\!\left(\mathrm{Sym}^2\!\left(\mathbb{R}^{d\geq 3}\right)\right)=\mathbb{Z}_2$, therefore there are two one-dimensional unitary representations. The trivial, corresponding to bosons, and the sign, corresponding to fermions.
On the other hand, $\pi_1\!\left(\mathrm{Sym}^2\!\left(\mathbb{R}^2\right)\right)=\mathbb{Z}$, so there are infinitely many one-dimensional unitary representations, corresponding to all possible statistics interpolating between bosons and fermions.
Finally, for the example of $\mathbb{S}^2$ that you were interested in, we have
$$\pi_1\!\left(\mathrm{Sym}^2\!\left(\mathbb{S}^2\right)\right) = \pi_1\!\left(\mathbb{S}^2\times\mathbb{RP}^2\right)=\mathbb{Z}_2,$$
so you cannot have anyons on a sphere.
