Can anyons exist on a torus without any additional conditions?

Question

While learning recently some more "advanced" stuff about path integral formalism I was introduced to the topological conditions that specify the process of construction of the propagator, i.e. the homotopy theorem for path integration Pierre Cartier, Cecile DeWitt-Morette - Functional Integration Action and Symmetries-Cambridge University Press (2007), Ch. 8.3.

There is an already similar more general question addresing the subject: Fermions, Bosons, Anyons on a 2-manifold. I wanted to check if my understanding is right in the case of the 2D torus - $T^2$. From what I can see the corresponding homotopy group here is trivial for k$\geq$2 and for k=1 it's $\mathbb{Z}^2$ https://topospaces.subwiki.org/wiki/Homotopy_of_torus. Doesn't that exclude the existence of anyons on such manifold? Or the topology is somehow modified by the interaction?

Answer 1

Anyons can indeed exist on a torus, and in the condensed matter setting this is often the manifold that we like to consider (see for example the paper "Fault-Tolerant Quantum Computation by Anyons" by A. Kitaev). I'm not too familiar with the book you mention, but I am familiar with the argument given in the answer to the other question.

I think that there are a couple of misconceptions here (which is no surprise as the topic of anyons is quite confusing at first!). Firstly, $\mathbb{Z}^2$ which you mention as the first homotopy group for the torus is not trivial (note that $\mathbb{Z}^2$ is the product of two copies of the infinite cyclic group, reflecting the two independent con-contractible cycles). Secondly, the argument given in the answer to the more general question you mention uses the homotopy group of the configuration space, not the manifold itself. What that means is that for two particles, the space to consider is the space of all possible positions for both particles (so the product of two copies of the torus) up to swaps of position (which is the quotient $/ \mathbb{Z}_2$ in that answer). Finally, the answer to the other question is itself incomplete because it assumes that we only have two identical particles on the sphere, which severely restricts which types of particle are allowed (due to anyon fusion rules). We should be sure to allow for the possibility of multiple particles, even if they are not involved in the braiding (in which case they can be treated as punctures on the manifold). This is why that answer claims that there are no anyons on the 2-sphere, which is not correct.