I've been reading the paper "Comments on operators with large Spin" (here) and I am having some trouble understanding the following:

In section 2, they begin by studying, in a conformal field theory on flat space, an operator of the form $$ \mathcal{O}=\left(\bar{q} \overleftrightarrow{D}^S q \right) $$ This operator is said to have spin $S$, and their aim is to check that it's anomalous dimension scales as $\log(S)$. To do this they will relate this operator with a quark-antiquark pair spinning on the cilinder, by saying two things. First:

In a conformal field theory, the anomalous dimension of an operator is equal to the energy of the corresponding state of the field theory on the cylinder $R × S^3$.

I understand that this comes from radial quantization, but my problem arises with their second claim:

On the cylinder, a high spin operator consists of two particles (or group of particles) that are moving very rapidly along a great circle of $S^3$. These particles are colored and the color field lines go between the two particles. For simplicity, let us first consider the case of a quark and an antiquark moving very fast along a great circle of the S^3, with color gauge fields joining them.

My problem is: Why is it that the operator $\mathcal{O}$ is related to a pair of particles spinning around the cylinder? More specifcally: The spin $S$ of $\mathcal{O}$ comes from the lorentz indices in the derivatives. On the other hand, the two particles are spinning around a circle on the $S^3$, but to me this kind of movement would give rise to ORBITAL angular momentum, not Spin (or intrinsic) angular momentum. For $\mathcal{O}$, the Spin operator is given by a combination of $\gamma$ matrices, and for the pair of particles on the cylinder it is said that the spin operator is $\partial_{\phi}$. How does the mapping of the spins work?

I've been thinking about these issues for a while but so far can't seem to get around it, and I would very much appreciate some help.


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