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I should preface this by saying that I am a physicist. My question pertains to the paper https://arxiv.org/abs/1802.09978, where $r$-spin structures are defined (Def. 2.1).

To orient the discussion, let me give my basic understanding of usual spin structures, before moving on to the $r$-spin generalization. I'm happy to restrict the discussion to Riemann surfaces of genus $g$ with no boundary, $\Sigma_g$. Heuristically, I think of a spin structure on $\Sigma_g$ as an assignment of a $\mathbb{Z}_2$ holonomy to each cycle in $\Sigma_g$, which gives $2^{2g}$ distinct choices of spin structure.

Here are two facts about $r$-spin structures, both found in Prop. 2.21 of the reference:

i) A $r$-spin structure exists on $\Sigma_g$ iff $2-2g = 0$ modulo $r$.

ii) When i) is met, there are $r^{2g}$ distinct $r$-spin structures on $\Sigma_g$.

Now here come the questions: Fact ii) makes me want to think of $r$-spin structures as assignments of $\mathbb{Z}_r$ valued holonomies to each cycle in $\Sigma_g$, in analogy with usual spin structures. Naively, this works for any pair $(r,g)$, in tension with fact i). I'm wondering if there is any easy way to see what the obstruction is in a concrete example, say $(r,g) = (3,2)$.

I should also mention that I have heard various explanations (and the paper mentions this) involving conditions under which one can take roots of the canonical line bundle, but these are not intuitive to me. Above all (and I realize this may not be doable), I would love a pictorial explanation in a simple case.

Thank you in advance.

johnny
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  • Do you understand the $r=2$ case? (Beyond "assigning of $Z_2$-holonomy", which is only a fragment of the story and does not explain the most interesting part.) – Moishe Kohan Apr 04 '24 at 11:58
  • Let me say some words which I am familiar with but don't claim to understand particularly deeply. I know that the $Spin$ group is a double cover of the special orthogonal group, and that one gets sign ambiguities in lifting a $SO(n)$ bundle to a $Spin(n)$ one. I also know there are global obstructions to the existence of $spin$ structures (second Stiefel-Whitney class). I have been told (this may be incorrect) that there is no analog of Stiefel-Whitney classes for the $r$-spin case. Any clarification of these aspects, or intuition about Stiefel-Whitney classes would be most welcome. – johnny Apr 04 '24 at 13:30
  • Then start by reading the last part of my answer here: https://math.stackexchange.com/questions/3961908/what-makes-spinors-mysterious/4110007#4110007 – Moishe Kohan Apr 04 '24 at 13:37
  • Thank you for this, I wasn’t aware of this perspective for Riemann surfaces. However, it would seem like in this setting the obstruction to having a $r$-spin structure would again boil down to listing conditions under which one can define roots of the canonical line bundle. – johnny Apr 04 '24 at 19:57
  • One can state it in terms of the tangent bundle or frame bundle. But appearance of some bundle in this context is inevitable. I can also define it in terms of the fundamental group, but it is just a way to hide a flat bundle (a bundle with a flat connection). – Moishe Kohan Apr 04 '24 at 20:06
  • I would be curious to hear about the definition in terms of the fundamental group, if you have a reference. – johnny Apr 04 '24 at 20:18
  • I do not have a reference, but I will write it as an answer later on. – Moishe Kohan Apr 04 '24 at 20:25

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