(I think the answer is no, but I'm not sure.)
In Hurwitz theory, one counts $n$-fold branched covers $\Sigma'\to\Sigma$ of a Riemann surface $\Sigma$ with fixed ramification data around each branch point. There are many variants of this definition, including
- spin Hurwitz numbers, where $\Sigma$ and $\Sigma'$ are given spin structures and the count is weighted by the Arf invariant of $\Sigma'$ (for more details see §1.3 of Gunningham, "Spin Hurwitz numbers and topological quantum field theory"); and
- $r$-spin Hurwitz numbers or Hurwitz numbers with completed cycles depending on an $r\in\mathbb N$, where the ramification data involves completed cycles in $\mathbb C[S_n]$ (for more details see §2 of Shadrin-Spitz-Zvonkine, "Equivalence of ELSV and Bouchard-Mariño conjectures for $r$-spin Hurwitz numbers").
The reason for the name "$r$-spin Hurwitz numbers" comes from their use in studying a moduli space of $r$-spin structures on Riemann surfaces. An $r$-spin structure is a generalization of a spin structure: instead of reducing the structure group of the tangent bundle across the double cover $\mathrm{Spin}_2\to\mathrm{SO}_2$, one lifts it across the $r$-fold connected cover $G\to\mathrm{SO}_2$; equivalently, a spin structure on a Riemann surface is a choice of a square root of the canonical bundle, and an $r$-spin structure is a choice of an $r^{\mathrm{th}}$ root. Thus a spin structure is an $r$-spin structure for $r = 2$.
Therefore a natural question is, are $r$-spin Hurwitz numbers for $r = 2$ the same as spin Hurwitz numbers?
It doesn't seem like $r$-spin structures enter the definition of $r$-spin Hurwitz numbers at all, which is a little strange to me, but maybe I'm missing something. Even then, what about the Arf invariant? So it seems like these two notions are different, but I'm not confident enough in my understanding of $r$-spin Hurwitz numbers to be certain.
