# Are spin Hurwitz numbers $r$-spin Hurwitz numbers?

(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

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.

• From what I can tell, I think you are right: they are not obviously related. $r$-spin Hurwitz numbers appear to be called so because they are related to intersections on the moduli of r-spin structures (which is a covering of the moduli of curves) by an analogue of the ELSV formula. But I also find this stuff confusing and would like to see a more authoritative confirmation! – Sam Gunningham Jun 21 '18 at 17:40