TITLE: Is the moduli space of unorientable Riemann surfaces with $pin^+$ structure orientable?
QUESTION [28 upvotes]: By a non-orientable Riemann surface ${\cal C}$, I mean a compact non-orientable two-manifold without boundary that is endowed with a conformal structure.
Such objects have a moduli space that is somewhat like the more familiar moduli space of oriented Riemann surfaces, but this moduli space is itself non-orientable.  (See corollary 2.3 of https://arxiv.org/abs/1309.0383, where this is proved with a simple explicit example.)
My question is this:  Suppose that ${\cal C}$ is endowed with a $pin^+$ structure.   (This is one of the two possible analogs of a spin structure in the non-orientable case, the other being a $pin^-$ structure.)  I've come to suspect that the moduli space of non-orientable Riemann surfaces with a $pin^+$ structure is itself orientable.  I wonder if this is a known result and where it might be found.

REPLY [4 votes]: This is not an answer to the question but rather a comment related to the motivation behind it that might be of interest. A more natural point of view from the perspective of GW theory is to consider symmetric Riemann surfaces i.e. orientable RS with an orientation reversing involution. The moduli of such curves is not orientable but the moduli of equivariant J-hol. maps to a symplectic manifold with an anti-symplectic involution of dimension 2(2n+1) often are and give rise to real GW invariants (in particular this is the case for target curves and CY 3-folds arXiv). The calculations in the target P^1 case suggest that these are related to matrix models for SO/Sp - I hope I'll be able to say something more precise soon.