Generalised spin structures on 2-dimensional orbifolds

Abstract

Generalised spin structures, or $r$-spin structures, on a $2$-dimensional orbifold $\Sigma$ are $r$-fold fibrewise connected coverings (also called $r$\textsuperscript{th} roots) of its unit tangent bundle $ST\Sigma$. We investigate such structures on hyperbolic orbifolds. The conditions on $r$ for such structures to exist are given. The action of the diffeomorphism group of $\Sigma$ on the set of $r$-spin structures is described, and we determine the number of orbits under this action and their size. These results are then applied to describe the moduli space of taut contact circles on left-quotients of the $3$-dimensional geometry $\widetilde{\mathrm{SL}}_{2}$.