Comments on M${}_{24}$ representations and $C{Y}_3$ geometries

We show using string dualities that Mathieu moonshine controls Gromov-Witten invariants and periods of the holomorphic 3-form $\mathrm{\Omega }$ for certain $C{Y}_3$ manifolds. We also discuss how the period vectors appear in flux compactifications on these $C{Y}_3$ manifolds and work out the connection between the sporadic group M${}_{24}$ and the Yukawa couplings in four dimensional theories that arise from heterotic string theory compactifications on these $C{Y}_3$ manifolds.