  We show using string dualities that Mathieu moonshine controls Gromov-Witten
invariants and periods of the holomorphic 3-form $\Omega$ for certain $CY_3$
manifolds. We also discuss how the period vectors appear in flux
compactifications on these $CY_3$ 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
$CY_3$ manifolds.
