Comments on M$_{24}$ representations and $CY_3$ geometries

We show using string dualities that Mathieu moonshine controls Gromov-Witten invariants and periods of the holomorphic 3-form $Ω$ 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.

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