On Mathieu moonshine and Gromov-Witten invariants

We show that a large number of $C{Y}_3$ manifolds are involved in an intricate way in Mathieu moonshine viz. their Gromov–Witten invariants are related to the expansion coefficients of the twined/ twisted–twined elliptic genera of $K3$. We use the string duality between CHL orbifolds of heterotic string theory on $K3\times T^2$ and type IIA string theory on $C{Y}_3$ manifolds to explicitly show this connection. We then work out two concrete examples where we exactly match the expansion coefficients on both sides of the duality.