On Mathieu moonshine and Gromov-Witten invariants

We show that a large number of \(CY_3\) 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 \(CY_3\) 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.