Abstract
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.
Highlights
Models which preserve only N = (0, 4) worldsheet supersymmetry and that are connected to N = (4, 4) non-linear sigma model with K3 target, have as their symmetry group the full M24 group [14, 15]
Of particular interest to us is [14], where it was shown that the elliptic genus of K3 appears in compactifications of the heterotic string theory and that, by duality, the Gromov-Witten invariants of certain CY3 manifolds are related to the Mathieu group M24
We review that by mirror symmetry this implies that for certain CY3 manifolds the holomorphic 3-form Ω is likewise connected to the Mathieu group M24
Summary
We first review Mathieu moonshine that was discovered in [1]. There the authors expand the elliptic genus of the K3 manifold and find that the expansion coefficients are sums of dimensions of irreducible representations of the largest Mathieu group M24. We use the duality between heterotic string theory compactifications on K3 × T 2 and type IIA compactifications on CY3 manifolds Xn that are elliptic fibrations over Fn to discuss (following [14]) how Mathieu moonshine is connected to the Gromov-Witten invariants of the Xn. Using mirror symmetry we connect Mathieu moonshine to the holomorphic 3-form Ω of Yn, that are the mirror CY3 manifolds of the Xn. We argue using the Greene-Plesser construction of mirror pairs that at least some of Xn and Yn exhibit a connection between M24 and both their Gromov-Witten invariants and their holomorphic 3-form Ω
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