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

TL;DR

This paper reveals how Mathieu moonshine, linked to the sporadic group M24, influences Gromov-Witten invariants, periods, and Yukawa couplings in certain Calabi-Yau threefolds via string dualities.

Contribution

It demonstrates the control of M24 moonshine over geometric and physical properties of specific Calabi-Yau threefolds through string dualities and flux compactifications.

Findings

Mathieu moonshine governs Gromov-Witten invariants.

Period vectors relate to flux compactifications.

Connection established between M24 and Yukawa couplings.

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.