Gravitational couplings in ${\cal N}=2$ string compactifications and Mathieu Moonshine

TL;DR

This paper computes gravitational couplings in heterotic string compactifications related to Mathieu Moonshine, revealing integer Gopakumar-Vafa invariants and the role of twisted sectors in conifold singularities.

Contribution

It provides explicit calculations of gravitational couplings linked to Mathieu group symmetries and predicts Gopakumar-Vafa invariants in dual Calabi-Yau models, including twisted sectors.

Findings

Gopakumar-Vafa invariants are integers across models.

Conifold singularities occur when twisted sector states become massless.

Strength of singularities is determined by genus zero invariants.

Abstract

We evaluate the low energy gravitational couplings, $F_g$ in the heterotic $E_8\times E_8$ string theory compactified on orbifolds of $K3\times T^2$ by $g'$ which acts as a $\mathbb{Z}_N$ automorphisim on $K3$ together with a $1/N$ shift along $T^2$. The orbifold $g'$ corresponds to the conjugacy classes of the Mathieu group $M_{24}$. The holomorphic piece of $F_g$ is given in terms of a polylogarithim with index $3-2g$ and predicts the Gopakumar-Vafa invariants in the corresponding dual type II Calabi-Yau compactifications. We show that low lying Gopakumar-Vafa invariants for each of these compactifications including the twisted sectors are integers. We observe that the conifold singularity for all these compactifications occurs only when states in the twisted sectors become massless and the strength of the singularity is determined by the genus zero Gopakumar-Vafa invariant at this point in the moduli space.