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

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 $g^{\prime }$ which acts as a ${\mathbb{Z}}_N$ automorphisim on $K3$ together with a $1/N$ shift along $T^2$. The orbifold $g^{\prime }$ 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.

54 pages, Mathematica code included in the source file, minor typos fixed, one reference added

References in corpus (12)

• Notes on the K3 Surface and the Mathieu group M_24
• Note on Twisted Elliptic Genus of K3 Surface
• Product Representation of Dyon Partition Function in CHL Models
• Dyon Spectrum in Generic N=4 Supersymmetric Z_N Orbifolds
• Non-geometric Calabi-Yau Backgrounds and K3 automorphisms
• Four-derivative couplings and BPS dyons in heterotic CHL orbifolds
• Topological Amplitudes in Heterotic Strings with Wilson Lines
• $\mathcal{N}=2$ heterotic string compactifications on orbifolds of $K3\times T^2$
• Dyon degeneracies from Mathieu moonshine
• Dualities in CHL-Models
• Threshold corrections in heterotic flux compactifications
• Calabi-Yau manifolds and sporadic groups

Cited by in corpus (5)

• Modular Forms as Classification Invariants of 4D N=2 Heterotic--IIA Dual Vacua
• On Mathieu moonshine and Gromov-Witten invariants
• Heterotic strings on $(K3\times T^2)/{\mathbb{Z}}_3$ and their dual Calabi-Yau threefolds
• Gravitational couplings in $\mathcal{N}=2$ heterotic compactifications with Wilson lines
• State counting on fibered CY-3 folds and the non-Abelian Weak Gravity Conjecture