${\cal N}=2$ heterotic string compactifications on orbifolds of $K3\times T^2$

TL;DR

This paper explores ${ m N}=2$ heterotic string compactifications on orbifolds of $K3 imes T^2$, analyzing their supersymmetric indices, gauge coupling corrections, and the role of Mathieu group symmetries, including standard and non-standard embeddings.

Contribution

It provides a detailed analysis of supersymmetric indices and gauge coupling corrections for heterotic compactifications on orbifolds of $K3 imes T^2$, linking them to Mathieu group symmetries and automorphic forms.

Findings

Supersymmetric index decomposes into twisted elliptic genus of $K3$.

One-loop gauge corrections are captured by automorphic forms from theta lifts.

Non-standard embeddings characterized by instanton numbers and spectrum differences.

Abstract

We study ${\cal N}=2$ compactifications of $E_8\times E_8$ heterotic string theory on orbifolds of $K3 \times T^2$ by $g'$ which acts as an $\mathbb{Z}_N$ automorphism of $K3$ together with a$1/N$ shift on a circle of $T^2$. The orbifold action $g'$ corresponds to the $26$ conjugacy classes of the Mathieu group $M_{24}$. We show that for the standard embedding the new supersymmetric index for these compactifications can always be decomposed into the elliptic genus of $K3$ twisted by $g'$. The difference in one-loop corrections to the gauge couplings are captured by automorphic forms obtained by the theta lifts of the elliptic genus of $K3$ twisted by $g'$. We work out in detail the case for which $g'$ belongs to the equivalence class $2B$. We then investigate all the non-standard embeddings for$K3$ realized as a $T^4/\mathbb{Z}_\nu$ orbifold with $\nu = 2, 4$ and $g'$ the $2A$ involution. We show that for non-standard embeddings the new supersymmetric index as well as the difference in one-loop corrections to the gauge couplings are completely characterized by the instanton numbers of the embeddings together with the difference in number of hypermultiplets and vector multiplets in the spectrum.