The moduli space of hyper-K{\"a}hler four-fold compactifications

TL;DR

This paper explores the moduli space of hyper-Kähler four-fold compactifications in string and M-theory, analyzing its dimension bounds, specific examples, and the role of fluxes in achieving various supersymmetries.

Contribution

It provides new insights into the structure and bounds of the moduli space for hyper-Kähler four-folds, with detailed examples and flux configurations for supersymmetry enhancement.

Findings

The moduli space dimension is strictly bounded from above.

Fluxes are necessary to achieve certain supersymmetries in specific examples.

Subtlety in the symmetric product limit $S^2(K3)$ is identified.

Abstract

I discuss some aspects of the moduli space of hyper-K{\"a}hler four-fold compactifications of type II and ${\cal M}$- theories. The dimension of the moduli space of these theories is strictly bounded from above. As an example I study Hilb$^2(K3)$ and the generalized Kummer variety $K^2(T^4)$. In both cases RR-flux (or $G$-flux in ${\cal M}$-theory) must be turned on, and we show that they give rise to vacua with ${\cal N}=2$ or ${\cal N}=3$ supersymmetry upon turning on appropriate fluxes. An interesting subtlety involving the symmetric product limit $S^2(K3)$ is pointed out.