Exotic holonomies $\E_7^{(a)}$

TL;DR

This paper proves that certain exceptional Lie groups, including $ ext{E}_7^{(a)}$, can serve as holonomy groups of torsion-free affine connections, revealing their finite-dimensional moduli spaces and local symmetries.

Contribution

It establishes the occurrence of specific $ ext{E}_7^{(a)}$ groups as holonomies and analyzes the structure and symmetries of the associated moduli spaces.

Findings

$ ext{E}_7^{(a)}$ groups occur as holonomies of torsion-free affine connections

Moduli spaces of these connections are finite dimensional

Each connection has a positive-dimensional local symmetry group

Abstract

It is proved that the Lie groups $\E_7^{(5)}$ and $\E^{(7)}_7$ represented in $\R^{56}$ and the Lie group $\E_7^{\C}$ represented in $\R^{112}$ occur as holonomies of torsion-free affine connections. It is also shown that the moduli spaces of torsion-free affine connnections with these holonomies are finite dimensional, and that every such connection has a local symmetry group of positive dimension.