Manifolds of G_2 Holonomy from N=4 Sigma Model

TL;DR

This paper constructs 7-dimensional G_2 holonomy manifolds as quotients of 8-dimensional hyper-Kahler spaces derived from 2D N=4 sigma models with ADE gauge symmetries, revealing geometric structures linked to Dynkin diagrams.

Contribution

It introduces a method to generate G_2 holonomy manifolds using 2D N=4 sigma models with ADE gauge charges, expanding the toolkit for constructing special holonomy spaces.

Findings

Explicit examples of G_2 manifolds as quotients of hyper-Kahler spaces.

Connection between ADE Cartan matrices and geometric structures of G_2 manifolds.

Description of the topology of the resulting G_2 spaces, including cone and bundle structures.

Abstract

Using two dimensional (2D) N=4 sigma model, with $U(1)^r$ gauge symmetry, and introducing the ADE Cartan matrices as gauge matrix charges, we build " toric" hyper-Kahler eight real dimensional manifolds X_8. Dividing by one toric geometry circle action of X_8 manifolds, we present examples describing quotients $X_7={X_8\over U(1)}$ of G_2 holonomy. In particular, for the A_r Cartan matrix, the quotient space is a cone on a $ {S^2}$ bundle over r intersecting $\bf WCP^2_{(1,2,1)}$ projective spaces according to the A_r Dynkin diagram.