G2 manifold

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The correct title of this article is G₂ manifold. It features superscript or subscript characters that are substituted or omitted because of technical limitations.

A G₂ manifold, also known as a Joyce manifold, is a seven-dimensional Riemannian manifold with holonomy group G₂. The group $G 2$ is one of the five exceptional simple Lie groups. It can be described as the automorphism group of the octonions, or equivalently, as a proper subgroup of SO(7) that preserves a spinor in the eight-dimensional spinor representation. G₂ manifolds are Ricci-flat. The name is for Dominic Joyce.

These manifolds are important in string theory. They break the original supersymmetry to 1/8 of the original amount. For example, M-theory compactified on a $G 2$ manifold leads to a realistic four-dimensional (11-7=4) theory with N=1 supersymmetry. The resulting low energy effective supergravity contains a single supergravity supermultiplet, a number of chiral supermultiplets equal to the third Betti number of the $G 2$ manifold and a number of U(1) vector supermultiplets equal to the second Betti number.