G2 manifold
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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.
See also: Calabi-Yau manifold, Spin(7) manifold
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