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G2 manifold

A G₂ manifold, also known as a Joyce manifold, is a seven-dimensional Riemannian manifold with holonomy group G₂. The group <math>G_2<math> is one 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.

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 <math>G_2<math> manifold leads to a realistic four-dimensional (11-7=4) theory with N=1 supersymmetry.

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