G2 manifold
Template:CS1 config In differential geometry, a G2 manifold or Joyce manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group <math>G_2</math> 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 special orthogonal group SO(7) that preserves a spinor in the eight-dimensional spinor representation or lastly as the subgroup of the general linear group GL(7) which preserves the non-degenerate 3-form <math>\phi</math>, the associative form. The Hodge dual, <math>\psi=*\phi</math> is then a parallel 4-form, the coassociative form. These forms are calibrations in the sense of Reese Harvey and H. Blaine Lawson,<ref>Template:Citation.</ref> and thus define special classes of 3- and 4-dimensional submanifolds.
PropertiesEdit
All <math>G_2</math>-manifold are 7-dimensional, Ricci-flat, orientable spin manifolds. In addition, any compact manifold with holonomy equal to <math>G_2</math> has finite fundamental group, non-zero first Pontryagin class, and non-zero third and fourth Betti numbers.
HistoryEdit
The fact that <math>G_2</math> might possibly be the holonomy group of certain Riemannian 7-manifolds was first suggested by the 1955 classification theorem of Marcel Berger, and this remained consistent with the simplified proof later given by Jim Simons in 1962. Although not a single example of such a manifold had yet been discovered, Edmond Bonan nonetheless made a useful contribution by showing that, if such a manifold did in fact exist, it would carry both a parallel 3-form and a parallel 4-form, and that it would necessarily be Ricci-flat.<ref>Template:Citation.</ref>
The first local examples of 7-manifolds with holonomy <math>G_2</math> were finally constructed around 1984 by Robert Bryant, and his full proof of their existence appeared in the Annals in 1987.<ref>Template:Citation.</ref> Next, complete (but still noncompact) 7-manifolds with holonomy <math>G_2</math> were constructed by Bryant and Simon Salamon in 1989.<ref>Template:Citation.</ref> The first compact 7-manifolds with holonomy <math>G_2</math> were constructed by Dominic Joyce in 1994. Compact <math>G_2</math> manifolds are therefore sometimes known as "Joyce manifolds", especially in the physics literature.<ref>Template:Citation.</ref> In 2013, it was shown by M. Firat Arikan, Hyunjoo Cho, and Sema Salur that any manifold with a spin structure, and, hence, a <math>G_2</math>-structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with <math>G_2</math>-structure.<ref name ="arikanetal">Template:Citation.</ref> In the same paper, it was shown that certain classes of <math>G_2</math>-manifolds admit a contact structure.
In 2015, a new construction of compact <math>G_2</math> manifolds, due to Alessio Corti, Mark Haskins, Johannes Nordstrőm, and Tommaso Pacini, combined a gluing idea suggested by Simon Donaldson with new algebro-geometric and analytic techniques for constructing Calabi–Yau manifolds with cylindrical ends, resulting in tens of thousands of diffeomorphism types of new examples.<ref>Template:Cite journal </ref>
Connections to physicsEdit
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. The resulting low energy effective supergravity contains a single supergravity supermultiplet, a number of chiral supermultiplets equal to the third Betti number of the <math>G_2</math> manifold and a number of U(1) vector supermultiplets equal to the second Betti number. Recently it was shown that almost contact structures (constructed by Sema Salur et al.)<ref name="arikanetal" /> play an important role in <math>G_2</math> geometry".<ref>Template:Citation</ref>