“G2 manifold”的意思、由来-开放百科全书

In differential geometry, a G₂ manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G₂. The group

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

, the associative form. The Hodge dual,

is then a parallel 4-form, the coassociative form. These forms are calibrations in the sense of Reese Harvey and H. Blaine Lawson,^[1] and thus define special classes of 3- and 4-dimensional submanifolds.

Properties

History

The fact that

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 then made an interesting 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.^[2]

The first local examples of 7-manifolds with holonomy

were finally constructed around 1984 by

Robert Bryant, and his full proof of their existence appeared in the Annals in 1987

Next, complete (but still noncompact) 7-manifolds with holonomy

were constructed by Bryant and Simon Salamon in 1989.^[4] The first compact 7-manifolds with holonomy

were constructed by Dominic Joyce in 1994, and compact

manifolds are sometimes known as "Joyce manifolds", especially in the physics literature.^[5] In 2013, it was shown by M. Firat Arikan, Hyunjoo Cho, and Sema Salur that any manifold with a spin structure, and, hence, a

-structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with

-structure.^[6] In the same paper, it was shown that certain classes of

-manifolds admit a contact structure.

Connections to physics

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

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

manifold and a number of U(1) vector supermultiplets equal to the second Betti number.

See also

References

1. ^{{citation | first = Reese | last = Harvey | first2 = H. Blaine | last2 = Lawson|authorlink2=H. Blaine Lawson | title = Calibrated geometries | journal = Acta Mathematica | volume = 148 | year = 1982 | pages = 47–157 | doi=10.1007/BF02392726 |mr=0666108}}.
2. ^{{citation | first =Edmond| last = Bonan| authorlink=Edmond Bonan| title = Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)| journal = Comptes Rendus de l'Académie des Sciences | volume =262| year = 1966 | pages = 127–129}}.
3. ^{Bryant, Rober L. (1987) Metrics with exceptional holonomy, Annals of Mathematics (2)126, 525–576.
4. ^{{citation | last1 = Bryant | first1 = Rober L. |authorlink1=Robert Bryant (mathematician)| first2 = Simon M. | last2 = Salamon | title = On the construction of some complete metrics with exceptional holonomy | journal = Duke Mathematical Journal | volume = 58 | year = 1989 | pages = 829–850 | doi = 10.1215/s0012-7094-89-05839-0|mr=1016448 }}.
5. ^{{citation | first = Dominic D. | last = Joyce | authorlink=Dominic Joyce| title = Compact Manifolds with Special Holonomy | series = Oxford Mathematical Monographs | publisher = Oxford University Press | isbn = 0-19-850601-5 | year = 2000}}.
6. ^{{citation | first = M. Firat | last = Arikan | first2 = Hyunjoo | last2 = Cho | first3 = Sema | last3 = Salur | title = Existence of compatible contact structures on

-manifolds | journal = Asian J. Math | issue = 2 | volume = 17 | year = 2013 | pages = 321–334 | doi = 10.4310/AJM.2013.v17.n2.a3 | publisher = International Press of Boston| arxiv = 1112.2951 }}.