G2 manifold

In differential geometry, a G₂ manifold or Joyce manifold is a seven-dimensional Riemannian manifold with holonomy group contained in 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 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 $\phi$ , the associative form. The Hodge dual, $\psi=*\phi$ is then a parallel 4-form, the coassociative form. These forms are calibrations in the sense of Reese Harvey and H. Blaine Lawson,{{citation | first1 = Reese | last1 = Harvey | first2 = H. Blaine | last2 = Lawson|author-link2=H. Blaine Lawson | title = Calibrated geometries | journal = Acta Mathematica | volume = 148 | year = 1982 | pages = 47–157 | doi=10.1007/BF02392726 |mr=0666108| doi-access = free }}. and thus define special classes of 3- and 4-dimensional submanifolds.

Properties

All $G_2$ -manifold are 7-dimensional, Ricci-flat, orientable spin manifolds. In addition, any compact manifold with holonomy equal to $G_2$ has finite fundamental group, non-zero first Pontryagin class, and non-zero third and fourth Betti numbers.

History

The fact that $G_2$ 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.{{citation | first =Edmond| last = Bonan| author-link=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}}.

The first local examples of 7-manifolds with holonomy $G_2$ were finally constructed around 1984 by Robert Bryant, and his full proof of their existence appeared in the Annals in 1987.{{citation | last = Bryant | first = Robert L. | author-link=Robert Bryant (mathematician)|title = Metrics with exceptional holonomy | journal = Annals of Mathematics | issue = 2 | volume = 126 | year = 1987 | pages = 525–576 | doi = 10.2307/1971360 | jstor = 1971360}}. Next, complete (but still noncompact) 7-manifolds with holonomy $G_2$ were constructed by Bryant and Simon Salamon in 1989.{{citation | last1 = Bryant | first1 = Robert L. |author-link1=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 | issue = 3 | pages = 829–850 | doi = 10.1215/s0012-7094-89-05839-0|mr=1016448 }}. The first compact 7-manifolds with holonomy $G_2$ were constructed by Dominic Joyce in 1994. Compact $G_2$ manifolds are therefore sometimes known as "Joyce manifolds", especially in the physics literature.{{citation | first = Dominic D. | last = Joyce | author-link=Dominic Joyce| title = Compact Manifolds with Special Holonomy | series = Oxford Mathematical Monographs | publisher = Oxford University Press | isbn = 0-19-850601-5 | year = 2000}}. In 2013, it was shown by M. Firat Arikan, Hyunjoo Cho, and Sema Salur that any manifold with a spin structure, and, hence, a $G_2$ -structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with $G_2$ -structure.{{citation | first1 = M. Firat | last1 = Arikan | first2 = Hyunjoo | last2 = Cho | first3 = Sema | last3 = Salur | author3-link=Sema Salur|title = Existence of compatible contact structures on $G_2$ -manifolds | journal = Asian Journal of Mathematics | issue = 2 | volume = 17 | year = 2013 | pages = 321–334 | doi = 10.4310/AJM.2013.v17.n2.a3 | arxiv = 1112.2951 | s2cid = 54942812 }}. In the same paper, it was shown that certain classes of $G_2$ -manifolds admit a contact structure.

In 2015, a new construction of compact $G_2$ 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.{{cite journal|last1=Corti|first1= Alessio|author1-link=Alessio Corti|last2= Haskins|first2= Mark|last3= Nordström|first3= Johannes|last4= Pacini|first4= Tommaso |year=2015|title={{math|G₂}}-manifolds and associative submanifolds via semi-Fano 3-folds|journal=Duke Mathematical Journal |volume=164|issue= 10|pages=1971–2092|doi= 10.1215/00127094-3120743|s2cid= 119141666|url= http://opus.bath.ac.uk/44698/1/g2m_duke_accepted.pdf}}

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 $G_2$ 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. Recently it was shown that almost contact structures (constructed by Sema Salur et al.) play an important role in $G_2$ geometry".{{citation

| last1 = de la Ossa | first1 = Xenia | author1-link = Xenia de la Ossa

| last2 = Larfors | first2 = Magdalena

| last3 = Magill | first3 = Matthew

| arxiv = 2101.12605

| doi = 10.4310/atmp.2022.v26.n1.a3

| issue = 1

| journal = Advances in Theoretical and Mathematical Physics

| mr = 4504848

| pages = 143–215

| title = Almost contact structures on manifolds with a {{math|G₂}} structure

| volume = 26

| year = 2022}}

G2 manifold

Properties

History

Connections to physics

See also

References

Further reading