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,^[1] 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.^[2]

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.^[3] Next, complete (but still noncompact) 7-manifolds with holonomy $G_{2}$ were constructed by Bryant and Simon Salamon in 1989.^[4] 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.^[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 $G_{2}$ -structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with $G_{2}$ -structure.^[6] 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.^[7]

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.)^[6] play an important role in $G_{2}$ geometry".^[8]

References

^ Harvey, Reese; Lawson, H. Blaine (1982), "Calibrated geometries", Acta Mathematica, 148: 47–157, doi:10.1007/BF02392726, MR 0666108.
^ Bonan, Edmond (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)", Comptes Rendus de l'Académie des Sciences, 262: 127–129.
^ Bryant, Robert L. (1987), "Metrics with exceptional holonomy", Annals of Mathematics, 126 (2): 525–576, doi:10.2307/1971360, JSTOR 1971360.
^ Bryant, Robert L.; Salamon, Simon M. (1989), "On the construction of some complete metrics with exceptional holonomy", Duke Mathematical Journal, 58 (3): 829–850, doi:10.1215/s0012-7094-89-05839-0, MR 1016448.
^ Joyce, Dominic D. (2000), Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-850601-5.
^ ^a ^b Arikan, M. Firat; Cho, Hyunjoo; Salur, Sema (2013), "Existence of compatible contact structures on $G_{2}$ -manifolds", Asian Journal of Mathematics, 17 (2): 321–334, arXiv:1112.2951, doi:10.4310/AJM.2013.v17.n2.a3, S2CID 54942812.
^ Corti, Alessio; Haskins, Mark; Nordström, Johannes; Pacini, Tommaso (2015), " $G 2$ -manifolds and associative submanifolds via semi-Fano 3-folds" (PDF), Duke Mathematical Journal, 164 (10): 1971–2092, doi:10.1215/00127094-3120743, S2CID 119141666
^ de la Ossa, Xenia; Larfors, Magdalena; Magill, Matthew (2022), "Almost contact structures on manifolds with a $G 2$ structure", Advances in Theoretical and Mathematical Physics, 26 (1): 143–215, arXiv:2101.12605, doi:10.4310/atmp.2022.v26.n1.a3, MR 4504848

v t e String theory
Background	Strings Cosmic strings History of string theory First superstring revolution Second superstring revolution String theory landscape
Theory	Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string Type IIA string Type IIB string Heterotic string N=2 superstring F-theory String field theory Matrix string theory Non-critical string theory Non-linear sigma model Tachyon condensation RNS formalism GS formalism
String duality	T-duality S-duality U-duality Montonen–Olive duality
Particles and fields	Graviton Dilaton Tachyon Ramond–Ramond field Kalb–Ramond field Magnetic monopole Dual graviton Dual photon
Branes	D-brane NS5-brane M2-brane M5-brane S-brane Black brane Black holes Black string Brane cosmology Quiver diagram Hanany–Witten transition
Conformal field theory	Virasoro algebra Mirror symmetry Conformal anomaly Conformal algebra Superconformal algebra Vertex operator algebra Loop algebra Kac–Moody algebra Wess–Zumino–Witten model
Gauge theory	Anomalies Instantons Chern–Simons form Bogomol'nyi–Prasad–Sommerfield bound Exceptional Lie groups (G₂, F₄, E₆, E₇, E₈) ADE classification Dirac string p-form electrodynamics
Geometry	Worldsheet Kaluza–Klein theory Compactification Why 10 dimensions? Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G₂ manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics) Twisted K-theory
Supersymmetry	Supergravity Eleven-dimensional supergravity Type I supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup
Holography	Holographic principle AdS/CFT correspondence
M-theory	Matrix theory Introduction to M-theory
String theorists	Aganagić Arkani-Hamed Atiyah Banks Berenstein Bousso Curtright Dijkgraaf Distler Douglas Duff Dvali Ferrara Fischler Friedan Gates Gliozzi Gopakumar Green Greene Gross Gubser Gukov Guth Hanson Harvey 't Hooft Hořava Gibbons Kachru Kaku Kallosh Kaluza Kapustin Klebanov Knizhnik Kontsevich Klein Linde Maldacena Mandelstam Marolf Martinec Minwalla Moore Motl Mukhi Myers Nanopoulos Năstase Nekrasov Neveu Nielsen van Nieuwenhuizen Novikov Olive Ooguri Ovrut Polchinski Polyakov Rajaraman Ramond Randall Randjbar-Daemi Roček Rohm Sagnotti Scherk Schwarz Seiberg Sen Shenker Siegel Silverstein Sơn Staudacher Steinhardt Strominger Sundrum Susskind Townsend Trivedi Turok Vafa Veneziano Verlinde Verlinde Wess Witten Yau Yoneya Zamolodchikov Zamolodchikov Zaslow Zumino Zwiebach

G₂ manifold

Properties

History

Connections to physics

See also

References

Further reading