A 3-dimensional submanifold $\Sigma_3 \hookrightarrow X_7$ of a G₂-manifold $X_7$ is called associative if it is a calibrated submanifold, hence if the canonical 3-form $\phi$ of the $G_2$-manifold (tangent space-wise the associative 3-form) restricts to the volume form on $\Sigma_3$.
Accordingly a 4-dimensional submanifold $\Sigma_4 \hookrightarrow X_7$ is called coassociative if it is a calibrated submanifold with respect to the coassociative 4-form $\star_X \phi$.
associative 3-manifolds appear as supersymmetric cycles relevant for membrane instantons for M2-branes wrapping them;
coassociative 4-manifolds appear as fibers of compactifications dual to F-theory on CY4-folds (Gukov-Yau-Zaslow 02).
Ian Weiner, Associative 3-manifolds in $\mathbb{R}^7$, 2001 (pdf)
Selman Akbulut, Sema Salur, Associative submanifolds of a $G_2$-manifold (pdf)
Damien Gayet, Smooth moduli spaces of associative submanifolds (arXiv:1011.1744)
On associative submanifolds of the 7-sphere:
Discussion of wrapped branes on associative 3-cycles is in
Further discussion specifically of M5-branes wrapped on associative 3-cycle which are either the 3-sphere or a hyperbolic 3-manifold is in
Bobby Acharya, Jerome Gauntlett, Nakwoo Kim, Fivebranes Wrapped On Associative Three-Cycles, Phys.Rev. D63 (2001) 106003 (arXiv:hep-th/0011190)
Sergei Gukov, Shing-Tung Yau, Eric Zaslow, Duality and Fibrations on $G_2$ Manifolds (arXiv:hep-th/0203217)
Last revised on July 18, 2024 at 11:31:39. See the history of this page for a list of all contributions to it.