nLab associative submanifold

Definition

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 $\star_X \phi$.

Application in string theory

In M-theory on G₂-manifolds,

• 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).

References

• 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:

• Jason Lotay, Associative Submanifolds of the 7-Sphere, Proc. London Math. Soc. (2012) 105 (6): 1183-1214 (arXiv:1006.0361)

Discussion of wrapped branes on associative 3-cycles is in

• Katrin Becker, Melanie Becker, D.R.Morrison, Hirosi Ooguri, Y.Oz, Z.Yin, Supersymmetric Cycles in Exceptional Holonomy Manifolds and Calabi-Yau 4-Folds, Nucl.Phys.B480:225-238,1996 (arXiv:hep-th/9608116)

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)