symmetric monoidal (∞,1)-category of spectra
The Cayley 4-form $\Phi$ (Harvey-Lawson 82) is a certain differential 4-form on the real 8-dimensional space $\mathbb{R}^8$.
which constitutes an exceptional calibration of $\mathbb{R}^4$ with its Euclidean geometry.
More generally, a Spin(7)-manifold carries a globalization of this 4-form calibration, then also called a Cayley-4-form.
The stabilizer subgroup inside GL(8) of the Cayley 4-from under the action given by pullback of differential forms is the subgroup Spin(7) inside SO(8).
(Harvey-Lawson 82, Prop. 1.36)
The Cayley 4-form constitutes a calibration of the Euclidean space $\mathbb{R}^8$ (Harvey-Lawson 82)
A calibrated submanifold for $\Phi$ is also called a Cayley 4-plane (not to be confused with the Cayley plane).
The space (moduli space) of Cayley 4-planes, denoted $CAY$ (Bryant-Harvey 89, (2.19)) or $CAYLEY$ (Gluck-Mackenzie-Morgan 95, (5.20)), is hence a topological subspace of the Grassmannian of all 4-planes in 8-dimensions:
This is of codimension 4 (Harvey-Lawson 82, below (5)). In fact, this space is homeomorphic to the coset space of Spin(7) by Spin(4).Spin(3) = Spin(3).Spin(3).Spin(3) = Sp(1).Sp(1).Sp(1):
(Harvey-Lawson 82, Theorem 1.38, see also Bryant-Harvey 89, (3.19), Gluck-Mackenzie-Morgan 95, (5.20))
Moreover, the coset space of Spin(6) by Spin(3).Spin(3) $\simeq$ SO(4)
is the Grassmannian of those Cayley 4-planes which are also special Lagrangian submanifolds (BBMOOY 96, p. 7 (8 of 17)).
See also at Spin Grassmannians.
classification of special holonomy manifolds by Berger's theorem:
$\,$G-structure$\,$ | $\,$special holonomy$\,$ | $\,$dimension$\,$ | $\,$preserved differential form$\,$ | |
---|---|---|---|---|
$\,\mathbb{C}\,$ | $\,$Kähler manifold$\,$ | $\,$U(n)$\,$ | $\,2n\,$ | $\,$Kähler forms $\omega_2\,$ |
$\,$Calabi-Yau manifold$\,$ | $\,$SU(n)$\,$ | $\,2n\,$ | ||
$\,\mathbb{H}\,$ | $\,$quaternionic Kähler manifold$\,$ | $\,$Sp(n).Sp(1)$\,$ | $\,4n\,$ | $\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,$ |
$\,$hyper-Kähler manifold$\,$ | $\,$Sp(n)$\,$ | $\,4n\,$ | $\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\,$ ($a^2 + b^2 + c^2 = 1$) | |
$\,\mathbb{O}\,$ | $\,$Spin(7) manifold$\,$ | $\,$Spin(7)$\,$ | $\,$8$\,$ | $\,$Cayley form$\,$ |
$\,$G₂ manifold$\,$ | $\,$G₂$\,$ | $\,7\,$ | $\,$associative 3-form$\,$ |
Reese Harvey, H. Blaine Lawson, Calibrated geometries, Acta Math. Volume 148 (1982), 47-157 (Euclid:1485890157)
Robert Bryant, Reese Harvey, Submanifolds in Hyper-Kähler Geometry, Journal of the American Mathematical Society Vol. 2, No. 1 (Jan., 1989), pp. 1-31 (jstor:1990911)
Herman Gluck, Dana Mackenzie, Frank Morgan, Volume-minimizing cycles in Grassmann manifolds, Duke Math. J. Volume 79, Number 2 (1995), 335-404 (euclid:1077285156)
Liviu Ornea, Paolo Piccinni, Cayley 4-frames and a quaternion-Kähler reduction related to $Spin(7)$, Proceedings of the International Congress of Differential Geometry in the memory of A. Gray, held in Bilbao, Sept. 2000 (arXiv:math/0106116)
Mikhail Katz, Steven Shnider, Cayley 4-form, comass, and triality isomorphisms (arXiv:0801.0283)
Kirill Krasnov, Dynamics of Cayley Forms [arXiv:2403.16661]
On a complex and Lorentzian variant:
Last revised on April 1, 2024 at 19:27:26. See the history of this page for a list of all contributions to it.