Cayley 4-form Comass and Triality Isomorphisms
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چکیده
The Cayley 4-form ωCa is of fundamental importance both in the calibration theory of R. Harvey and H. B. Lawson [HL82], who calculated its comass ‖ωCa‖ using the non-associative algebra of the Cayley numbers, and in exceptional holonomy of type Spin(7) studied by R. Bryant [Br87] and D. Joyce [Jo00]. The Cayley form is self-dual and the triality property of SO(8) relates the representation on self-dual 4-forms to the representation on traceless symmetric matrices. Following an idea of J. Dadok, R. Harvey, and F. Morgan, we use this relation to calculate ‖ωCa‖. The comass of a 4-form is the maximum of the pairing with decomposable 4-forms of norm 1, that is, the maximum over an SO(8) orbit. Therefore ‖ωCa‖ can be calculated by evaluating the pairing of
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تاریخ انتشار 2008