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The twistor theory of the Hermitian Hurwitz pair \((\mathbb C^4(I_{2,2}), \mathbb R^5(I_{2,3}))\). (English) Zbl 0917.15022

The authors find an analogue of the twistor theory. The so-called duality theorem for Hurwitz twistors is proved: For this special Hermitian Hurwitz pair mentioned in the title a \(1-1\)-mapping between twistors on \((1,3)\)-space and \((2,2)\)-space is deduced. As one of the main results it is shown that every solution of the spinor equation with \(\text{spin}\, n/2\) on \((2,2)\)-space can be represented as a \(\partial\)-harmonic 1-form with respect to the Hermitian indefinite metric. Further a relationship between complex analysis and spinor theory on a suitable open set is pointed out (locally). In this way some kind of a semi-global version of the Penrose theory is obtained. A global version of the original Penrose theory can be also deduced applying the Penrose transformation. The authors introduce the notions of the Dirac-Hurwitz spinor of \(\text{spin}\,1/2\) and the Weyl-Hurwitz spinor related to the Hurwitz spinor. The results in this contribution can be used to get a relative de Rham form in order to describe the correspondence between the Dolbeault cohomology groups and spinor functions.

MSC:

15A66 Clifford algebras, spinors
58A12 de Rham theory in global analysis
55N99 Homology and cohomology theories in algebraic topology
81R25 Spinor and twistor methods applied to problems in quantum theory
32L25 Twistor theory, double fibrations (complex-analytic aspects)
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