Bochner-Martinelli Formula for Higher Spin Operators of Several ℝ⁶ Variables

Abstract

The higher spin operator of several ℝ⁶ variables is an analogue of the $\overline \partial$ operator in theory of several complex variables. The higher spin representation of ${\mathfrak{s}\mathfrak{o}_6}(\mathbb{C})$ is ⊙ ^kℂ⁴ and the higher spin operator ${{\cal D}_k}$ acts on ⊙ ^kℂ⁴-valued functions. In this paper, the authors establish the Bochner-Martinelli formula for higher spin operator ${{\cal D}_k}$ of several ℝ⁶ variables. The embedding of ℝ⁶ⁿ into the space of complex 4n × 4 matrices allows them to use two-component notation, which makes the spinor calculus on ℝ⁶ⁿ more concrete and explicit. A function annihilated by ${{\cal D}_k}$ is called k-monogenic. They give the Penrose integral formula over ℝ⁶ⁿ and construct many k-monogenic polynomials.

Keywords

Higher spin operator / k-Monogenic / Bochner-Martinelli formula / Penrose integral formula

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