# The twistor theory of the Hermitian Hurwitz pair (ℂ^{4}(*I* _{2,2}),ℝ(*I* _{2,3}))

- 32 Downloads
- 1 Citations

## Abstract

An analogue of the twistor theory is given for the Hermitian Hurwitz pair(ℂ^{4}(*I* _{2,2}),ℝ(*I* _{2,3})). In Sect. 2 a concept of Hurwitz twistors is introduced and a counterpart of the Penrose correspondence is obtained. It is proved that there exists a one-to-one correspondence between the twistors on the (1,3)-space and the (2,2)-space, which is called the duality theorem for Hurwitz twistors (Theorem 1). In Sect. 3. a concept of spinor equations is introduced for an Hermitian Hurwitz pair (abbreviated as HHP) and the duality theorem for solutions of the spinor equations is proved (Theorem 2). In Sect. 4 we give an elementary proof of the Penrose theory on the base of our Key Lemma. Then we can give the desired correspondence explicitly. In sect. 5 we consider the Penrose theory in the context of HHPs. At first we give a local version. It is proved that every solution of the spinor equation on the (2,2)-space can be represented as a ∂-harmonic one-form. By use of this result, we can get a direct relationship between the complex analysis and spinor theory on some open set*M* ^{+}, which is called as “semi-global version” of the Penrose theory (Theorem 7). Moreover, we can get the original Penrose theory by use of the Penrose transformation (Theorem 5).

## 1991 Mathematics Subject Classification

32L25 Secondary 346C20## Key words

Clifford analysis Hurwitz pair twistor space with an indefinite scalar product## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Furuoya I., S. Kanemaki, J. Ławrynowicz, and O. Suzuki, Hermitian Hurwitz pairs in: “Deformations of Mathematical Structures II. Hurwitz-Type Structures and Applications to Surface Physics”, J. Ławrynowicz (ed.). Kluwer Academic, Dordrecht 1994, pp. 135–154Google Scholar
- [2]Griffith P. A. and J. Harris “The Principles of Algebraic Geometry” Wiley, New York 1978Google Scholar
- [3]Ławrynowicz J. and O. Suzuki, The duality theorem for the Hurwitz pairs of bidimension (8,5) and the Penrose theory, in: “Deformations of Mathematical Structures II. Hurwitz-Type Structures and Applications to Surface Physics”, J. Ławrynowicz (ed.), Kluwer Academic, Dordrecht 1994, pp. 209–212Google Scholar
- [4]Ławrynowicz J. and O. Suzuki, Hurwitz-type and space-time-type duality theorems for Hermitian Hurwitz pairs, in “Complex Analysis and Its Applications”, E. Ramirez de Arellano, M. Shapiro and N. Vasilevski (eds.), Kluwer Academic, Dordrecht 1998, to appearGoogle Scholar
- [5]Ławrynowicz J. and O. Suzuki, Hurwitz duality theorems for Fueter and Dirac equations,
*Advances Appl. Clifford Algebras***7**(2) 1997, pp 113–132MATHCrossRefGoogle Scholar - [6]
- [7]Ward R., O. Raymond and R. O. Wells, Jr. “Twistor Geometry and Field Theory”, Cambridge Univ. Press, Cambridge, 1990Google Scholar
- [8]Wells R. O. Jr., Complex manifolds and mathematical physics,
*Bull. Amer. Math. Soc.***1**no. 2, 1979Google Scholar - [9]Wells R. O. Jr., “Complex Geometry in Mathematical Physics,” Séminaire de Mathématiques supérieures — Université de Montréal, Les presses de l’Université de Montréal, Montréal 1982Google Scholar