Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 183

Kaehlerian E-spinors

K.-D. Kirchberg

If $M^{2m}$ is a K"ahler manifold with complex structure $J$, scalar curvature $R ot= 0$, and spinor bundle $S$, then, for $r in {0,1,...,m}$ and $psi in Gamma (S)$, the differential equations of the second order [ abla_X Dą{pm} psi + frac{R}{8(2r-1)} (X mp iJX) cdot psi = 0 hspace{4cm} (E^r_{pm}) ] are considered. They must be satisfied for each real vector field $X$. The equations $E^{r}_{+}$ and $E^{r}_{-}$ are equivalent in the sense that there is a canonical antilinear isomorphism between the corresponding $C!!!I$-vector spaces of solutions $epsilon^r_{+}$ and $epsilon^r_{-}$. The non-trivial elements of these spaces are called K"ahlerian $E$-spinors of type $(r,+)$ and $(r,-)$, respectively. There is a basic relation between K"ahlerian $E$-spinors and K"ahlerian twistor spinors that is essential for the proofs of all results of the paper.

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