Green's function for Dirac operator on $S^4$

Question

Let $S^4$ be a round sphere of radius 1 (with the standard Riemannian round metric), and let $D_\text{F} \equiv \gamma^\mu \nabla_\mu$ be the Dirac operator on $S^4$, acting on the usual spinors for the Riemannian metric.

I wonder what is its Green's function $G_\text{F}$ on $S^4$, given by the equation $ D_\text{F} G_\text{F}(x)_{\alpha \beta} = \delta_{S^4}(x) \delta_{\alpha \beta}?$

I happen to know the Green's function for the conformally coupled scalar, namely Green's function $G_\text{B}$ for $$\Delta_\text{scalar} \equiv - \nabla^\mu \nabla_\mu + 2$$ acting on scalar functions. But I think $\gamma^\mu \nabla_\mu \gamma^\nu \nabla_\nu$ slightly differs from $\Delta_\text{scalar}$ by a constant proportional to the $S^4$ scalar curvature, so it's not obvious to me how to get $G_\text{F}$ from $G_\text{B}$ (which we do all the time on flat space).

So more generally, I'd like to know the Green's function for the Dirac operator on $S^n$.

References and suggestions will be great!

Answer 1

The spinor bundle on the standard sphere $S^n$ can be trivialized by Killing spinors as explained e.g. in an article by Baer. This means we have spinors $\psi_1,...,\psi_N$ such that $$ \nabla_X\psi_j=\mu X\cdot\psi_j $$ for all tangent vectors $X$ on $S^n$, where $\mu\in\{\pm\frac{1}{2}\}$, and every spinor $\Phi$ can be globally written as $\Phi=\sum_{j}f_j\psi_j$ with complex scalar functions $f_j$. Let $D$ be the Dirac operator. Analogously to Lemma 3 in the article mentioned above one gets $$ (D+\mu)^2\Phi=\sum_{j=1}^{N}\Big(\Delta_g f_j+\frac{(n-1)^2}{4}f_j\Big)\psi_j $$ and thus $$ D(D+2\mu)\Phi=\sum_{j=1}^{N}\Big(\Delta_g f_j+\frac{n(n-2)}{4}f_j\Big)\psi_j. $$ On the right hand side we have the conformal Laplace operator acting on the functions $f_j$. Therefore, if $f_j$ are multiples of the Green function for the conformal Laplace operator at a point $y\in S^n$, then $(D+2\mu)\Phi$ is a Green function for the Dirac operator at $y$.