In this research paper, it states a math-physical description of the particular instance of Atiyah-(Patodi)-Singer index theorem. In a footnote 2, it says:
If one conformally compactifies $S^4 \times R$ to the five-sphere $S^5$, then on $S^5$ one can choose the instanton field to be invariant (up to a gauge transformation) under an SU(3) subgroup of the symmetry group $O(6)$ of the five-sphere. The fermion zero mode is then the unique SU(3) invariant spinor field that can be defined.
I am hoping to understand the mathematical side of the story given above thus I post here in Math.SE (instead of Physics.SE).
Question:
What is the precise form of the instanton field living on the $S^5$?
What is the unique SU(3) invariant spinor field and how that explicitly relates to the counting of fermion zero modes?
Attempt: I suppose that we can find the instanton field in terms of SU(3) group (with gauge field $A$), and the SU(3) invariant spinor field $\psi$ ouples to the $A$, while integrating out the spinor field, we get the fermion determinant of the spinor operator (like Dirac operator $D$). The zero eigenvalues of the determinant correspond to the zero modes.
