A moduli space in spinorial geometry

Question

I consider a spin manifold $M$, with the Dirac operator $D$ and spinors $\psi$. I consider the differential equation:

$$ D \psi = \frac {1} {2} (\frac {d < \psi, \psi >}{<\psi, \psi>})^* .\psi $$

with the Clifford multiplication of the vectors. 

Then it can be showed that the real gauge group ${\cal C }^{\infty} (M,R^*)$ acts over the solutions. Have I a nice moduli space (finite dimensional and compact)?