The Dirac moduli space

Question

Let $(M,g)$ be a spin manifold, the Dirac equations are acting on $(X,\psi)$, a vector field and a spinor:

$$X.\nabla_X \psi = \alpha \nabla_X X.\psi +\nu ||X||^2 \psi$$

$${\cal D}(X.\psi)=(\alpha +1) dX.\psi + \mu X.\psi$$

with $\cal D$ the Dirac operator, $\alpha,\nu,\mu$ are constants..

The gauge group is ${\cal G}=C^{\infty}(M,{\bf R}^*_+)$ and acts:

$$f.(X,\psi)=(f^b X, f^a \psi)$$

with $\alpha =a/b$.

The moduli space is ${\cal M}(M)= S(X,\psi)/{\cal G}$.

Is the moduli space finite dimensional and can we define invariants?