The Dirac moduli space

Question

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

$$X.\nabla_Y \psi = (a/b) \nabla_Y X.\psi $$

$${\cal D}(X.\psi)=((a/b) +1) dX.\psi $$

$$Y.\nabla_X \psi =(a/c) \nabla_X Y.\psi$$

$${\cal D}(Y.\psi)=((a/c)+1) dY.\psi$$

with $\cal D$ the Dirac operator.

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

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

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

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