Seiberg-Witten equations for spinorial connection

Question

Let $(M,g)$ be a spin manifold with spinorial connection $\nabla$. The Seiberg-Witten equations for spinorial connection are:

$$\nabla_Y \psi = -i g(Y,X) \psi$$

$$dX^*_+= i < (YZ-ZY).\psi ,\psi >$$

With $\psi$ a spinor and $X$ a vector field, $Y,Z$ are variable vector fields.

The gauge group is $f: M \rightarrow S^1$, it acts on the solutions:

$$f.(X,\psi )= (X+i \frac{df^*}{f}, f \psi)$$

The moduli space is the quotient by the action of the gauge group.

Can we make Seiberg-Witten theory for spinorial connection?

Comments

I don't understand the question. There's already a spinorial connection in the equations you provide...