I understand that given a (pseudo-) Riemannian manifold admitting a spin structure, we may write the Schrodinger-Licnerowiz Formula in terms of the Dirac and Covariant derivative operators:
$$D^{*}D\psi=\nabla^{*}\nabla\psi+\frac{1}{4}R\psi$$
Where R is the scalar curvature and $\psi$ is a section of our spinor bundle.
Now suppose our manifold admits a $Spin_{c}=Spin(n)\times U(1)$ (complex) structure. We then have the Schrodinger-Licnerowiz Formula:
$$D^{*}D\psi=\nabla^{*}\nabla\psi+\frac{1}{4}R\psi+\frac{1}{2}<F^{+},\psi>$$
Where now $F^{+}$ is the self dual part of the curvature of the $U(1)$ part of the connection.
As I understand it, In general there can exist $Spin_{G}$ structures Where $G$ can be a compact Lie group. What is, or How do I find the associated formula For such generalized Spin structures? I would expect it to be similar to the $spin_c$ case but possibly with a term coming from the nonabelian nature of a more general $G$.
I would be happy just to find the paper for which the $Spin_c$ formula was derived (which I've been unable to do)