Lichnerowicz Formula

$$D^*D\psi=\nabla^*\nabla\psi+{\textstyle{1\over 4}}R\psi-{\textstyle{1\over 2}}F_L^+(\psi),$$

where $D$ is the Dirac operator $D:\Gamma(W^+)\to\Gamma(W^-)$, $\nabla$ is the Covariant Derivative on Spinors, $R$ is the Curvature Scalar, and $F_L^+$ is the self-dual part of the curvature of $L$.

See also Lichnerowicz-Weitzenbock Formula

References

Donaldson, S. K. ``The Seiberg-Witten Equations and 4-Manifold Topology.'' Bull. Amer. Math. Soc. 33, 45-70, 1996.