Lichnerowicz formula

The Lichnerowicz formula (also known as the Lichnerowicz–Weitzenbock formula) is a fundamental piece of Seiberg–Witten theory and gauge theory. It is used in a great many proofs on the subject, and gives a relationship between spinors, scaler curvature and connection through the Dirac operator and covariant derivative.

Given a complex spin structure on a manifold "M", a spinor bundle "W"^± with section $$phi, and a connection "A" on its determinant line bundle "L", the Lichnerowicz formula is

:$$D_{A}^{*}D_{A}phi= abla _A^{*} abla_{A}phi+frac{1}{4}Rphi+frac{1}{2}langle F_{A}^{+},phi angle.

Here, $$D_A is the Dirac operator $$D_A:Gamma (W^+) o Gamma (W^-), and $$abla _A is the covariant derivative associated with the connection A, $$abla _A: Gamma (W^+) o Gamma(W^+otimes T_M^*). $$R is the usual scalar curvature (a contraction of the Ricci tensor) and $$F_A^+ is the self-dual part of the curvature of A. The asterisks denote the adjoint of the quantity and the brackets $$langle , angle denote the Clifford action.

References

*
*