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.

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