In Hutchings and Taubes lecture note on Seiberg-Witten equation HERE, above equation (4.20) the authors claim that there is a version of Weitzenbock formula reads (where $\beta \in \Omega^{0,2}(M, E)$, M is a symplectic manifold with compatible $J$, $E$ is a line bundle with $U(1)$ connection $a$)

\begin{equation} \int {{{\left| {\bar \partial _a^*\beta } \right|}^2}} = \frac{1}{2}\int {\left( {{{\left| {\nabla _a^*\beta } \right|}^2} - i\left\langle {\omega ,{F_a}} \right\rangle {{\left| \beta \right|}^2}} \right)} \end{equation}

I try to prove it, by first guessing that it might come from something like a Kahler identity \begin{equation} {{\bar \partial }_a}\bar \partial _a^*\beta = \frac{1}{2}{\nabla _a}\nabla _a^*\beta + \left( {...} \right) \end{equation} and try to figure out if the ... matches the formula in the note. But I could not reproduce the result: the ... I got is of the form $\Lambda \left( {F_a^{2,0} \wedge \beta } \right)$, where $\Lambda = (\omega\wedge)^*$, and a (2,0)-piece of $F$ rather than a (1,1) piece shows up.

So I wonder how to prove the Weitzenbock formula above? Any reference is welcome; I google for a while but may be I am not looking at the write place.