I'm studying Aigner's paper 'Existence of the Ginzburg-Landau Vortex Number' (2001) and I have some difficulties to prove the equality (3.1) , which is
$(|F_A|^2+|d_A\phi|^2+|\frac12(1-|\phi|^2)|^2)dxdy=(2|\bar\partial_A\phi|^2+|\star F_A-\frac12(1-|\phi|^2)|^2)dxdy+F_A-d(i\bar\phi d_A\phi),$
where $\phi$ is a complex-valued function in $\mathbb{R^2}$ with $d_A\phi=\partial_A\phi+\bar\partial_A\phi$, $A$ is a complex 1-form in $\mathbb{R^2}$ and $F_A=dA$. In other words, $A$ is the connection and $F_A$ the curvature. How can I prove the equality?
