This question concerns the boundary conditions that A-branes solve.
Consider the following problem: Suppose that an A-brane wraps a submanifold $Y$ of $X$. Let $\mathcal{L} \rightarrow Y$ be a rank one Chan-Paton bundle over $Y$ equipped with a unitary connection. What are the allowed boundary conditions on the endpoints of an open string with embeding functions $\Phi$, worldsheet fermions $\psi$ and whose endpoints lie on $Y$?.
According to the paper "Remarks on A-branes, Mirror Symmetry, and the Fukaya category" (equation 4, page 8) the answer is that at $z=\bar{z}$ (if the open string worldsheet is the upper-half complex plane), $\Phi$ and the fermions $\psi, \bar{\psi}$ should obey $$\partial_{z}\Phi = R[\partial_{\bar{z}}\Phi] \ \ \ , \ \ \ \psi= R[\bar{\psi}].$$ Where the operator $R$ is defined as $$R=(-1)_{NY}\oplus (g-F)^{-1}(g+F).$$
and a local decomposition of $TX = NY \oplus TY$ is assumed, g is the restriction of the metric of $X$ to $Y$, and $F$ is the curvature tensor of the unitary connection on $\mathcal{L} \rightarrow Y$.
Question: How can I derive that answer?