# A-brane boundary conditions

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?

I noticed that in fact my question was straightforward, and not specific to topological string theory. It solves the general problem of an open string (with the worldsheet topology of a disk) with endpoints on a stack of branes.

Sketch of solution:

Impose Dirichlet boundary conditions for $$\Phi$$ on $$NY$$. The computation is developed in the texbook "String theory on a nutshell" (equation 4.16.3); up to a constant it reads $$(\partial-\bar{\partial})\Phi=0,$$ on $$z=\bar{z}.$$

The correct boundary conditions for the directions $$TY$$ are $$[g(\partial-\bar{\partial})-F(\partial+\bar{\partial})]\Phi=0.$$

That expression can be found in String Theory and Noncommutative Geometry (equations 2.2 and 2.3); or explicitly worked on the textbook "A first course in string theory" ( second edition, chapter 16).

Then the operator that impose the correct boundary conditions on the whole $$TX$$ is $$[ (\partial-\bar{\partial}) \oplus (g(\partial-\bar{\partial})-F(\partial+\bar{\partial})) ] \Phi=0,$$

the required solution.