# Why boundary conditions of an open string involve the time derivative?

I am trying to understand the boundary conditions of an open string stretching from one brane to another, in TIIA theory. Let's consider to D6-branes which spans a line along the $(x_4,x_5)$ plane and forms the angle $\theta_1$, then it is said that the boundary conditions are:

$$\partial_\sigma X^4(\tau,\sigma)|_{\sigma=0}=0\\ \partial_\tau X^5(\tau,\sigma)|_{\sigma=0}=0$$

The first constraint seems reasonable since it's just a Neumann boundary condition, but I don't understand why the second constraint involves a time derivative. I would expect a Dirichlet boundary condition since the string would be stuck along a line spanned by the D6-brane. So what is the explanation?

I'm familiar with the derivation of the possible boundary conditions computing the variation of the action, but in that case we fix the value of $X^\mu$ at the initial and final values of $\tau$, leaving $\partial_\tau X^\mu$ unconstrained. I'm guessing the problem involves T-duality but I don't know how.

• For Dirichlet b.c. you fix the values of $X^\mu$ at the initial and final values of $\sigma$. This is equivalent to the requirement that $\partial_\tau X^\mu$ vanishes on the initial and final values of $\sigma$. – Prahar Jul 4 '17 at 16:10