Dp Brane Open String Boundary Condition

In flat Type II theory a static Dp-brane is the timelike affine plane $W={\mathbb{R}}^{1,p}\times \{X^a=y_0^a\}$ in ten-dimensional Minkowski space with metric $\eta =\mathrm{diag}(-1,+1,\dots ,+1)$, where $\alpha =0,\dots ,p$ are tangent indices, $a=p+1,\dots ,9$ are normal indices, $0\le p\le 9$, $p$ is even in Type IIA and odd in Type IIB, and $y_0^a$ are fixed target-space coordinates of length dimension.

For the open strip $\mathrm{\Sigma }=I\times [0,\pi ]$, where $I\subseteq \mathbb{R}$ is an open time interval, with dimensionless worldsheet coordinates $(\tau ,\sigma )$ and a map $X$ that is $C^2$ up to the timelike boundary, the conformal-gauge endpoint data are Neumann, ${\mathrm{\partial }}_{\sigma }{X}^{\alpha }(\tau ,0)={\mathrm{\partial }}_{\sigma }{X}^{\alpha }(\tau ,\pi )=0$, tangent to $W$, and Dirichlet, $X^a(\tau ,0)=X^a(\tau ,\pi )=y_0^a$, normal to $W$. This is boundary data for the free open-string coordinate fields only; it does not specify brane dynamics, RR charge, or a backreacted solution.

Flat Bosonic String Model

Open String Boundary Variation

The strip picture records the domain and endpoint map. The two long edges map into $W$; the diagram is a topological schematic and does not represent a target-space metric or a brane trajectory.