How do branes come into existence according to string theory? 
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*How do branes come into existence according to string theory?

*Also what are branes made out of?
 A: I am not working on string theory and have not given a careful study on string theory either. Here is my understanding.
When we study an open string, we must assign boundary conditions for its end points.  An open string in $D$-dimensional spacetime can be described by $D$ fields, $X^\mu(\tau,\sigma)$ with $\mu=0,...,D-1$. Suppose the end points of the string have $\sigma=0,\pi$. There are two types of boundary conditions


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*Neumann boundary conditions: you fix the velocity of the end points, $\frac{\partial X^\mu}{\partial\tau}|_{\sigma=0,\pi}$ for $\mu=0,1,..., p$. 

*Dirichlet boundary conditions: you fix the position of the end points, $X^\mu(\tau)|_{\sigma=0,\pi}$ for $\mu=p+1,..., D-1$.


The first type boundary condition simply means that the string end points move freely along the directions $X^\mu$ with $\mu=1,...,p$ (note for the time $X^0$, we must have Neumann boundary condition). The second type boundary condition means that the string end points are fixed at some position for $X^\mu$ with $\mu=p+1,...,D-1$.
Now, for the latter, what does the string end points attach to? There must be something for the strings to end at. Further study can show that the object carry energy-momentum in order to have energy-momentum conservation. This is the $p$-dimensional D-brane. The dimension of the brane is equal to the number of directions that the string end points can move freely.
What are branes made out of? I would say they are made of the string field. In string field theory, strings are excitations of string field. As the normal field theory, one can have extended solutions, like solitons, in the string field theory which I think are the D-branes.     
