# What are strings with endpoint boundary conditions NN, DD, ND and DN?

http://en.wikipedia.org/wiki/D-brane says, "The equations of motion of string theory require that the endpoints of an open string (a string with endpoints) satisfy one of two types of boundary conditions: The Neumann boundary condition, corresponding to free endpoints moving through spacetime at the speed of light, or the Dirichlet boundary conditions, which pin the string endpoint. Each coordinate of the string must satisfy one or the other of these conditions. There can also exist strings with mixed boundary conditions, where the two endpoints satisfy NN, DD, ND and DN boundary conditions. If p spatial dimensions satisfy the Neumann boundary condition, then the string endpoint is confined to move within a p-dimensional hyperplane. This hyperplane provides one description of a Dp-brane."

First, I am confused by this paragraph. The sentence, "There can also exist strings with mixed boundary conditions, where the two endpoints satisfy NN, DD, ND and DN boundary conditions," seems to indicate there are four mixed boundary conditions; in which case, prior sentences describe strings with two Neumann boundary conditions and strings with two Dirichlet boundary conditions that are not NN and DD. Does this paragraph descirbe four or six kinds of strings based on endpoint boundary conditions?

Second, does D-brane theory characterize what one might expect to see, if a large enough collider could be made to detect strings with ends having these various boundary conditions?

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## Answers and Replies

Chalnoth
Science Advisor
It's two possible boundary conditions for each dimension. Since most string theories are 10-dimensional, that's 2^10 = 1024 possible combinations of boundary conditions. Of course, most of these are the exact same boundary conditions just with a rotation of coordinates.

It's two possible boundary conditions for each dimension. Since most string theories are 10-dimensional, that's 2^10 = 1024 possible combinations of boundary conditions. Of course, most of these are the exact same boundary conditions just with a rotation of coordinates.
OK, what does it mean to us when a string ends as an N or D in one of our four dimensions? Can we observe such an end?

Chalnoth
Science Advisor
OK, what does it mean to us when a string ends as an N or D in one of our four dimensions? Can we observe such an end?
Well, if it has Dirichlet boundary conditions in one dimension, then that means the endpoints are only free to move in three dimensions. So if it's free to move in two spatial dimensions and one time dimension, then that means its endpoints are confined to a two-dimensional spatial surface. That surface could be a plane stretching through all of space. Or it could be a cylinder. Or it could be the surface of a sphere.

And this surface we would call a 2-brane.