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

• edearl
In summary, D-branes, also known as Dirichlet-branes, are hypersurfaces in spacetime on which strings can end. These hypersurfaces can have various boundary conditions, including Neumann and Dirichlet, which dictate the movement of the string endpoints. In most string theories, there are 2^10 = 1024 possible combinations of these boundary conditions. D-branes can have different dimensionalities, with a 2-brane being a surface in three dimensions where the string endpoints are confined to move. These branes could potentially be observed in a large enough collider that can detect strings with different endpoint boundary conditions.

#### edearl

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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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.

Chalnoth said:
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?

edearl said:
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.

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

Strings with endpoint boundary conditions NN, DD, ND, and DN refer to the different ways in which strings (one-dimensional objects) can be attached or fixed at their endpoints. These boundary conditions determine how the string behaves at its endpoints and play an important role in string theory.

## 2. What does NN boundary condition mean?

NN boundary condition stands for Neumann-Neumann boundary condition. This means that both endpoints of the string are free to move and their derivatives (rates of change) are also free. In other words, the string can move and stretch freely at both ends.

## 3. What is the difference between DD and NN boundary conditions?

DD stands for Dirichlet-Dirichlet boundary condition, which means that both endpoints of the string are fixed and cannot move. In contrast, NN boundary condition allows for free movement of both endpoints. This is the main difference between DD and NN boundary conditions.

## 4. What does ND boundary condition imply?

ND boundary condition refers to Neumann-Dirichlet boundary condition, which means that one endpoint of the string is free to move while the other is fixed. This creates an asymmetry in the behavior of the string, with one end being able to move and the other being fixed.

## 5. How do the different boundary conditions affect the behavior of a string?

The different boundary conditions affect the behavior of a string by determining its allowed modes of vibration and the energy levels of these modes. For example, NN boundary condition allows for the existence of zero-point energy, while DD boundary condition does not. In general, different boundary conditions can lead to different physical properties and behaviors of a string.