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P-brane Action  (in progress)

4/6/2021

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A p-brane is a p-dimensional manifold embedded in a D dimensional space-time where, by definition, \(D > p\). For example, a particle has \(p = 0\) and a string in string theory has \(p = 1\). Strings can be closed (circles) or open (lines).

Suppose we have a string in our everyday 4-dimensional space-time, that is, 3 spatial dimensions and 1 time dimension. Then we can describe the trajectory of the string by \(X^{\mu}(\sigma, \tau)\) where \(\sigma\) parameterizes the \(D - 1\) spatial coordinates of the string at time \(\tau\). If we let a given string evolve over time...
Picture
Picture
it will sweep out some \(p + 1\) dimensional surface (a line for \(p = 0\), area for \(p = 1\), volume for \(p > 1\)).
Picture
The world line, area, or (most generally) volume, can be determined by integrating the change in each of the \(D\) coordinates of \(X^{\mu}\) in relation to each of its parameters. I'll return to this shortly.

The most general form of the p-brane action is given by \(S_{p} = -T_{p}V\) where \(V\) is the world-volume to be minimized by the least action principal and \(T_{p}\) is described as a tension or energy density.

Since time is standardly written as the first coordinate of a D-dimensional space-time, we can consider the example of a morphed string shown before with an added time axis in the vertical direction which would produce a morphing column over time similar to the examples shown in the image taken from Wikipedia at the start of this post.
Picture


To the left is my attempt to draw such a surface. Working with the coordinates shown in this image requires that we multiply \(X^{\mu}\) by the Minkowski metric which is the D-dimensional (in a D-dimensional space-time) identity matrix except with -1 in the upper, left-most entry.
The generalized volume factor in \(S_{p} = -T_{p}V\) for any p-brane in \(D\) dimensions is given by \[ V = \int\sqrt{-det(g_{\mu\nu}(X)\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}}d^{p + 1}\sigma\] where \(X^{0}\) corresponds to the axis \(\tau\), \(X^{1}\) to \(\sigma^{1}\), and \(X^{2}\) to \(\sigma^{2}\). We can note that \(X^{0} = \sigma^{0} = \tau\) thus \(\partial_{\sigma}X^{0} = 0\) and likewise, ...(CHECK). \(\alpha\) and \(\beta\) range over the values \(0, ..., p\) and \(\mu\), \(\nu\) range over the values \(0, ..., D-1\). The determinant serves to permute all nonzero combinations into a "single line" (or non-matrix form) equation and \(d^{p + 1}\sigma\) indicates the infinitesimal elements of the parameters \(\sigma\) and \(\tau\).

Specifically, for a string (p = 1) the action \(S_{1} = -TV\) simplifies to the Nambu-Goto action: \[S_{NG} = -T\int\sqrt{(\dot{X}\cdot X')^{2} - \dot{X}^{2}\cdot X'^{2}}d\sigma d\tau\] where \(\dot{X}^{\mu} = \frac{\partial X^{\mu}}{\partial\tau}\) and \(X'^{\mu} = \frac{\partial X^{\mu}}{\partial\sigma}\) with \(A\cdot B = \eta_{\mu\nu}A^{\mu}B^{\nu}\) for a flat space-time.
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