Matrix coordinates of D branes

Main Question or Discussion Point

Can someone explain to me how is it possible for D-branes to be parametrized with matrix coordinates? I mean, D-brane is a surface embedded in ordinary space, no? And the coordinates of ordinary space are vectors. So how can those vector coordinates suddenly turn into matrix ones on a D-brane?

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haushofer
I'm no expert, but maybe page 7 of Zarembo's "An introduction to matrix superstring models" (pdf online) gives more insight (section 2.3, "How matrices arise"). The coordinates describing the transverse fluctuations of the D-brane can be identified with components of the vector field living on the D-brane, and if these fields are charged under SU(N) they are matrix-valued one forms.

haushofer
You're welcome.

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Can someone explain to me how can D-brane coordinates take matrix form? After all, D-brane is embedded into 10-dimensional space. So if the 10 coordinates are numbers rather than matrixes, how is it possible for D-brane coordinates to suddenly be matrices? Or are you saying that some of those 10 coordinates of space are matrices as well?

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jambaugh
$$(x,y) \to x+iy \to xI + yJ = \left(\begin{array}{cc} x & -y \\ y & x\end{array}\right)$$