Matrix coordinates of D branes

[EternalStudent]

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?

 

[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.

 

[EternalStudent]

<Moderator's note: post merged to this existing thread on the same topic>

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?

 

[jambaugh]

Consider x,y coordinates. The ordered pair (x y) is a 1x2 matrix. You can also express complex numbers in matrix form:
$$(x,y) \to x+iy \to xI + yJ = \left(\begin{array}{cc} x & -y \\ y & x\end{array}\right)$$

So that's some of how it *can* be done. Contrawise a matrix is a Vector in the abstract sense and lives in a vectors space of some dimensions.

These are general Linear Algebraic observations and you'll have to look up exactly how and why someone might do that for D-brane coordinates or someone else may be familiar with the specifics.