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  • I would in general like to understand how to derive the low energy metrics to describe D-brane configurations. Any pedagogic reference which explains the method?

In particular I have in mind these two facts,

(1) For N coincident D3 branes in IIB it seems that the metric is,

$ds^2 = (1+ \frac{L^4}{y^4})^{-0.5}\eta_{ij}dx^idx^j + (1+ \frac{L^4}{y^4} )^{0.5}(dy^2+y^2d\Omega_5^2) $

where $L^4 = 4\pi g_s N (\alpha')^2$ ($L$ being called the radius of the Dbrane)

(2) A "shell" of D3 branes is apparently described by,

$ds^2 = h^{-1}(r)[-dt^2 + d\chi^2 + dx_1^2 + dx_2^2 ] + h(r)[dr^2 + r^2d\Omega_5^2 ]$

where $\chi$ is a compactified dimension of length $L$ and $h(r) = \frac{R^2}{r^2}$ for $r>r_0$ and $h(r) = \frac{R^2}{r_0^2}$ for $r<= r_0$

  • How are these two related? What exactly is a "shell of Dbranes"? How are these two derived?

  • What exactly is meant by "radius of the Dbrane" (L) in the first one? Is that the coordinate in the spacetime which was compactified and T-dualized and is that the same as the $L$ in the second metric?

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They're different solutions of the equations of type IIB SUGRA – and different configurations in the full type IIB string theory – that "locally agree" on certain things but they're not quite identical. In both cases, there is the same flux over the 5-sphere surrounding the D3-branes. Locally, the objects and charges that are responsible for the discontinuities or singular points are the same ones – the D3-branes and their RR charges.

But the "shell of D3-branes" is meant to be spherical - literally a sphere that is "made out of" the D3-brane stuff. People use this shell in thought experiments where they allow them to collapse to create the black hole/brane. Inside, $r\lt r_0$, one has a flat 10-dimensional Minkowski space, and outside, we have $AdS_5\times S^5$. These two 10D geometries are "nonsmoothly" connected at $r=r_0$. The shell has this extra $r_0$ parameter, the radius. Naively, you could think that the limit $r_0\to 0$ of the shell gives you the stack because you eliminate the flat region inside but you actually make the branes infinitely curved so I think that you're getting away from the flat stack.

You have written all the formulae that define (and answer the question) what is meant by the "radius of the D-branes" $L$. $L$ is also the curvature radius of the AdS space, and the curvature radius (or ordinary radius) of the five-sphere. It is the length scale or a value of the $y$ coordinate at which the two components of the metric tensor (e.g. the redshift) change substantially. You have also written the radius $L$ in terms of $N,g_s,\alpha'$. The more branes you have or the stronger the string coupling is, the stronger the gravitational field is around them, and the larger $L$ is – the further the stack of D3-branes may affect the gravitational field.

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Motl Thanks for the reply! Can you give some references and or explain how people come up with these geometries? How are these metrics invented? Can one go back and forth - like given a metric figure out what D-brane configuration would generate it in its low energy approximation? – user6818 Feb 1 '14 at 2:05

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