the basis of gauge/gravity duality.

One important example is D3-branes which can also be seen as solutions of ten dimensional type IIB supergravity at low energies, with metric of the form \[62\],

$$ds^2 = \left( 1 + \frac{L^4}{r^4} \right)^{-1/2} \left[ -dt^2 +
d\vec x^2 \right] + \left( 1 + \frac{L^4}{r^4} \right)^{1/2} \left[
dr^2 + r^2\, d\Omega_5^2 \right]\, ,$$

where $$L^4 = 4 \pi g_s N^2 \alpha' ,$$ here, $g_s$ is the string coupling constant which is related to the constant dilaton as ($g_s = e^\Phi$). Also, there is $N_c$ units of $F_{[5]}$ flux. $L$ is the only length scale in the solution. This metric interpolates between a throat geometry and a ten dimensional Minkowski region.

If we take the [near horizon limit]{} of the solution given in eq. (1), $r \ll L$, and redefine $z = L^2/r$, we can completely decouple the Minkowski region and are left with a throat geometry which is given by $$ds^2 = \frac{L^2}{z^2} \left[ -dt^2 + d\vec x^2 + dz^2 \right] +
L^2\, d\Omega_5^2 \, ,$$ which is the Poincaré wedge of the direct product of five dimensional anti-de-Sitter space and a five sphere (AdS$_5 \times$ S$^5$). The isometry group of this space is given 