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I want to know the metric form of $AdS_5 \times S^5$.

I know there are two forms (maybe more?) Poincare patch and global patch.

And what is the difference between these two patches?

Can you state the two forms (?) of the metric and explain the difference?

I want to know, for example, Poincare patch is used in some areas due to something and so on.


I noticed for global patch, identifying $AdS_5$ and $S_5$ have the same radius $R$, I have \begin{align} ds^2 = R^2 \left( - \cosh^2(\rho) dt^2 + d\rho^2 + \sinh^2(\rho) d \Omega_3^2 + \cos^2(\theta) d \phi^2 + d\theta^2 + \sin^2(\theta) d \tilde{\Omega}_3^2 \right) \end{align} where $d\Omega_3^2, d\tilde{\Omega}_3^2$ denote the two separate $3-$sphere.

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    $\begingroup$ Do you know how to write the metric for a product space? Is there anything in wikipedia pages on Anti-de Sitter spacetime and n-sphere you do not understand? $\endgroup$
    – A.V.S.
    Jan 27 '21 at 7:50
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    $\begingroup$ The global form of the AdS metric is used for the study of closed strings, while the Poincare patch is used for the study of open string states. This does not mean, of course, that you cannot study open strings in the global coordinates, it's just more conveniently done in the Poincare patch and vice versa. There are many excellent places in the literature that show the embedding precisely and the different uses. This is a good starting point I think arxiv.org/abs/1012.3986 $\endgroup$ Jan 27 '21 at 10:20
  • $\begingroup$ @DiSp0sablE_H3r0, I see thanks for the detailed explanation and reference! $\endgroup$
    – phy_math
    Jan 27 '21 at 12:00
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For the ADS parts, the global patch is given by \begin{align} ds^2 = R^2 \left( -\cosh^2(\rho) d\tau^2 + d\rho^2 + \sinh^2(\rho) d \Omega_{n-2}^2 \right) \end{align} This covers the entire hyperboloid with $\tau\in [0,2\pi]$ and $\rho \in \mathbb{R}^+$.

And, the Poincare patch is given by \begin{align} ds^2 = \frac{R^2}{z^2} \left( dz^2 + dx_\mu dx^\mu \right) \end{align} where $dx_\mu dx^\mu = - dt^2 + d\vec{x}^2$.

Now the $S^5$ can be described by \begin{align} ds^2 &= d \theta^2 + \cos^2(\theta) d \phi^2 + \sin^2(\theta) d \tilde{\Omega}_3^2 \\ &=d \theta^2 + \cos^2(\theta) d \phi^2 + \sin^2(\theta) \left( d \phi_1^2 + \cos^2(\phi_1) d \phi_2^2 + \sin^2(\phi_1) d \phi_3^2 \right) \end{align}

So $AdS_5 \times S^5$ metric in global patch is given by \begin{align} ds^2 = R^2 \left( - \cosh^2(\rho) d\tau^2 + d\rho^2 + \sinh^2(\rho) d \Omega_3^2 + \cos^2(\theta) d \phi^2 + d\theta^2 + \sin^2(\theta) d \tilde{\Omega}_3^2 \right) \end{align} In Poincare coordinates, we have \begin{align} ds^2 = R^2 \left( \frac{dz^2}{z^2} - \frac{dt^2}{z^2} + \frac{d\vec{x}^2}{z^2}+ \cos^2(\theta) d \phi^2 + d\theta^2 + \sin^2(\theta) d \tilde{\Omega}_3^2 \right) \end{align} where $d\vec{x}^2 = dx_1^2 + dx_2^2 + dx_3^2$.

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