Index Concordia

Updated roadmap

Posted in AdS/CFT, Relativity, String Theory by Index Guy on July 13, 2008

I have become more comfortable with the “T-dual” transformation and now I am confident on what to consider.

Exact adS can be described by the metric

ds^2 = \displaystyle\frac{R^2}{r^2} \left(\eta_{mn}dy^m dy^n + dr^2\right).

After the T-transformation the metric returns to the same form but now r = R^2 / \rho,

ds^2 = \displaystyle\frac{R^2}{\rho^2} \left(\eta_{mn}dx^m dx^n + d\rho^2\right).

Both of these spaces can be described by embedding a hyper-surface in R_{2,d}. They both have scale invariance.

There are two extra-dimensional spaces that one can obtain by embedding a hyper-surface in R_{3,d} and reduce to adS in some limit.

Case I: C^2 = 0

For this case anti-de Sitter space is obtained by setting the extra coordinate equal to R. The metric in the original Poincaré coordinates looks like:

ds^2 = \displaystyle\frac{R^2}{r^2} \eta_{mn}dy^m dy^n + \displaystyle\frac{z^2}{r^2}dr^2 + dz^2 + 2\frac{z}{r}dzdr.

After the T-transformation (leaving the z coordinate intact) one obtains

ds^2 = \displaystyle\frac{R^2}{\rho^2} \eta_{mn}dx^m dx^n + \displaystyle\frac{z^2}{\rho^2}d\rho^2 + dz^2 - 2\frac{z}{\rho}dz d\rho.

Case II: C^2 = -R^2

Now adS is reached when the extra coordinate vanishes. In Poincaré coordinates,

\displaystyle\frac{R^2}{r^2} \left(\eta_{mn}dy^m dy^n + dr^2\right) + \displaystyle\frac{z^2}{r^2}dr^2 + dz^2 + 2\frac{z}{r}dzdr.

While in the T-coordinates we obtain

\displaystyle\frac{R^2}{r^2} \left(\eta_{mn}dx^m dx^n + dr^2\right) + \displaystyle\frac{z^2}{r^2}dr^2 + dz^2 - 2\frac{z}{r}dzdr.

Note that both of these cases have scale-invariance in the coordinates (r, y) and (\rho, x). These two cases are atractive because they can be constructed straight from a higher-dimensional reduction. On the other hand one can just modify the adS metric and see what happens. Here are some of the modifications.

  • Soft wall: ds^2 = \displaystyle\frac{R^2}{r^2} \eta_{mn}dy^m dy^n + \displaystyle\frac{R^2}{r^2}\left( 1 + A^2(r)\right)dr^2
  • Hard wall: ds^2 = \displaystyle\frac{R^2}{r^2} \left(\eta_{mn}dy^m dy^n + dr^2\right)\theta(r - r_{0})
  • Cutoff: ds^2 = \displaystyle\frac{R^2}{r^2} \eta_{mn}dy^m dy^n + \displaystyle\frac{R^2}{r^2 - a^2}dr^2

I might not want to keep the scale invariance so obvious for the y coordinates.


2 Responses

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  1. Index Guy said, on July 15, 2008 at 1:19 pm

    Why insist so much in leaving the extra-coordinate z unchanged? The metric actually diagonalized when one uses \xi = Rz/r.

    For case I one obtains

    \displaystyle\frac{R^2}{r^2}\eta_{mn}dy^m dy^n + \frac{r^2}{R^2}d\xi^2.

    For case II we have

    \displaystyle \displaystyle\frac{R^2}{r^2}(\eta_{mn}dy^m dy^n + dr^2) + \frac{r^2}{R^2}d\xi^2.

    Notice that for case I the r coordinate will not contain derivatives in the action…

  2. Index Guy said, on July 15, 2008 at 1:28 pm

    The change of variables \xi = Rz/r should be done in the T-dual space.

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