July « 2008 « Index Concordia

Back to zero

Posted in Organizational by Index Guy on July 21, 2008

It turns out that all that I have been doing for the past few weeks has been slightly off-track. Back to the main source: Alday and Maldacena.

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Alday & Roiban give us 100 pages of goodness

Posted in Sparkling on the arXiv by Index Guy on July 14, 2008

Today I found this:

Luis Fernando Alday and Radu Roiban, Scattering amplitudes, Wilson loops and the Strings/Gauge Theory correspondence, arXiv:0807.1889.

At this moment it looks really helpful.

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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.

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Yet another candidate

Posted in Relativity by Index Guy on July 9, 2008

Today I found yet another space that looks promising. This one looks even more nicer than the previous cases. To obtain the metric one can start with the product of 3-d Minkowski and d-dimensional Minkowski space. The coordinates can be taken as

$C^{M} = (Z^A, z, Y^m),$

with $A$ a two-dimensional Minkowski index and $m$ a d-dimensional Minkowski index. Let us write the metric as:

$ds^2 = -2dZ_{+}dZ_{-} + dz + \eta_{mn}dY^{m}dY^{n}.$

Now one imposes the condition:

$C^2 = 0,$

and introduces “Poincaré coordinates” that satisfy this constrain. These can be defined as:

$Y^m = \displaystyle\frac{R}{r}y^m,$ $\sqrt{2}Z_{+} = \displaystyle\frac{R^2}{r},$ $\sqrt{2}Z_{-} = \displaystyle \frac{r z^2}{R^2} + \displaystyle\frac{\eta_{mn}y^m y^n}{r}.$

Then writing the metric in terms of these coordinates one obtains:

$ds^2 = \displaystyle\frac{R^2}{r^2}dy^2 + \frac{z^2}{r^2}dr^2 + \frac{2z}{r}drdz + dz^2.$

One can see that when we set $z = R$ we obtain exact d-dimensional anti-de Sitter space. This metric is very atractive. It contains more symmetry than the other cases that I have consider previously. Besides the first term being invariant under Poincaré transformations, one also has an invariance under the rescalling of the $(y^m , r)$ coordinates. I shall explore this space further.

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Three roads to anti-de Sitter space

Posted in Relativity by Index Guy on July 8, 2008

I want to summarize the three different routes I am pursuing to reach anti-de Sitter space.

First is the direct route. This means to start with anti-de Sitter space in the first place. Alday and Maldacena followed this path. For future references the metric for adS space in Poincaré coordinates looks like

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

Second is to deform the exact adS metric by introducing some arbitrary function $A(r)$ of the fifth coordinate. Right now my main struggle is what form of the deformation should I use. Here are some favorites:

$ds^2 = \displaystyle\frac{R^2 - A^2}{r^2}\left(\eta_{mn}dx^m dx^n + dr^2\right)$
$ds^2 = \left(\displaystyle\frac{R^2 - A^2}{r^2}\right)dr^2 + \displaystyle\frac{R^2}{r^2}\eta_{mn}dx^m dx^n$
$ds^2 = \displaystyle\frac{R^2}{r^2}\left(1 + A^2\right)dr^2 + \displaystyle\frac{R^2}{r^2}\eta_{mn}dx^m dx^n$

Lastly we can try something completely different. Along this road it was suggested to start with an extra spatial dimensions and impose some constraints. One gets a six-dimensional space. In Poincaré coordinates this looks like:

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

After the so-called “T-dual” transformation, the metric obtains the following form:

$ds^2 = \displaystyle\frac{R^2}{\rho^2}\left(\eta_{mn}dy^m dy^n + d\rho^2\right) + \left(dz - \displaystyle\frac{z}{\rho}d\rho\right)^2 .$

This latter approach is the most attractive in my humble opinion. It contains adS in the limit when the coordinate $z$ vanishes. We also have an invariance under a scale transformation on the coordinates $(x^m , r)$ or $(y^m, \rho).$ What am I suppose to do with this metric? I want to consider classical strings propagating in any of these particular backgrounds.

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In search for anti-de Sitter space (III)

Posted in Relativity by Index Guy on July 2, 2008

Let us start by considering the space consisting of the product of a 3-d Minkowski manifold and $d$ -dimensional Minkowski manifold. The metric reads as:

$ds^2 = -dZ_{0}^2 + dZ^{2}_{1} + dZ^{2}_{2} - dY_{0}^2 + dY_{1}^2 + ... + dY_{d - 1}^2.$

After imposing $X^2 = -R^2$ one can use coordinates:

$\sqrt{2} Z_{+} = Z_{0} + Z_{1} = \displaystyle\frac{R^2}{r}$ , $\sqrt{2} Z_{-} = Z_{0} - Z_{1} = r + \displaystyle\frac{r z^2}{R^2} + \frac{y^2}{r}$ , $Z_{2} = z$ and $Y^{m} = \displaystyle\frac{R}{r}y^{m}$ .

Once we plug-in these coordinates, the metric takes a surprisingly simple form:

$ds^2 = \left(\displaystyle\frac{R^2 + z^2}{r^2}\right)dr^2 + \displaystyle\frac{R^2}{r^2}dy^2 + \displaystyle\frac{2 z}{r}dz dr + dz^2.$

Now we concentrate in the case where $d = 4$ . Then the space described by the above metric is six-dimensional. We can obtain a five-dimensional space by fixing one of the coordinates. It turns out that by fixing $z = R_{0}$ one gets some interresting results.

For $R_{0} \neq 0$ we have a funky version of 5-d anti-de Sitter space:

$ds^2 = \displaystyle \frac{R^2}{r^2}dy^2 + \frac{\rho^{ 2}}{r^2} dr^2$ with $\rho^{2} = R^2 + R_{0}^2.$

This could be useful for something. Exact 5-d adS is obtained when $R_{0} = 0.$

The metric is certainly more attractive than my previous blunder. But I still do not know how to understand the off-diagonal terms.

In any case, this metric still needs some work. In the work of Alday and Maldacena, they do a “T-dual” transformation on the $y$ coordinates. This transformation is defined by:

$\partial_{a} x^{m} = i \displaystyle\frac{R^2}{r^2}\epsilon_{ab}\partial_{b}y^{m}.$

(Alday & Maldacena use $x$ for the initial coordinates and $y$ for the T-dually-transformed coordinates. In my work I use the opposite convention.) Furthermore, after redefining $\rho = R^{2}/r$ we can write the metric in AdS form again, with a slight change in the second term:

$ds^2 = \displaystyle\frac{R^2}{\rho^2}\left(dx^2 + d\rho^2\right) + \left(dz - \displaystyle\frac{\rho z}{R^2}d\rho\right)^{2}.$

In this form it certainly looks interesting. Alas, I do not understand this T-duality transformation. A quote from the Alday-Maldacena paper:

In the regime under consideration the T-dual coordinates are real and the worldsheet is Euclidean. In addition, the boundary condition for the original coordinate $y^{m}$ , which is that the carry momentum $k^{m}$ , translates into the condition that $y^{m}$ has “winding” $\Delta y^{m} = 2 \pi k^{m}.$