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.

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.

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.

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.

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.

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}.

Indeed.