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.
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.
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
After the T-transformation the metric returns to the same form but now ,
Both of these spaces can be described by embedding a hyper-surface in . They both have scale invariance.
There are two extra-dimensional spaces that one can obtain by embedding a hyper-surface in and reduce to adS in some limit.
For this case anti-de Sitter space is obtained by setting the extra coordinate equal to . The metric in the original Poincaré coordinates looks like:
After the T-transformation (leaving the coordinate intact) one obtains
Now adS is reached when the extra coordinate vanishes. In Poincaré coordinates,
While in the T-coordinates we obtain
Note that both of these cases have scale-invariance in the coordinates and . 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:
- Hard wall:
I might not want to keep the scale invariance so obvious for the coordinates.
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
with a two-dimensional Minkowski index and a d-dimensional Minkowski index. Let us write the metric as:
Now one imposes the condition:
and introduces “Poincaré coordinates” that satisfy this constrain. These can be defined as:
Then writing the metric in terms of these coordinates one obtains:
One can see that when we set 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 coordinates. I shall explore this space further.
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
Second is to deform the exact adS metric by introducing some arbitrary function of the fifth coordinate. Right now my main struggle is what form of the deformation should I use. Here are some favorites:
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:
After the so-called “T-dual” transformation, the metric obtains the following form:
This latter approach is the most attractive in my humble opinion. It contains adS in the limit when the coordinate vanishes. We also have an invariance under a scale transformation on the coordinates or What am I suppose to do with this metric? I want to consider classical strings propagating in any of these particular backgrounds.
Let us start by considering the space consisting of the product of a 3-d Minkowski manifold and -dimensional Minkowski manifold. The metric reads as:
After imposing one can use coordinates:
, , and .
Once we plug-in these coordinates, the metric takes a surprisingly simple form:
Now we concentrate in the case where . 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 one gets some interresting results.
For we have a funky version of 5-d anti-de Sitter space:
This could be useful for something. Exact 5-d adS is obtained when
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 coordinates. This transformation is defined by:
(Alday & Maldacena use for the initial coordinates and for the T-dually-transformed coordinates. In my work I use the opposite convention.) Furthermore, after redefining we can write the metric in AdS form again, with a slight change in the second term:
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 , which is that the carry momentum , translates into the condition that has “winding”