In search for anti-de Sitter space (III)
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”