Three roads to anti-de Sitter space
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.