Let us start by considering the direct product of a 3d Minkowski space with a 4d Minkowski space. The coordinates for this space then go as
If we impose the condition we obtain the following relation:
In this form the scale and boost transformations are more explicit. But these coordinate come for the case of exact anti-de Sitter space and I want to consider the metric with
with being a function of the extra (fifth) coordinate only. This space will not have the same isometries as AdS, so we should not use the same parameterizations mentioned above. By in the UV limit we should obtain AdS space, so the parameterization should not be that different.
Maybe I should start with the 3d case. The metric goes as
We expect the parameterization of the worldsheet to change. But what if we keep it the same and look for a constant solution? The action will still diverge. [To be started…]