A certain background for strings (II)

Today I will approach the problem I started here from a different angle. In the work of Alday and Maldacena, the authors used the following parameterization when calculating the action near a cusp:

   ,      and  

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...]