I would like to consider the gravity background:
with the case that We saw previously that the equations of motion were given by:
Now we take the warp factor to have the form . Then we have
We will introduce a new symbol,
with a function of the worldsheet coordinates.
The differential equation now looks like:
We now assume that can be factored into functions of each coordinate, . Then we can solve this PDE when
with a constant.
Separation of variables gives the following differential equation:
Inserting this into Mathematica gives:
Notice that the case is interesting: for the constraints we have used the solution is an exponential function of a quadratic polynomial.
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.
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…]