## Solving equations of motions in some gravity background

I would like to consider the gravity background:

with the case that We saw previously that the equations of motion were given by:

and

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:

with

and

Inserting this into Mathematica gives:

which means

Notice that the case is interesting: for the constraints we have used the solution is an exponential function of a quadratic polynomial.

## A curved lagrangian in terms of a flat one

Let us consider the following gravitational background:

and the Polyakov lagrangian in conformal gauge with Euclidean Lorentzian signature:

The Euler-Lagrange equations can be found to be:

and

We look at this and think how can we make these equations simpler. The first equation can be solved with:

This is a “T-duality” transformation. For the second equation we see that if then we get

with the flat-space Polyakov lagrangian in terms of the new coordinates :

We can integrate this equation to find an expression for in terms of the solutions of the equations of motion and the flat-space lagrangian:

Then, pluging this back into the classical action we get

Finally we can write

so we can write the expression in the exponential as a sum of integrals over the worldsheet coordinates.

On the other hand, if instead we have then we can write:

with

The lagrangian can be writen as

We have found an expression for the classical lagragian in some (not-so) arbitrary gravitational background in terms of the coordinates and the flat-space lagrangian. The problem is we do not get any information about the solution for the equations of motion.

## Minimal area in arbitrary background (I)

Let us start with the following background metric:

where the functions and are functions of the extra coordinate . The case of anti-de Sitter space corresponds to

Under the “T-duality” transformation

and

the metric in the dual space takes the form

but now

and

Since we can always bring the metric in this form, we will just consider the initial case and see what can we do with it. Note that it could be the case that the change of variables between could be of the general form

which means

This case could be more complicated… For now we will just stick with the first change of variables introduced.

## Some progress towards something

On Thursday night I was reading Polyakov’s contribution to the book *50 years of Yang-Mills theory*. This is my version of a bed time story. 😉

Polyakov mentions some of his earlier work and how he solved different problems. I actually got tired after a while, and decided to work and solve my own problem. I had been sort of running away from it with feelings of overwhelming difficulty. There is also a chance that reading this post on the (now dead) string coffee table provided some motivation. In any case, here we go…

## Updated roadmap

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.

**Case I:**

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

**Case II:**

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:
- Cutoff:

I might not want to keep the scale invariance so obvious for the coordinates.

## 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…]

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