Index Concordia

Solving equations of motions in some gravity background

Posted in AdS/CFT, String Theory by Index Guy on August 31, 2008

I would like to consider the gravity background:

$ds^2 = A dx^2 + B dr^2 ,$

with the case that $A = B^{-1}.$ We saw previously that the equations of motion were given by:

$\partial_{i}\left(A\partial_{i}x^a\right) = 0$     and     $-2\partial_{i}^{2}r = \displaystyle\frac{d \left(\ln{B}\right)}{d r}\left(-\eta_{ab}\partial_{i}y^{a}\partial_{i}y^{b} +\partial_{i}r\partial_{i}r\right).$

Now we take the warp factor to have the form $B = r^{\beta}$. Then we have

$-2 r \partial_{i}^{2}r = \beta\displaystyle\left(-\eta_{ab}\partial_{i}y^{a}\partial_{i}y^{b} +\partial_{i}r\partial_{i}r\right).$

We will introduce a new symbol,

$\eta_{ab}\partial_{i}y^{a}\partial_{i}y^{b} = -W^{2}$     with $\displaystyle W(u_{1}, u_{2})$ a function of the worldsheet coordinates.

The differential equation now looks like:

$2 r \partial_{i}^{2}r + \beta \partial_{i}r\partial_{i}r = \beta W^{2}.$

We now assume that $r$ can be factored into functions of each coordinate, $r = T(u_{1}) S(u_{2})$. Then we can solve this PDE when

$k^2 = \displaystyle\frac{\beta W^2}{2 T^{2} S^{2}}$     with $k$ a constant.

Separation of variables gives the following differential equation:

$\displaystyle\frac{d^2 Y}{du_{i}^2} + \alpha \left(\displaystyle\frac{d Y}{d u_{i}}\right)^{2} = k_{i}^{2},$

with

$\alpha = 1 + \displaystyle\frac{\beta}{2}$     and     $T = \exp{Y}.$

Inserting this into Mathematica gives:

$Y(u) = C_{1} + \displaystyle\frac{1}{\alpha}\ln{\left(\cosh{\left(k_{i}\sqrt{\alpha}\left[u_{i} + C_{2}\right]\right)}\right)}.$     which means     $T(u) = C_{1} \left(\cosh{\left(k_{i}\sqrt{\alpha}\left[u_{i} + C_{2}\right]\right)} \right)^{1/\alpha}.$

Notice that the case $\beta = -2$ is interesting: for the constraints we have used the solution is an exponential function of a quadratic polynomial.

Posted in AdS/CFT, Relativity, String Theory by Index Guy on July 13, 2008

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

$ds^2 = \displaystyle\frac{R^2}{r^2} \left(\eta_{mn}dy^m dy^n + dr^2\right).$

After the T-transformation the metric returns to the same form but now $r = R^2 / \rho$,

$ds^2 = \displaystyle\frac{R^2}{\rho^2} \left(\eta_{mn}dx^m dx^n + d\rho^2\right).$

Both of these spaces can be described by embedding a hyper-surface in $R_{2,d}$. They both have scale invariance.

There are two extra-dimensional spaces that one can obtain by embedding a hyper-surface in $R_{3,d}$ and reduce to adS in some limit.

Case I: $C^2 = 0$

For this case anti-de Sitter space is obtained by setting the extra coordinate equal to $R$. The metric in the original Poincaré coordinates looks like:

$ds^2 = \displaystyle\frac{R^2}{r^2} \eta_{mn}dy^m dy^n + \displaystyle\frac{z^2}{r^2}dr^2 + dz^2 + 2\frac{z}{r}dzdr.$

After the T-transformation (leaving the $z$ coordinate intact) one obtains

$ds^2 = \displaystyle\frac{R^2}{\rho^2} \eta_{mn}dx^m dx^n + \displaystyle\frac{z^2}{\rho^2}d\rho^2 + dz^2 - 2\frac{z}{\rho}dz d\rho.$

Case II: $C^2 = -R^2$

Now adS is reached when the extra coordinate vanishes. In Poincaré coordinates,

$\displaystyle\frac{R^2}{r^2} \left(\eta_{mn}dy^m dy^n + dr^2\right) + \displaystyle\frac{z^2}{r^2}dr^2 + dz^2 + 2\frac{z}{r}dzdr.$

While in the T-coordinates we obtain

$\displaystyle\frac{R^2}{r^2} \left(\eta_{mn}dx^m dx^n + dr^2\right) + \displaystyle\frac{z^2}{r^2}dr^2 + dz^2 - 2\frac{z}{r}dzdr.$

Note that both of these cases have scale-invariance in the coordinates $(r, y)$ and $(\rho, x)$. 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: $ds^2 = \displaystyle\frac{R^2}{r^2} \eta_{mn}dy^m dy^n + \displaystyle\frac{R^2}{r^2}\left( 1 + A^2(r)\right)dr^2$
• Hard wall: $ds^2 = \displaystyle\frac{R^2}{r^2} \left(\eta_{mn}dy^m dy^n + dr^2\right)\theta(r - r_{0})$
• Cutoff: $ds^2 = \displaystyle\frac{R^2}{r^2} \eta_{mn}dy^m dy^n + \displaystyle\frac{R^2}{r^2 - a^2}dr^2$

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

A certain background for strings (II)

Posted in AdS/CFT, Relativity, String Theory by Index Guy on June 25, 2008

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:

$X^{0} = e^{\tau} \sinh{\sigma}$   ,   $X^{1} = e^{\tau}\cosh{\sigma}$   and   $r = e^{\tau} w(\tau).$

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 $G_{mn} = Y^{2}\eta_{mn}$ with

$Y^2 = \displaystyle\frac{R^2- A^2}{r^2} ,$

with $A$ 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

$ds^2 = Y^2 \left( - dX_{0}^2 + dX_{1}^2 + dr^2\right).$

We expect the parameterization of the worldsheet to change. But what if we keep it the same and look for a constant $w$ solution? The action will still diverge. [To be started…]

Scattering amplitudes à la Alday-Maldacena (II)

Posted in AdS/CFT, Gauge Theory, String Theory by Index Guy on June 20, 2008

Continuing what we started before, I would like to discuss the four-point scattering amplitude using AdS/CFT methods.

Scattering amplitudes à la Alday-Maldacena (I)

Posted in AdS/CFT, Gauge Theory, String Theory by Index Guy on June 19, 2008

I would like to summarize the work of Luis Fernando Alday and Juan Maldacena on gluon scattering amplitudes at strong coupling via AdS/CFT methods. I will follow mostly (i.e. verbatim) these two preprints from the arXiv: