Index Concordia

In search for anti-de Sitter space (II)

Posted in Relativity by Index Guy on June 30, 2008

I want to continue the discussion started here about a metric that may yield anti-de Sitter space as some limit.

In search for anti-de Sitter space (I)

Posted in Relativity by Index Guy on June 29, 2008

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

$X^{m} = \left(Z_{0}, Z_{1}, Z_{2}, Y_{0}, Y_{1}, Y_{2}, Y_{3}\right).$

If we impose the condition $X^2 = 0$ we obtain the following relation:

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:

A certain background for strings (I)

Posted in Classical Mechanics, Relativity, String Theory by Index Guy on June 11, 2008

We would like to consider bosonic strings in the following gravitational background:

$ds^2 = Y^2 (\eta_{ij}dx^i dx^j + dz^2).$

What we have in mind is something along the lines of

$Y^2 = \displaystyle\frac{R^2}{z^2} - A^2 = (\displaystyle\frac{R}{z} - A)(\displaystyle\frac{R}{z} + A).$