Index Concordia

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).

In this way, when we look at the UV limit (very small z), we obtain five-dimensional anti-de Sitter space. This form might be particular since it restricts the function A (which is only a function of the fifth coordinate z). For spacelike or timelike events we must have either

A > R/z   or   A < R/z.

We consider the Nambu-Goto action:

S = T\displaystyle\int d^2 \tau \sqrt{-\det{G_{ab}}},

with the spacetime metric entering through G_{ab} = G_{mn}\partial_{a}X^{m}\partial_{b}X^{n}. My convention is the usual: \partial_{\tau} X^{m} = \dot{X}^{m} and \partial_{\sigma}X^{m} = X^{\prime m}.

We first notice that the metric can be written as G_{mn} = Y^2 \eta_{mn}. Then the determinant can be written as -\det{G_{ab}} = Y^4 \eta_{mn}\eta_{pq}X^{mnpq} with the four-index tensor given by

X^{mnpq} = \dot{X}^{m}X^{\prime n} \dot{X}^{p}X^{\prime q} - \dot{X}^{m}\dot{X}^{n}X^{\prime p} X^{\prime q}.

Notice that this tensor is antisymmetric in the middle indices. The next big thing we can do is to obtain the Euler-Lagrange equations:

\displaystyle\frac{\partial L}{\partial X^{l}} = \displaystyle\frac{\partial \mathcal{P}^{\tau}_{l}}{\partial \tau} + \displaystyle\frac{\partial \mathcal{P}^{\sigma}_{l}}{\partial \sigma}.

We have used the usual symbols for the conjugate momenta. The left hand side of the Euler-Lagrange equations can be calculated to give

\displaystyle\frac{\partial L}{\partial X^{l}} = 2 (...)^{-1/2}[(\dot{X}\cdot X^{\prime})^2 - \dot{X}^2 X^{\prime 2}] Y^3\displaystyle\frac{\partial Y}{\partial X^{l}}

It turns out that this expresion vanishes exept for the l = 4 component (X^4 = Z). The right-hand side of the Euler-Lagrange equations is less compact. We use the following symbols:

\beta = \dot{X}\cdot X^{\prime} = \eta_{mn}\dot{X}^m X^{\prime n},

\xi^{-1} = \sqrt{Y^4 \left( \beta^2 - \dot{X}^2 X^{\prime 2}\right)}.

The conjugate momenta can be written now:

\mathcal{P}_{l}^{\tau} = Y^4 \xi \left( \beta X^{\prime}_{l} - X^{\prime 2} \dot{X}_{l}\right)   and   \mathcal{P}_{l}^{\sigma} = Y^4 \xi \left( \beta \dot{X}_{l} - \dot{X}^{2} X_{l}^{\prime}\right).

Next we must calculate the derivatives of the conjugate momenta. Using the above symbols we get

\displaystyle\frac{\partial \mathcal{P}_{l}^{\tau}}{\partial \tau} = 4 Y^3 \dot{Y}\xi \left(...\right) + Y^4 \dot{\xi}\left(...\right) + Y^4 \xi \left( \dot{\beta}X^{\prime}_{l} + \beta \dot{X}^{\prime}_{l} - 2 X^{\prime}\cdot \dot{X}^{\prime}\dot{X}_{l} - X^{\prime 2} \ddot{X}_{l}\right)

and similarly for the other momentum conjugate to \sigma. Other things that we need include

\displaystyle\frac{\partial Y}{\partial Z} = -2\left(\frac{R^2}{Z^3} + A\frac{\partial A}{\partial Z}\right)

\dot{Y} = \displaystyle\frac{\partial Y}{\partial Z} \dot{Z}   and   Y^{\prime} = \displaystyle\frac{\partial Y}{\partial Z}Z^{\prime}

\dot{\beta} = \ddot{X}\cdot X^{\prime} + \dot{X}\cdot \dot{X}^{\prime}   and   \beta^{\prime} = \dot{X}^{\prime} \cdot X^{\prime} + \dot{X} \cdot X^{\prime \prime}.

From here we can try some things. I have a pretty good feeling this problem does not have a nice solution…

We can separate variables into parts that depend only on each of the worldsheet parameters. We have to make some gauge choices to fix the parametrization of the worldsheet, since the equation of motions should be reparameterization invariant. One such gauge choice that comes to my mind is the “static gauge”,

X^{0} = \tau.

This is not Lorentz invariant. Maybe a light-cone gauge would be better. We also have to fix the \sigma parametrization. When we do this fixing we have to take into account the constraints that come with them.

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4 Responses

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  1. Index Guy said, on June 17, 2008 at 3:51 pm

    I am currently very confuse. If I am going to apply this to the work of Alday and Maldacena then I need to understand better type IIB string theory. But the AM paper works with the Nambu-Goto action…

  2. Index Guy said, on June 17, 2008 at 4:11 pm

    So far this will describe non-interacting strings. To fix this, can we just add the term
    \displaystyle\sum_{l}k_{l}^{m}X_{m}^{l}
    to the action?

  3. Index Guy said, on June 19, 2008 at 7:12 pm

    I tried the 3d AdS case with the same parameterization that Alday and Maldacena used in their paper and got an expression for the modified metric. The area of the worldsheet still diverges, so there is not much improvement.

  4. […] Relativity, String Theory — Index Guy @ 9:28 pm 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 […]


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