Notes on string theory: String action in the light-cone

In the last post we entered into the light-cone. The next thing we want to do is engage with a quick study of the string action using light-cone coordinates. Recall the S_{P} action written in full,

    \[ S_{P} = -\frac{T}{2} \int d^{2}\sigma \sqrt{-\gamma}\gamma^{\alpha \beta} \partial_{\alpha} X^{\mu}\partial_{\beta}X^{\nu} \eta_{\mu \nu} \]

Of course, we can calculate the determinate for \gamma. From the definition for the metric given previously,

    \[ \sqrt{-\gamma}\gamma^{\alpha \beta} \partial_{\alpha} X^{\mu}\partial_{\beta}X^{\nu} \eta_{\mu \nu} \]

    \[ = - \sqrt{\frac{1}{2}} h^{-+} \partial_{+}X^{\mu}\partial_{-}X^{\nu} \eta_{\mu \nu} - \sqrt{\frac{1}{4}} h^{-+}\partial_{-}X^{\mu}\partial_{+}X^{\nu} \eta_{\mu \nu} -2 \partial_{+}X^{\mu}\partial_{-}X^{\nu} \eta_{\mu \nu} \]

And so it follows that the Polyakov action can be written in the form,

    \[ S_{P} = T \int d^{2}\sigma \partial_{+}X^{\mu}\partial_{-}X^{\nu} \eta_{\mu \nu} \]

Just as before, we can study the equations of motion where we eventually arrive at the wave equation. Noting \delta \frac{\partial X^{\mu}}{\partial \sigma^{\pm}} = \frac{\partial(\delta X^{\mu})}{\partial \sigma^{\pm}} we can compute,

    \[ \delta S_{P} = \delta T \int d^{2}\sigma \partial_{+}X^{\mu} \partial_{-}X^{\nu}\eta_{\mu \nu} \]

    \[ = T \int d^{2}\sigma \delta (\partial_{+}X^{\mu} \partial_{-}X^{\nu})\eta_{\mu \nu} \]

    \[ = T \int d^{2}\sigma \partial_{+}(\delta X^{\mu})\partial_{-}X^{\nu})\eta_{\mu \nu} + T \int d^{2}\sigma \partial_{+}X^{\mu} \partial_{-}(\delta X^{\nu})\eta_{\mu \nu} \]

After integrating by parts and dropping the boundary terms, we arrive at

    \[ \delta S_{P} = -T \int d^{2}\sigma (\delta X^{\mu})\partial_{+}\partial_{-}X^{\nu}\eta_{\mu \nu} - T \int d^{2}\sigma \partial_{-}\partial_{+}X^{\mu}(\delta X^{\nu})\eta_{\mu \nu} \]

Since it must be the case that \delta S_{P} = 0, we arrive at a very nice result,

    \[ \partial_{+}\partial_{-}X^{\mu} = 0 \]

This is the wave equation. 

The open string

For the open string it is important to note that we have the interval,

    \[-\infty \leq \tau \leq \infty \ \  \ 0 \leq \sigma \leq l \]

Recall, from the introduction of this chapter, that the action S_{P} has redundancies. It is possible to compute the amount of redundancies. For example,

    \[ \tau \rightarrow \tau^{\prime}(\tau, \sigma) \]

    \[ \sigma \rightarrow \sigma^{\prime}(\tau, \sigma) \]

What we see is that we already have two arbitrary functions. But there is a third, and this third redundancy is given by way of Weyl invariance. What this means is that we have 3 conditions to fix diff + Weyl symmetries.

The more important question now concerns how we go about this task of fixing these invariances. The answer is that there are various ways to approach the issue. The easiest way to motivate the reasoning and logic behind the ‘best choice’ is to show the calculation. Recalling, analogously, our previous discussion about the flat metric, we start with:

    \[ X^{+} = \tau \]

    \[ \partial_{\sigma} \gamma_{\sigma \sigma} = 0 \]

    \[ det \gamma = -1 \]

These are gauge fixing conditions implying LC gauge fixing for the string.

As an aside, one can always choose LC gauge fixing locally.

Due to spatial constraints, I will exclude the explicit calculations that show these workings. In the next post, we will dive right into a study of the equations of motion, wave equation and centre of mass position.

References

Joseph Polchinski. (2005). “String Theory: An Introduction to the Bosonic String“, Vol. 1.