Exploring the Symmetries of Nambu-Goto Action in String Theory?

[CompuChip]

Hi.

[introduction]
First let me recall that there are two equivalent classical bosonic string actions, the Nambu-Goto action
$$S_\mathrm{NG} = - T \iint \mathrm d\sigma \, \mathrm d\tau \sqrt{ (\dot X X')^2 - \dot X^2 {X'}^2}$$
and the Polyakov action
$$S_\mathrm{Pol} = - \frac{T}{2} \iint \mathrm d\sigma \, \mathrm d\tau \sqrt{ -h} h^{\alpha\beta} \partial_\alpha X^\mu \partial_\beta X_\mu,$$
where the worldsheet metric $h_{\alpha\beta}$ is considered as an independent field.

Now I understand that people usually work with the latter, because it is more convenient, and that this has a reparametrization invariance ($X^\mu \to X^\mu + \xi^\alpha \partial_\alpha X^\mu + \mathcal O(\xi^2)$ with corresponding transformation on the metric) and Weyl invariance ($h_{\alpha\beta} \to \exp(-2\Lambda) h_{\alpha\beta}$).
[/introduction]

My question is what the symmetries of the Nambu-Goto action are. My lecture notes give just the reparametrization invariance
$$\delta X^\mu = \xi^\alpha \partial_\alpha X^\mu$$
but is that all? Is the presence of the Weyl symmetry a consequence of introducing an extra independent field which is not really independent?

In other words, if I want to fix all gauge freedom using the Nambu-Goto action, it would suffice to specify $\xi^\alpha$ ?

 

[xepma]

The whole purpose of introducing the Polyakov action is to get rid of the non-linearity associated with the NG action. The "price" you pay is the introduction of this new field h. Upon setting this field to satisfy its classical equations of motion the Polyakov action reduces to the NG action.

It would not be a good thing if the Polyakov action would introduce other symmetries to the remaining fields, or, for that matter, get rid of some of the existing ones. For example, this could mean that some Noether currents and conserved charges would be present in one but absent in the other.

 

[CompuChip]

Thanks for your reply xepma.
So as I suspected, there is only one symmetry associated with the NG-action, and the other symmetries (reparametrization of h and Weyl symmetry) arise because of the introduction of the "auxiliary" field h.

 

[Alwi]

Well the introduction of the world-sheet metric is to simplify the problem at hand. One can begin with the Nabu-Goto action and then impose Weyl symmetry etc. However it will be ugly and terribly messy owing to the non-polynomial nature of the Nambu-Goto action.

However - in the Nambu-Goto framework, one can define the linking number etc for knotted strings. I have yet to see such an extension in the Polyakove framework. perhaps because in the Polyakov formalism, one has to deal with Riemann surfaces right from the start and that rules out other more exotic objects like Algebraic Surfaces spanned by a knotted string.

Best Regards
Alwi

 

[buddychimp]

Hi! I have little questions about symmetries. I begin in the field, so...

First about conformal symmetry. As I studied, in 2-d, a transformation $$(\tau, \sigma) \to (\tau', \sigma')$$ changing the metric by a multiplicative factor implies that the transformation $$(\tau, \sigma) \to (\tau', \sigma')$$ satisfies Cauchy-Riemann equations : $$\partial_\tau \tau' (\tau, \sigma) = \partial_\sigma \sigma'(\tau, \sigma)$$ and $$\partial_\sigma \tau' (\tau, \sigma) = - \partial_\tau \sigma'(\tau, \sigma)$$. Under such a transformation $$(\tau, \sigma) \to (\tau', \sigma')$$, one can verify that the Polyakov action remains unchanged and we say the action is conformally invariant. (Correct?)

What is not clear to me is the following. We also have reparametrization invariance. But I would be tempted to say that reparametrization implies conformal symmetry since it seems to be more general: we still start from a transformation $$(\tau, \sigma) \to (\tau', \sigma')$$, but without the constraints $$\partial_\tau \tau' (\tau, \sigma) = \partial_\sigma \sigma'(\tau, \sigma)$$. I'm wrong somewhere, but I can't figure out where.

Thanks for your help.

 

[crackjack]

@ Alwi
By imposing a Weyl symmetry on Nambu-Goto action, are you imposing that on the space-time metric? In that case, wouldn't such a symmetry just add to the existing (flat space-time) symmetries of Polyakov action?
I think CompuChip was rather talking of Weyl symmetries of world-sheet metric which are purely from the addition of the auxiliary field right?

edit: just now looked at the date of the posts! @buddychimp: why would you post such a question here??