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[introduction]

First let me recall that there are two equivalent classical bosonic string actions, the Nambu-Goto action

[tex]S_\mathrm{NG} = - T \iint \mathrm d\sigma \, \mathrm d\tau \sqrt{ (\dot X X')^2 - \dot X^2 {X'}^2}[/tex]

and the Polyakov action

[tex]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,[/tex]

where the worldsheet metric [itex]h_{\alpha\beta}[/itex] 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 ([itex]X^\mu \to X^\mu + \xi^\alpha \partial_\alpha X^\mu + \mathcal O(\xi^2)[/itex] with corresponding transformation on the metric) and Weyl invariance ([itex]h_{\alpha\beta} \to \exp(-2\Lambda) h_{\alpha\beta}[/itex]).

[/introduction]

Myquestionis what the symmetries of the Nambu-Goto action are. My lecture notes give just the reparametrization invariance

[tex]\delta X^\mu = \xi^\alpha \partial_\alpha X^\mu[/tex]

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 [itex]\xi^\alpha[/itex] ?

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# Symmetries of NG action?

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