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I have a question about the static gauge in string theory, in which one sets

[tex]

\tau = X^0

[/tex]

I understand that in the usual approach for strings in a flat target space, after the gauge fixing of the worldsheet metric gamma,

[tex]

\gamma_{\alpha\beta} = \eta_{\alpha\beta}

[/tex]

one still has enough gauge symmetry left to choose the static gauge. See e.g. (2.3.7) of Green, Schwarz, Witten. My question is the following.

If one considers classical strings in curved target spaces, the wave equation for the embedding coordinates becomes

[tex]

\gamma^{\alpha\beta}(\partial_{\alpha} \partial_{\beta} X^{\rho} + \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} \Gamma^{\rho}_{\mu\nu}) = 0

[/tex]

Here alpha,beta,... are 0,1, while mu,nu,rho run from 0,...,D-1. First, one can choose the conformal gauge,

[tex]

\eta^{\alpha\beta}(\partial_{\alpha} \partial_{\beta} X^{\rho} + \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} \Gamma^{\rho}_{\mu\nu}) = 0

[/tex]

My question is: is one still able to choose after this conformal gauge for the worldsheet metric the static gauge for tau? The question arises, because the remaining gauge symmetries for tau and sigma show that the transformed tau obeys a free wave equation, see (2.3.7) of Green, Schwarz, Witten. However, X^0 obeys the equation above, which isnota free wave equation.

Any suggestions? :)

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# Static gauge for strings in curved background

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