Static gauge for strings in curved background

[haushofer]

Hi,

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

$$\tau = X^0$$

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

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

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

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

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

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

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 is not a free wave equation.

Any suggestions? :)

 

[haushofer]

For general geometries, one apparently cannot choose this static gauge anymore (only in a point) because of the inhomogenous term in the geodesic equation. For the same reason the lightcone gauge cannot be chosen, so it seems to me that the usual argument that only the transverse coordinates $X^I$, where I=2,3,...,D-1 are "true degrees of freedom" might break down, although this seems a bit counterintuitive to me.

So my question is: does a classical string in a curved spacetime only have transverse degrees of freedom, as in the flat case?

 

[haushofer]

Nobody here knows of a general statement that one can always (or: for arbitrary backgrounds) gauge away the longitudinal coordinates such that the transverse coordinates $X^I$ stay as "true degrees of freedom"?

Is this just a silly question, or is it beyond textbook?

 

[crackjack]

See:
http://arxiv.org/abs/hep-th/9402106
http://prd.aps.org/abstract/PRD/v42/i6/p1950_1 (referred to in above paper)

So, as far as I understand, conformal gauge and light-cone gauge can be simultaneously applied only if there is a covariantly constant null vector.