People frequently use fluctuating string in AdS black hole (see 1,2,3 etc) to study dynamics of a "free external quark" in quark-gluon-plasma (QGP) at finite temperature. The background black hole geometry induces a black hole in the string world sheet. Many interesting results about the boundary theory can be obtained, at least qualitatively, just studying the Nambu-Goto action.
My question is : Why do people usually choose just one of the transverse spatial directions along which the world sheet can fluctuate? Suppose there is a background magnetic field in addition. Shouldn't one need at least two spatial directions for the world sheet to fluctuate to observe/study its effect on the "quark"?
EDIT : Please look at the paragraph just below equation (2.7) in ref-1. The authors are considering only one coordinate namely $x$ which is non-vanishing. This is the only direction along which the string world sheet can fluctuate.