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There seems to be some confusion. In the referenced paper there is no temporal direction, and even though there is curvature there, it has nothing to do with gravity (there is no time). Where as when you compare with $2D$ gauge theory, one of those $2$ dimensions is temporal in deed. So if you wanted to carry the analogy you should consider two dimensions, ...

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There are a few different ways to see that the bosonic string lives in $D=26$. This, by the way, is known as the critical dimension of the theory. I'll give a brief sketch the answer, a more complete one can be found in any textbook, but in particular Polchinski's. Classically, the Polyakov action has 3 main symmetries. These are: 1) Lorentz invariance of ...

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In the representation theory of Virasoro algebra there is a mathematically strict theorem: http://en.wikipedia.org/wiki/Goddard%E2%80%93Thorn_theorem for bosonic string theory, and similarly for superstring theory. Intuitively the critical dimensions come from zeta function regularization.

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