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Suppose you define a Euclidean two-dimensional CFT for a scalar field $X$ of dimension length, $$ S = \int d^2x h{}_{\mu\nu} \partial^\mu X \partial{}^\nu X \, , $$ and $h{}_{\mu\nu}$ is the metric of that Euclidean space. This action is invariant under the infinite-dimensional conformal transformations, and hence we can employ techniques from 2D CFTs. In particular, in string theory, one can think of the scalar fields X (or, rather, $N$ copies of them) as target space coordinates. Avoiding the conformal anomaly after Faddeev--Popov gauge fixing then demands that N=26 (in the bosonic case).

My question: if the scalar fields $X$ are interpreted as the target space coordinates, then what do the worldsheet coordinates correspond to?

A different way to ask the same question: the worldsheet, as far as I understand, is a submanifold of our spacetime, because we can observe strings (at least theoretically) floating around. But if I need to introduce 26 scalar fields to cancel the conformal anomaly, and I take this number to be the critical dimensionality for bosonic string theory, don't I miss the two coordinates of the worldsheet?

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  • $\begingroup$ The relationship between worldsheet coordinates and target space, is like the relationship between latitude and longitude (coordinates for Earth's surface) and three-dimensional space. $\endgroup$ – Mitchell Porter Mar 29 at 3:38
  • $\begingroup$ This is precisely my question. I understand what a submanifold is but I am not sure whether this notion makes sense always. You see: you need N=26 scalar fields to cancel the conformal anomaly, and this fact is then interpreted as an information about the dimensionality of full spacetime. But if you define your CFT for these 26 coordinate scalar fields on a Euclidean geometry, is it considered part of spacetime? Because then spacetime should be 26+2=28-dimensional. In other words: I understand what you are saying, but what you say is just one of the options and not by itself an explanation. $\endgroup$ – Jens Mar 29 at 16:53

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