# Critical dimension from the symmetries of the string action

(Related: This post and this post.)

In this thesis it is said (on page 13) that just by assuming that we have some general action with the same symmetries as the Polyakov action (Poincare invariance, reparametrization invariance and Weyl invariance) one can deduce that the conformal symmetry (that I believe can be assumed to exist in this general case simply because of Weyl symmetry) must have central charge $$c=26$$.

The usual way I see $$c=26$$ being derived uses the ghost action (which has $$c=-26$$). So my question can be reformulated as: Is this ghost action always the same for any general model with the same symmetries of the Polyakov model? How exactly are the symmetries used in constructing the ghost action?

• In general terms, the Faddeev-Popov determinant is fixed once you know the form of the gauge symmetry and gauge fixing functional. The rest of the action is irrelevant. – AccidentalFourierTransform Oct 5 '19 at 11:42
• @AccidentalFourierTransform Nice. Do you know if this is discussed in any of the usual String theory textbooks? – Soap Oct 5 '19 at 11:43
• Not sure, but probably. I think Polchinski discusses gauge fixing in general QFTs. In any case, any QFT book should do; say, Weinberg's Vol. I. – AccidentalFourierTransform Oct 5 '19 at 11:46
• @AccidentalFourierTransform Revisiting Tong's lecture notes on String theory, the way he constructs the ghost action really seems completely independent of the form of the Polyakov action. It seems that Weyl and diffeo invariance are important just so that gauge fixing (and thus the introduction of the Fadeev Popov determinant) make sense in the first place. – Soap Oct 5 '19 at 12:07
• @AccidentalFourierTransform It seems that this is also true for the superstring, in which the Fadeev-Popov procedure fixes $N=1$ superconformal gauge symmetry instead of simply the conformal gauge symmetry. But does the Faddeev-Popov determinant depend on $N$? I see statements like: "Consider an $N=2$ superstring theory. We know that we must have $c=15$." But again this is only valid if the central charge of the ghost action is the same for $N=1$ and $N=2$. Related: this post – Soap Oct 9 '19 at 12:59