Is string theory self-consistent? (Conformal anomaly)

Question

Recently I attended a very short course on string theory. We went through the standard presentation in light-cone gauge for brevity. We ‘derived’ the Einstein field equation in the following manner. Partial gauge fixing in the conformal gauge leaves a residual (local) conformal symmetry. Since gauge (local) symmetries must be preserved at the quantum level for self-consistency, the beta function (which generically breaks scale and thus conformal symmetry by introducing anomalous dimensions c.f. Callan-Symanzik equation) must be zero (after fixing the dimension to be critical).

The claim is that $\beta=0$ gives you the Einstein field equations. On the other hand, $\beta=0$ is not usually considered as a dynamical equation. E.g. the Standard Model is a chiral gauge theory whose (gauge) chiral anomaly is zero due to cancellation from the charges of the fields. The lecturer told me some dodgy answer about how the induced effective theory had the Einstein field equations as dynamical equations, but I’m interested in considering string theory as a fundamental theory. The effective theory is irrelevant.

It seems to me we are merely pruning the configuration space of dynamically allowed fields to ensure self-consistency. I.e. string theory is not self-consistent: we need something extra to impose the Einstein field equations.

Why can we set $\beta =0$ when this is never done in any other field theory?

Answer 1

I think the point is the following:

from the point of view of the fields in the string worldsheet (in which the values of the fields are the coordinates of the string "points" in the target spacetime), $\beta = 0$ is not a dynamical equation but it is the condition on the target spacetime for the field theory in the worldsheet to be consistent (i.e., anomaly free).

Said in a different way, strings only "like to live" in a spacetime wich satisfies Einstein's Field Equations (EFE). As pointed out by Kosm, a string living and oscilating in a spacetime not satisfying EFE is not "confortable", it cannot live there, the QFT on the worldsheet is not conformally invariant.

For string theory to be consistent the target spacetime has to obey EFE. But this does not mean that $\beta = 0$ is a dynamical equation, it is only a condition for consistency. In the case $\beta \neq 0$, string theory is not self-consistent.