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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?

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  • $\begingroup$ Chiral anomaly is not zero in the SM. Gauge anomaly is. $\endgroup$
    – Kosm
    Apr 24, 2019 at 20:38
  • $\begingroup$ Although I doubt anyone would think I'm talking about the (global) chiral anomaly, I've now specified that I'm talking about the (gauge) chiral anomaly. $\endgroup$
    – thedoctar
    Apr 25, 2019 at 15:03
  • $\begingroup$ What's wrong with "pruning configuration space to ensure self-consistency"? $\endgroup$
    – Kosm
    Apr 26, 2019 at 14:45
  • $\begingroup$ @Kosm It's not done in any other gauge theory and doesn't give a dynamical explanation of gravity/local conformal symmetry. The point is we need to assume self-consistency, which isn't really self-consistency. Self-consistency means no contradiction without extra assumptions. If string theory were truly the fundamental theory, then it would seem that it is more than a gauge theory (hence the pruning). If it were just a gauge theory, it would not be self-consistent. $\endgroup$
    – thedoctar
    Apr 29, 2019 at 13:28
  • $\begingroup$ Let's put it this way. There are infinitely many string theories, and among those we choose only self-consistent ones. For self-consistent theories gravitons obey EFE. $\endgroup$
    – Kosm
    Apr 29, 2019 at 14:02

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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.

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  • $\begingroup$ Thanks for your answer. I agree with everything you say, but the statement: > In the case β≠0, string theory is not self-consistent. irks me. Doesn't this mean string theory is not automatically self-consistent? How does one actually construct a self-consistent string theory? $\endgroup$
    – thedoctar
    Feb 19 at 4:01

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