# Is string theory self-consistent? (Conformal anomaly)

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?

• Chiral anomaly is not zero in the SM. Gauge anomaly is. – Kosm Apr 24 at 20:38
• 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. – thedoctar Apr 25 at 15:03
• What's wrong with "pruning configuration space to ensure self-consistency"? – Kosm Apr 26 at 14:45
• @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. – thedoctar Apr 29 at 13:28
• 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. – Kosm Apr 29 at 14:02