The Polyakov action, $S\sim \int d^2\sigma\sqrt{\gamma}\, \gamma_{ab}\partial^a X^\mu \partial ^b X_\mu$, has the well known classical symmetries of world sheet diffeomorphism invariance, world sheet Weyl invariance and target space Poincare invariance, the last two symmetries being $\gamma_{ab}\to e^{2\omega(\sigma)}\gamma_{ab}$ and $X^\mu\to \Lambda^\mu{}_\nu X^\nu+c^\mu$, respectively.

Treated quantum mechanically, it's well know that these last two symmetries are anomalous in general dimensions. They are only preserved in $d=26$.

Is it true that an anomaly in one symmetry implies an anomaly in the other? If so, what is a simple argument showing that this is the case?

It certainly does not seem to be a coincidence that both symmetries are anomalous away from the critical dimension, but I don't see the precise link between the two symmetries in the case of the Polyakov action.

While the two anomalies appear related, they also don't seem to be on entirely equal footing, at least to me. The loss of target space Poincare invariance is a true disaster, but the loss of world sheet Weyl invariance is not obviously so bad. Are generic CFT's on curved spacetimes pathological? If not, then I don't see why the loss of Weyl invariance in the present context is any worse than its loss in the case of a generic CFT on curved space, except to the extent that the Weyl anomaly might necessarily imply the loss of Poincare invariance, too, for our particular action.

This thought leads to my final question.

Imagine I could invent a different theory with generally anomalous world sheet Weyl and target space Poincare symmetries, but with the property that there exist two different critical dimensions, one where Weyl is non-anomalous ($d_{Weyl}$) and one where Poincare is non-anomalous ($d_{Poincare}$) with $d_{Weyl}\neq d_{Poincare}$. Is this hypothetical theory always pathological, or would it be healthy in $d=d_{Poincare}$ dimensions?

  • $\begingroup$ In string theory the conformal symmetry is really a gauge symmetry, so a conformal anomaly makes the theory inconsistent. On the other hand I'm not sure why you think an anomaly in the Poincaré symmetry would be such a disaster. From the strings point of view this is just an internal flavour symmetry $\endgroup$
    – Olof
    Jul 22, 2015 at 11:17
  • $\begingroup$ A conformally coupled scalar also has a classical Weyl gauge symmetry, which is broken by anomalies. Is such a theory inconsistent? Serious question; I'm not sure of the answer here. I would think it's fine since the Weyl symmetry just gives us the freedom to choose the conformal factor which isn't a degree of freedom anyway since it's removed by diffeomorphism invariance, which is preserved. Losing Weyl invariance doesn't seem to lead to extra degrees of invariance and pathologies in the same way that losing other gauge symmetries does. $\endgroup$
    – user26866
    Jul 22, 2015 at 16:49
  • $\begingroup$ As for the second question, when $d\neq 26$ the first excited string states don't belong to any representation of the appropriate little group of Lorentz and so can't be thought of as particles, which seems like a disaster. $\endgroup$
    – user26866
    Jul 22, 2015 at 16:51
  • $\begingroup$ The Weyl invariance is needed to get rid of one of the components of the worldsheet metric, so without it you do indeed get a spurious degree of freedom. See eg Lubos answer on this related question: physics.stackexchange.com/q/6792 $\endgroup$
    – Olof
    Jul 22, 2015 at 18:42
  • $\begingroup$ I'm not sure what the corresponding statement is in the case of a conformally coupled scalar. As for the Poincaré invariance: what you say is of course correct, but my point is that this is a problem for the spacetime interpretation of the string theory, not an inconsistency of the worldsheet theory. In principle one could have found that there was a different spacetime interpretation in terms of a non-trivial background. $\endgroup$
    – Olof
    Jul 22, 2015 at 18:48

1 Answer 1


Quantum anomalies usually result from a conflict between too many requirements that we brought from classical mechanics to quantum mechanics. For instance, in the case of bosonic string, we have a conflict between the Lorentz symmetry and the reparametrization symmetry of the worldsheet. One can start with the Nambu-Goto action. $$ L=d\tau d\sigma\sqrt{\det(\partial_{i}x^{m}\partial_{j}x^{n}\eta_{mn})},\qquad i,j=(\sigma,\tau) $$ Assuming reparametrization symmetry allows us to gauge fix the action above to the light-cone gauge. After that, the action becomes quadratic, and there is no square root. Since it is a free action (ignoring string interaction), one can quantize easily. Quantization will imply a break for the Lorentz symmetry if $d\neq 26$.

Alternatively, one might insist on Lorentz symmetry. To make sense of the square root, we introduce an intrinsic worldsheet metric $h_{ij}$ and write $$ L=d\tau d\sigma \sqrt{h}h^{ij}\partial_{i}x^{m}\partial_{j}x^{n}\eta_{mn}\,, $$ which adds a new gauge symmetry given by Weyl transformations $\delta h_{ij}=\Lambda h_{ij}$. One can use Weyl transformations to gauge the metric to become flat in some patch of the worldsheet. This will give the so-called conformal gauge where the worldsheet action is again free and quantizable. Quantization will break reparametrization invariance if $d\neq 26$.

I should mention that depending on how you formulate the conformal symmetry, the anomaly might be interpreted as a Weyl anomaly. For example, instead of using the Weyl transformation to fix the worldsheet intrinsic metric to be flat, one can use reparametrizations such that the intrinsic metric becomes $h_{ij}=e^{\phi}\delta_{ij}$. The anomaly will be present in the Weyl transformation if $d\neq 26$.

This means we are looking at the same anomaly from different points of view.


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