# Target Space Lorentz Invariance vs. World Sheet Weyl Invariance

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?

• 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
– Olof
Jul 22, 2015 at 11:17
• 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. Jul 22, 2015 at 16:49
• 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. Jul 22, 2015 at 16:51
• 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
– Olof
Jul 22, 2015 at 18:42
• 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.
– Olof
Jul 22, 2015 at 18:48

Quantum anomalies usually results from a conflict between to much requirements that we bring from classical mechanics to quantum mechanics. For the case of bosonic string we have a conflict between the Lorentz symmetry and the reparametrization symmetry of the worldsheet. One can start from 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 that it is possible to regularize and renormalize the theory such that reparametrization symmetry survive allow us to gauge fix the action above to light cone gauge: $$x^{+}=p^{+}\tau$$. After that the action becomes quadratic and there is no square root, however there will be an anomaly for the Lorentz symmetry if $$d\neq 26$$.
Alternatively, one might insist in Lorentz symmetry. In order to make sense of the square root we introduce a 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 (classically) given by Weyl transformations $$\delta h_{ij}=\Lambda h_{ij}$$. One can regularize and renormalize the theory such that the Weyl and Lorentz symmetry is preserved and gauge fix the metric to become flat in some patch of the worldsheet (i.e. locally). There will be an anomaly for the reparametrization symmetry if $$d\neq 26$$.
One can also regularize and renormalize in such a way that reparametrization and Lorentz symmetry is preserved, and gauge fix the coordinates such that the intrinsic metric becomes $$h_{ij}=e^{\phi}\delta_{ij}$$. The anomaly will now be present in the Weyl transformation if $$d\neq 26$$.