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  Conformal group in 2D being a subgroup of Diff/Weyl - Polchinksi's 'String Theory'

+ 2 like - 0 dislike
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In the appendix on page 364 of 'String Theory', Polchinski defines the conformal group (Conf) in two dimensions to be the set of all holomorphic maps. On page 85 he explains how Conf is a subgroup of the direct product of the diffeomorphism (diff) and Weyl groups, denoted as (diff $\times$ Weyl) (here, diffeomorphisms refer to general coordinate transformations).

This is shown by first showing that Conf is a subgroup of Diff, by choosing the transformation function, $f$ to be holomorphic ($f(z)$). This is followed by showing that specific Weyl transformations with Weyl function \begin{equation} \omega=\textrm{ln}|\partial_zf| \end{equation}
can undo the conformal transformation.

This seems to imply that Conf is a subgroup of Weyl. In other words, Conf is a subgroup of diff, and Conf is a subgroup of Weyl. This then implies that Conf is a subgroup of (diff $\times$ Weyl).

However, in this post, Lubos Motl mentions that the conformal group is NOT a subgroup of the Weyl group. Why is there this inconsistency?

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user Mtheorist
asked Jul 27, 2016 in Theoretical Physics by Mtheorist (100 points) [ no revision ]
1. Weyl group does not mean what you think it means. 2. I don't understand how you think that the fact that the effect of conformal transformations on the metric can be undone by Weyl transformations implies they are a subset of Weyl transformations, which explicitly only act on the metric, while a conformation transformation is always a coordinate transformation/diffeomorphism by definition.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user ACuriousMind
I understand that the conformal transformations are coordinate transformations, while Weyl transformations are local rescalings of the metric. My point is, why say that conformal transformations are a subgroup of diff x Weyl, when they are a subgroup of diff alone, as you say.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user Mtheorist
If $H\subset G$ is a subgroup, then $H\times\{1\} \subset G\times K$ is a subgroup of $G\times K$ regardless of $K$.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user ACuriousMind
As explained on page 542 of Nakahara's Geometry, Topology and Physics, the Conformal Killing Vectors which generate the conformal group can be identified with the OVERLAP of Diff_0 and Weyl, hence, it is highly unlikely that Weyl plays a trivial role here.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user Mtheorist
Conformal transformations are coordinate transformations and as such are a symmetry of theory which are invariant under general coordinate transformations. On the other hand if you consider a theory on a fixed background metric then these conformal transformations will not be a symmetry. For this to be the case you need more: you need that the original diffeomorphism invariant theory is invariant under the Weyl symmetry. This allows you to compensate the transformation of the background metric and as a consequence to get an invariance of the action just in term of spacetime symmetries.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user Harold
I would suggest these two papers that bring interesting insights on these questions: hep-th/9607110, 1510.08042.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user Harold
While the Weyl transformations are conceptually different from conformal transformations, one can identity a subset of Weyl transformations with conformal transformations- notice that in order for $w$ to be able to be written as $\log |\partial_z f|$, $w$ needs to be harmonic. On the other hand, conformal transformations can only give rise to $e^{2w}$ with harmonic $w$. With this identification one thinks conformal transformations as a subgroup as Weyl transformations.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user user110373
@ACuriousMind In this case, the picture should be that $H$ is defined as a subgroup of $G$ but is isomorphic to a subgroup of $K$.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user user110373
@user110373 I agree with this, which brings me back to my original question, why the discrepancy with Lubos Motl's statement?

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user Mtheorist
@Harold Then there is an inconsistency in Polchinski's definitions, on one hand the conformal group includes holomorphic reparametrizations together with Weyl transformations which preserve the unit metric, on the other hand he also defines it to be the set of all holomorphic reparametrizations. I will take a look at the references, thanks.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user Mtheorist
@Mtheorist I realised I said a wrong statement: while it is certainly true that conformal transformations can be identified as a subset of Weyl transformations, it is not a subgroup. They compose differently. Also there is no discrepancy with Lubos Motl's statement as he did not claim there is not a map that identifies conformal transformations as Weyl transformations. By its definition, conformal transformations are not Weyl transformations.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user user110373
@ACuriousMind Please ignore my previous comment.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user user110373
@Harold From your reference, arxiv.org/abs/1510.08042, it is mentioned on page 2 that 'It is clear that Weyl invariance implies conformal invariance, but not the other way around...'. So it seems that conformal transformations are indeed a subset of Weyl transformations, as user110373 has also asserted.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user Mtheorist
@Mtheorist The distinction is similar to the one between the change of coordinate matrix and the automorphism of the vector space represented by the same matrix in some coordinate.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user user110373
@user110373 Okay, so I think you have answered the question. Weyl transformations can be identified as a SUBSET but not a SUBGROUP of conformal transformations, since the composition law is different. I hope you can answer the question, hopefully with a little more mathematical detail for clarity, once this question is no longer on 'hold'.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user Mtheorist
@Mtheorist: I don't understand this sentence in this way: the Weyl transformations are useful to obtain a conformal theory from a diffeomorphism invariant theory, but there conformal theory can be built in more general ways. In particular Weyl transformations cannot be a subset of conformal transformations since the latter are coordinate transformations while the first ones are not. In some way you can understand conformal transformations as the isometries of a diffeo invariant theory up to Weyl transformations.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user Harold

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