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

Question

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

Comments

@ACuriousMind Please ignore my previous comment.

This post imported from StackExchange Physics at 2016-07-29 21:43 (UTC), posted by SE-user user110373

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@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

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@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

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@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

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@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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@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

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@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