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