Re: D!=26 anomalies

torre@xxxxxxxxxx wrote:

Conformal transformations is not subgroup of diffeomorphisms.

Actually, it is.

Given a smooth manifold, diffeomorphisms form a transformation group
of the manifold. Given a metric on that manifold, one can ask for the
set of diffeomorphisms which act (by pull-back) on the metric such
that the pulled-back metric is proportional to the original metric.
It is easy to see that such transformations form a subgroup of the
diffeomorphism group.

We have here a different treatment of term "conformal transformation".
Manifold with metric can be viewed as pair (manifold,metric).
Morphism also can be viewed as pair (diffeomorphism, ...)
Conformal transformation can be defined as such morphism wich
only deform metric.
But we know that the usual diffeomorphism induce metric transformation.
In same cases it gives metric "proportional to the original metric".
In this picture conformal transformations explore possible symmetries
of our space.
In string theory world*** metric is independent variable. And we
must study metric changes wich is not diffeomorphisms.

For a two-dimensional
manifold with simple topology (e.g., the plane, or the cylinder) this
subgroup is non-trivial; it is infinite-dimensional in fact. It is
this group I was calling "the conformal group" - using "the Weyl
group" to denote the group of rescalings of the metric by functions.

Hm. I always thinks that by definition:
local conformal group = Weyl group
conformal group is subgroup of local conformal group.

.

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