Wolfram Alpha:

```Introduction to Conformal Field Theory
--------------------------------------

Conformal Group
---------------

The conformal group of a space is the group of
transformations from the space to itself that
preserve angles.

Associativity:  (Ω1.Ω2).Ω3 = Ω1.(Ω2).Ω3

(Ω1 + Ω2) + Ω3 = Ω1 + (Ω2) + Ω3

Inverse:  Ω1.Ω-1 = I

Ω1 + Ω-1 = 0

Closure:  Ω1.Ω2 = Ω3

Ω1 + Ω2 = Ω3

All conformal groups are Lie groups.

Rescaling versus Coordinate Transformations
-------------------------------------------

Coordinates Transformations
---------------------------

The equation:

gμν(x) = (∂x'ρ/∂xμ)(∂x'σ/∂xν)gρσ(x')

represents an active coordinate transformation of
the metric from frame x' -> frame x.  This means
that the components of the metric change under the
transformation.

Weyl Transformation
-------------------

The equation:

gμν(x) -> Ω(x)gμν

Where Ω(x) is real-valued positive scalar function
defined on the manifold, represents a scaling of
the metric that changes the proper distances at each
point by a factor and the factor may depend on the place
– but not on the direction of the line whose proper
distance we measure (because Ω is a scalar).  In
other words, the Weyl transformation takes us to a
coordinate system where the metric has the same form
as the one we started with, but the points have all
been moved around and pushed closer together or further
apart depending on the scale factor (aka conformal
factor).  To summarize, it is a transformation which
acts on the metric tensor itself and not on the
coordinates.

Conformal Transformation
------------------------

A conformal transformation can now be defined as a
coordinate transformation which acts on the metric as
a Weyl transformation.   In other words, it is a
coordinate transformation that produces the same result
as scaling the metric by Ω(x).  Therefore,

(∂x'ρ/∂xμ)(∂x'σ/∂xν)gρσ(x') = Ω(x)gμν(x)
^                       ^
|                       |
Coordinate transformation  Weyl transformation

Metrics obeying this equation are said to be conformally
equivalent.

Consider the infinitesimal transformation x'μ = xμ + εμ(x)

Expanding the LHS using ∂xμ/∂xν = δμν:

∂x'ρ/∂xμ = ∂x'ρ/∂xμ + ∂ερ/∂xμ

= δρμ + ∂ερ/∂xμ

Likewise,

∂x'σ/∂xν = δσν + ∂εσ/∂xν

Therefore,

gρσ(δρμ + ∂ερ/∂xμ)(δσν + ∂εσ/∂xν) = Ωgμν

∴ gρσ(δρμδσν + δρμ(∂εσ/∂xν) + (∂ερ/∂xμ)δσν + O(ε2)) = Ωgμν

∴ gμν + gρσδρμ(∂εσ/∂xν) + (∂ερ/∂xμ)gρσδσν) = Ωgμν

∴ gμν + (∂εμ/∂xν) + (∂εν/∂xμ) = Ωgμν

Therefore,

gμν + Δgμν = Ωgμν ... 1a.

Where,

Δgμν = (∂εμ/∂xν) + (∂εν/∂xμ) ... 1b.

Now, (∂εμ/∂xν) + (∂εν/∂xμ) must be proportional to gμν

Therefore,

(∂εμ/∂xν) + (∂εν/∂xμ) = kgμν   ... 2.

Now taking the trace of both sides gives:

gμν((∂εμ/∂xν) + (∂εν/∂xμ)) = gμνkgμν

∴ gμν((∂εμ/∂xν) + (∂εν/∂xμ)) = kδμμ

∴ gμν(∂νεμ + ∂μεν) = kδμμ

∴ ∂μεμ + ∂νεν = kδμμ

∴ 2∂μμε = kd

Now ∂μμε = ∂0ε0 + ∂1ε1 + ... + ∂dεd ≡ ∂.ε

Therefore,

2(∂.ε) = kd

or,

k = 2(∂.ε)/d

Therefore 2. becomes,

∂νεμ + ∂μεν = (2∂.ε)/d)gμν = Δgμν ... 3.

2-dimensions
------------

The case where d = 2 is of special interest. If we
let gμν = δμν we can write 3. as:

(∂μεν + ∂νεμ) = (∂.ε)δμν

μ = ν gives:

∂1ε1 = ∂2ε2

and μ ≠ ν gives:

∂1ε2 = -∂2ε1

In other words, εμ satisfies the Cauchy-Riemann
equations.

Combining 3. with 1a. and 1b. gives:

gμν + gμν(2/d)∂.ε = Ωgμν

∴ gμν(1 + (2/d)∂.ε) = Ωgμν

Or, by comparing:

Ω = 1 + (2/d)∂.ε

Note that:

ds2 -> ds2 + (∂μεν + ∂νεμ)dxμdxν

∂μεν + ∂νεμ = (2/d)(∂.ε)gμν and take the
derivative of both sides w.r.t ∂ν.

∂ν(∂μεν + ∂νεμ) = ∂ν((2/d)(∂.ε)gμν)

∴ ∂μ∂νεν + ∂ν∂νεμ = (2/d)∂μ(∂.ε)

∴ ∂μ(∂.ε) + ∂ν∂νεμ = (2/d)∂μ(∂.ε)

Now take the derivative of both sides of this
w.r.t. ∂ν.

∂ν(∂μ(∂.ε) + ∂ν∂νεμ) = (2/d)∂ν∂μ(∂.ε)

∴ ∂μ∂ν(∂.ε) + ∂ν∂ν∂νεμ = (2/d)∂ν∂μ(∂.ε)

∴ ∂μ∂ν(∂.ε) + □∂νεμ = (2/d)∂ν∂μ(∂.ε) ... 4.

Where □ is the d'Alembert operator = ∂ν∂ν.

Now, ∂νεμ = (1/2)(∂νεμ + ∂μεν)

= (1/d)(∂.ε)gμν

Using this, 4. becomes:

∂μ∂ν(∂.ε) + □(1/d)(∂.ε)gμν = (2/d)∂ν∂μ(∂.ε)

Which simplifies to:

∂μ∂ν(∂.ε)d + □(∂.ε)gμν = 2∂ν∂μ(∂.ε)

Rearranging:

gμν(□ + (d - 2)∂μ∂ν)(∂.ε) = 0

Conformal Manifolds
-------------------

The Weyl Tensor
---------------

The Riemann tensor, Rabcd, keeps track of how much
a vector that is parallel transported around a
small parallelogram deviates from the its initial
direction after returning to its original position.
All the information about the curvature of a
Riemann manifold is contained in the Riemann
tensor.

The Ricci tensor represents the amount by which the
volume of an object differs from its volume in
Euclidean space in the presence of matter.  The
Ricci tensor Rbd is obtained by contracting in the
first and third indeces of the Riemann tensor.  The
energy/matter source of Ricci curvature is the
stress-energy tensor, Tμν.

The Weyl tensor is the part of the curvature that
exists if there is no matter present.  It is basically
the Riemann tensor with all of its contractions removed.
It differs from the Ricci tensor in that it does not
convey information on how the volume of the object
changes, but rather only how the shape of the body
is distorted by tidal forces.  Therefore,

(i)  If the Ricci tensor is zero but Weyl tensor is
not, the object will be deformed by tidal forces
without changing its volume.

(ii) If the Weyl tensor is zero but the Ricci tensor
is not, the object will only change its volume
but not its shape.

The Weyl tensor measures the deviation of a
Riemannian manifold from conformal flatness.  If
it vanishes, the manifold is locally conformally
equivalent to a flat manifold.  The fancy way
of saying this is that a manifold is conformally
flat if each point has a neighborhood, U, that can
be mapped to flat space by a conformal transformation,

gab -> Ω2ηab
----
^
|
conformally
flat form

So,

This does not mean that the manifold itself is
necessarily flat.

The Weyl tensor has the special property that it
is invariant under conformal changes.  That is:

C'abcd = Ω2Cabcd

For this reason the Weyl tensor is also called the
conformal tensor.

The Weyl tensor can be shown to be 0 for 2 and 3
dimensions.  Therefore, any 2D or 3D Riemannian
manifold is conformally flat.  For example, the
2-sphere can be stereographically projected onto
a 2D plane as shown below.  Therefore, the 2-sphere
is locally conformally flat but is not itself flat
for obvious reasons.

```