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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: (Ω_{1}.Ω_{2}).Ω_{3} = Ω_{1}.(Ω_{2}).Ω_{3}
(Ω_{1} + Ω_{2}) + Ω_{3} = Ω_{1} + (Ω_{2}) + Ω_{3}
Inverse: Ω_{1}.Ω_{1} = I
Ω_{1} + Ω_{1} = 0
Closure: Ω_{1}.Ω_{2} = Ω_{3}
Ω_{1} + Ω_{2} = Ω_{3}
All conformal groups are Lie groups.
Rescaling versus Coordinate Transformations

Coordinates Transformations

The equation:
g_{μν}(x) = (∂x'^{ρ}/∂x^{μ})(∂x'^{σ}/∂x^{ν})g_{ρσ}(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:
g_{μν}(x) > Ω(x)g_{μν}
Where Ω(x) is realvalued 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}/∂x^{μ})(∂x'^{σ}/∂x^{ν})g_{ρσ}(x') = Ω(x)g_{μν}(x)
^ ^
 
Coordinate transformation Weyl transformation
Metrics obeying this equation are said to be conformally
equivalent.
Consider the infinitesimal transformation x'^{μ} = x^{μ} + ε^{μ}(x)
Expanding the LHS using ∂x^{μ}/∂x^{ν} = δ^{μ}_{ν}:
∂x'^{ρ}/∂x^{μ} = ∂x'^{ρ}/∂x^{μ} + ∂ε^{ρ}/∂x^{μ}
= δ^{ρ}_{μ} + ∂ε^{ρ}/∂x^{μ}
Likewise,
∂x'^{σ}/∂x^{ν} = δ^{σ}_{ν} + ∂ε^{σ}/∂x^{ν}
Therefore,
g_{ρσ}(δ^{ρ}_{μ} + ∂ε^{ρ}/∂x^{μ})(δ^{σ}_{ν} + ∂ε^{σ}/∂x^{ν}) = Ωg_{μν}
∴ g_{ρσ}(δ^{ρ}_{μ}δ^{σ}_{ν} + δ^{ρ}_{μ}(∂ε^{σ}/∂x^{ν}) + (∂ε^{ρ}/∂x^{μ})δ^{σ}_{ν} + O(ε^{2})) = Ωg_{μν}
∴ g_{μν} + g_{ρσ}δ^{ρ}_{μ}(∂ε^{σ}/∂x^{ν}) + (∂ε^{ρ}/∂x^{μ})g_{ρσ}δ^{σ}_{ν}) = Ωg_{μν}
∴ g_{μν} + (∂ε_{μ}/∂x^{ν}) + (∂ε_{ν}/∂x^{μ}) = Ωg_{μν}
Therefore,
g_{μν} + Δg_{μν} = Ωg_{μν} ... 1a.
Where,
Δg_{μν} = (∂ε_{μ}/∂x^{ν}) + (∂ε_{ν}/∂x^{μ}) ... 1b.
Now, (∂ε_{μ}/∂x^{ν}) + (∂ε_{ν}/∂x^{μ}) must be proportional to g_{μν}
Therefore,
(∂ε_{μ}/∂x^{ν}) + (∂ε_{ν}/∂x^{μ}) = kg_{μν} ... 2.
Now taking the trace of both sides gives:
g^{μν}((∂ε_{μ}/∂x^{ν}) + (∂ε_{ν}/∂x^{μ})) = g^{μν}kg_{μν}
∴ g^{μν}((∂ε_{μ}/∂x^{ν}) + (∂ε_{ν}/∂x^{μ})) = kδ^{μ}_{μ}
∴ g^{μν}(∂_{ν}ε_{μ} + ∂_{μ}ε_{ν}) = kδ^{μ}_{μ}
∴ ∂^{μ}ε_{μ} + ∂^{ν}ε_{ν} = kδ^{μ}_{μ}
∴ 2∂^{μ}_{μ}ε = kd
Now ∂^{μ}_{μ}ε = ∂^{0}ε_{0} + ∂^{1}ε_{1} + ... + ∂^{d}ε_{d} ≡ ∂.ε
Therefore,
2(∂.ε) = kd
or,
k = 2(∂.ε)/d
Therefore 2. becomes,
∂_{ν}ε_{μ} + ∂_{μ}ε_{ν} = (2∂.ε)/d)g_{μν} = Δg_{μν} ... 3.
2dimensions

The case where d = 2 is of special interest. If we
let g_{μν} = δ_{μν} we can write 3. as:
(∂_{μ}ε_{ν} + ∂_{ν}ε_{μ}) = (∂.ε)δ_{μν}
μ = ν gives:
∂_{1}ε_{1} = ∂_{2}ε_{2}
and μ ≠ ν gives:
∂_{1}ε_{2} = ∂_{2}ε_{1}
In other words, ε_{μ} satisfies the CauchyRiemann
equations.
Combining 3. with 1a. and 1b. gives:
g_{μν} + g_{μν}(2/d)∂.ε = Ωg_{μν}
∴ g_{μν}(1 + (2/d)∂.ε) = Ωg_{μν}
Or, by comparing:
Ω = 1 + (2/d)∂.ε
Note that:
ds^{2} > ds^{2} + (∂_{μ}ε_{ν} + ∂_{ν}ε_{μ})dx^{μ}dx^{ν}
We now need one more equation. We start with
∂_{μ}ε_{ν} + ∂_{ν}ε_{μ} = (2/d)(∂.ε)g_{μν} and take the
derivative of both sides w.r.t ∂^{ν}.
∂^{ν}(∂_{μ}ε_{ν} + ∂_{ν}ε_{μ}) = ∂^{ν}((2/d)(∂.ε)g_{μν})
∴ ∂_{μ}∂^{ν}ε_{ν} + ∂^{ν}∂_{ν}ε_{μ} = (2/d)∂_{μ}(∂.ε)
∴ ∂_{μ}(∂.ε) + ∂^{ν}∂_{ν}ε_{μ} = (2/d)∂_{μ}(∂.ε)
Now take the derivative of both sides of this
w.r.t. ∂_{ν}.
∂_{ν}(∂_{μ}(∂.ε) + ∂^{ν}∂_{ν}ε_{μ}) = (2/d)∂_{ν}∂_{μ}(∂.ε)
∴ ∂_{μ}∂_{ν}(∂.ε) + ∂^{ν}∂_{ν}∂_{ν}ε_{μ} = (2/d)∂_{ν}∂_{μ}(∂.ε)
∴ ∂_{μ}∂_{ν}(∂.ε) + □∂_{ν}ε_{μ} = (2/d)∂_{ν}∂_{μ}(∂.ε) ... 4.
Where □ is the d'Alembert operator = ∂^{ν}∂_{ν}.
Now, ∂_{ν}ε_{μ} = (1/2)(∂_{ν}ε_{μ} + ∂_{μ}ε_{ν})
= (1/d)(∂.ε)g_{μν}
Using this, 4. becomes:
∂_{μ}∂_{ν}(∂.ε) + □(1/d)(∂.ε)g_{μν} = (2/d)∂_{ν}∂_{μ}(∂.ε)
Which simplifies to:
∂_{μ}∂_{ν}(∂.ε)d + □(∂.ε)g_{μν} = 2∂_{ν}∂_{μ}(∂.ε)
Rearranging:
g_{μν}(□ + (d  2)∂_{μ}∂_{ν})(∂.ε) = 0
Conformal Manifolds

The Weyl Tensor

The Riemann tensor, R^{a}_{bcd}, 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 R_{bd} is obtained by contracting in the
first and third indeces of the Riemann tensor. The
energy/matter source of Ricci curvature is the
stressenergy tensor, T_{μν}.
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,
g_{ab} > Ω^{2}η_{ab}

^

conformally
flat form
So,
ds^{2} = Ω^{2}δ_{ab}dx^{a}dx^{b}
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} = Ω^{2}C_{abcd}
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
2sphere can be stereographically projected onto
a 2D plane as shown below. Therefore, the 2sphere
is locally conformally flat but is not itself flat
for obvious reasons.