Redshift Academy


Introduction to Conformal Field Theory 
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Conformal Group
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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
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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 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/∂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_μδ^σ_ν + δ^ρ_μ(∂ε^σ/∂x^ν) + (∂ε^ρ/∂x^μ)δ^σ_ν + O(ε²)) = Ωg_μν

∴ g_μν + g_ρσδ^ρ_μ(∂ε^σ/∂x^ν) + (∂ε^ρ/∂x^μ)g_ρσδ^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^μνkg_μν

∴ g^μν((∂ε_μ/∂x^ν) + (∂ε_ν/∂x^μ)) = kδ^μ_μ

∴ g^μν(∂_νε_μ + ∂_με_ν) = kδ^μ_μ

∴ ∂^με_μ + ∂^νε_ν = kδ^μ_μ

∴ 2∂^μ_με = kd

Now ∂^μ_με = ∂⁰ε₀ + ∂¹ε₁ + ... + ∂^dε_d ≡ ∂.ε

Therefore,

2(∂.ε) = kd

or,

k = 2(∂.ε)/d

Therefore 2. becomes,

∂_νε_μ + ∂_με_ν = (2∂.ε)/d)g_μν = Δg_μν ... 3.

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

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

(∂_με_ν + ∂_νε_μ) = (∂.ε)δ_μν

μ = ν gives:

∂₁ε₁ = ∂₂ε₂ 

and μ ≠ ν gives:

∂₁ε₂ = -∂₂ε₁

In other words, ε_μ satisfies the Cauchy-Riemann 
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:

ds² -> ds² + (∂_με_ν + ∂_νε_μ)dx^μdx^ν

We now need one more equation.  We start with 
∂_με_ν + ∂_νε_μ = (2/d)(∂.ε)g_μν and take the 
derivative of both sides w.r.t ∂^ν.

∂^ν(∂_με_ν + ∂_νε_μ) = ∂^ν((2/d)(∂.ε)g_μν)

∴ ∂_μ∂^νε_ν + ∂^ν∂_νε_μ = (2/d)∂_μ(∂.ε)

∴ ∂_μ(∂.ε) + ∂^ν∂_νε_μ = (2/d)∂_μ(∂.ε)

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

∂_ν(∂_μ(∂.ε) + ∂^ν∂_νε_μ) = (2/d)∂_ν∂_μ(∂.ε)

∴ ∂_μ∂_ν(∂.ε) + ∂^ν∂_ν∂_νε_μ = (2/d)∂_ν∂_μ(∂.ε)

∴ ∂_μ∂_ν(∂.ε) + □∂_νε_μ = (2/d)∂_ν∂_μ(∂.ε) ... 4.

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

Now, ∂_νε_μ = (1/2)(∂_νε_μ + ∂_με_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 
stress-energy 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 -> Ω²η_ab  
       ----
         ^
         |
     conformally
     flat form

So,

ds² = Ω²δ_abdx^adx^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 = Ω²C_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 
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.