Conformal Algebra
Conformal Algebra
Conformal Algebra
Dana Faiez
• Conformal Transformation/Generators
• 2D Conformal Algebra
• Virasoro Algebra
ds 2 = gµν (x)dx µ dx ν
Change of coordinates
0
x ⇒ x (x)
∂x µ 0 ρ
dx µ = dx
∂x 0 ρ
This coordinate transformation keeps distance between two nearby
points ds invariant.
∂x µ ∂x ν 0 ρ 0 σ 0 0 0 0
ds 2 = gµν (x) dx dx = gρσ (x )dx ρ dx σ
∂x 0 ρ ∂x 0 σ
Under a change of coordinates, the metric transforms according to
0 0 ∂x µ ∂x ν
gρσ (x ) = gµν (x)
∂x 0 ρ ∂x 0 σ
Ex: Scale transformation
(for λ real number)
0µ
xµ → x = λx µ
then g transforms like:
0 0 0
gρσ (x ) = gρσ (λx) = Ω2 (x)gµν
such that:
ωx
Ω(x) ≡ e 2
0 0 1
where w = −2lnλ ⇒ gρσ (x ) = g (x)
λ2 µν
More generally, we could have
0
gρσ (x 0 ) = Ω2 (x)gµν (x)
0
Transformations x → x (x) that satisfy this condition
are known as conformal transformations.
Conformal Group
gµν = ηµν
∂ρ ζσ + ∂σ ζρ + κηρσ = 0
−2(∂.ζ) = dκ
d∂ 2 ζσ = (2 − d)∂σ (∂.ζ)
translation
Lorentz scale Special conformal
z}|{ z }| { z}|{ z }| {
ζµ = a µ
+ bν x + cx µ + dν (η µν x 2 − 2x µ x ν )
µ ν
[J MN , J PQ ] = −η MP J NQ − η NQ J MP + η NP J MQ + η MQ J NP
D = J d,d+1
P µ and K µ L.C . of J µ,d and J µ,d+1
Guess:
1
J µ,d = (K µ + P µ )
2
1
J µ,d+1 = (K µ − P µ )
2
These can be checked using commutation relations of Js.
∂0 ζ0 = +∂1 ζ1 , ∂0 ζ1 = −∂1 ζ0
ln = −z n+1 ∂z , l n = −z n+1 ∂z ,
[lm , ln ] = (m − n)lm+n
[l m , l n ] = (m − n)l m+n
[lm , l n ] = 0
z
c l1 z = −cz 2 is the infinitesimal version of z → cz+1 .
az+b
z→ cz+d with a, b, c, d ∈ C requiring ad − bc 6= 0
SL(2,C)
The conformal group of S 2 ' C ∪ ∞ is the Mobius group Z2 .
Conformal Field Theory
A field theory that is invariant under conformal transformations.
⇒ The physics of the theory looks the same at all length scales
(theory can’t have any preferred length scale. There can not be
anything like a mass, Compton wavelength etc.)
Tµµ = 0
Consider 2D Euclidean:
∂x α ∂x β
Tµν transforms as Tµν = ∂x µ ∂x ν Tαβ .
∂f 2 c
T 0 (z) = ( ) T (f (z)) + S(f (z), z)
∂z 12
c is called the central charge and S(w, z) denotes the Schwarzian
derivative.
1
S(w , z) = (∂z w )2
((∂z w )(∂z3 w ) − 32 (∂z2 w )2 ).
c
[Lm , Ln ] = (m − n)Lm+n + (m − m3 )δm+n,0
12
Identity with h = h = 0,
1
”spin field σ” with h = h = 16 ,
h 1
T (z)φ(w , w ) = 2
φ(w , w ) + ∂w φ(w , w ) + ...
(z − w ) (z − w )
h 1
T (z)φ(w , w ) = 2
φ(w , w ) + ∂w φ(w , w ) + ...
(z − w ) (z − w )
c/2 2T (w ) ∂T (w )
T (z)T (w ) = + + + ...
(z − w )4 (z − w )2 (z − w )
Want to show that even though φ(z) itself does not have a
conformal dimension, the fields ∂z φ(z) have definite dimensions ,
i.e. they transform as primary fields:
We need:
1
∂φ(z)∂φ(w ) = − (z−w )2
+ ...
∂φ(w ) 1
Tφ (z)∂φ(W ) ≈ 2
+ ∂ 2 φ(w ).
(z − w ) (z − w )
From this we conclude that a single free chiral boson has c=1.
CFTs in higher dimensions