Pen and Paper/Mathematica
CFT Basics - Problem I
Conformal Symmetry and Correlators
Christopher Beem
Recall from the first lecture that a special conformal transformation acts on coordinates in
Rd as
xµ + bµ x2
x0µ = eiK·b xµ =
,
1 + 2(b · x) + (b · b)(x · x)
while an inversion acts according to
x0µ = I(x)µ =
xµ
.
x·x
(i ) Show that this transformation can be produced by conjugating a finite translation
with inversions,
eiK·b = I ◦ eiP ·b ◦ I .
(ii ) Show that the distance between two points in Rd transforms under a special conformal
transformation according to
|x01 − x02 | =
|x1 − x2 |
1/2 1/2
,
γ1 γ2
where γi := 1 + 2(b · xi ) + (b · b)(xi · xi ).
(iii ) Now show that under an arbitrary conformal transformation, the distance between
two points transforms according to
|x01 − x02 | = Ω(x1 )1/2 Ω(x2 )1/2 |x1 − x2 | ,
where Ω(x) is the local scale factor for the conformal transformation at x,
δµν
∂x0µ ∂x0ν
= Ω2 (x)δαβ .
∂xα ∂xβ
Now consider a correlation function of conformal primary scalar operators. Such a correlator will obey the conformal Ward identities
∆n
0
0
1
hO1 (x1 ) · · · On (xn )i = Ω∆
1 · · · Ωn hO1 (x1 ) · · · On (xn )i ,
where the ∆i are the scaling dimensions of the operators and Ωi := Ω(xi ).
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Pen and Paper/Mathematica
(iv ) Prove that the two-point function for primary scalar operators with dimensions ∆1
and ∆2 is
κ12 δ∆1 ,∆2
hO1 (x1 )O2 (x2 )i =
,
1
x2∆
12
where we define xij = |xi − xj |.
(v ) Prove that the three-point function of primary scalar operators with dimensions ∆i
is
λ123
hO1 (x1 )O2 (x2 )O3 (x3 )i = ∆123 ∆231 ∆312 ,
x12 x23 x31
where
∆ijk := ∆i + ∆j − ∆k .
(vi ) Prove that the most general form allowed by the conformal Ward identities for the
four-point function of identical scalar primary operators of scaling dimension ∆ is
hO(x1 )O(x2 )O(x3 )O(x4 )i =
1
F (u, v) ,
2∆
x2∆
12 x34
where u and v are conformal cross ratios that you should define. These are combinations of the coordinates xi that are invariant under arbitrary conformal transformations, so the function F can in principle be arbitrary.
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Pen and Paper/Mathematica
CFT Basics - Problem II
The Operator Product Expansion
Christopher Beem
Consider the operator product expansion of scalar primaries,
X
(a ···a )
O1 (x)O2 (0) ∼
C12k,a1 ···a` (x, P )Ok 1 ` (0) ,
k
where k runs over all conformal primaries and ∆k and ` refer to the scaling dimension and
spins of those operators.
(i ) Show that scale invariance (i.e., acting on both sides with the dilatation operator D)
implies that the differential operator C must obey
C12k,a1 ···a` =
f12k
F12k,a1 ···a` (x, P ) ,
|x|∆12k +`
where F12k,a1 ···a` obeys the homogeneity condition
F12k,a1 ···a` (λx, λ−1 P ) = λ` F12k,a1 ···a` (x, P ) .
(ii ) Specializing to the case where Ok is a scalar operator, we have an expansion in
derivatives,
F12k (x, P ) = 1 + α xµ Pµ + β (x · x)(P · P ) + γ xµ xν Pµ Pν + . . . .
Further specializing to the case where O1 and O2 both have scaling dimension ∆φ ,
use special conformal invariance to determine the numerical coefficients α, β, and γ.
(iii ) Derive the same result by applying the OPE inside the correlation function
hO1 (x)O2 (0)Ok (w)i =
f12k
∆
∆
12k
|x|
|w| 2k1 |x
− w|∆k12
.
(1)
and evaluating both sides in a series expansion.
(iv ) Now specializing to the case of one dimension (d = 1, you may assume that x < 0 <
w), show that the differential operator F12k (x, ∂) obeys
2∆k ∆k12
w
w
F12k (x, ∂y )
=
w−y
w−x
y=0
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(v ) Using Mathematica, compute the coefficients in the expansion
F12k (x, ∂) =
∞
X
cn xn ∂ n .
n=0
for n 6 20. See if you can guess an analytic formula for the cn . Can you prove the
formula?
(vi ) [HARD] Re-do parts (iv ) and (v ) for the case of two-dimensional CFTs. In this
case the relevant differential equation is
2∆k ∆k12
w
w
F12k (x, ∂y )
=
w−y
w−x
y=0
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Pen and Paper/Mathematica
CFT Basics - Problem III
Exercises from Lecture
Christopher Beem
This is a collection of small calculations mentioned during the lectures that were left as
exercises.
(i ) Derive the differential form of the Ward identity for conservation of the stress tensor,
h∂µ T
µν
(x)O1 (x1 ) · · · On (xn )i = −
n
X
δ(x − xi )
i=1
∂
hO1 (x1 ) · · · On (xn )i .
∂(xi )ν
Recall that this follows from diffeomorphism invariance.
(ii ) Derive the corresponding Ward identity for rotations, i.e., for conservation of ν (x)T µν (x)
where is the Killing vector that generates rotations.
(iii ) In two dimensions there are infinitely many solutions to the conformal Killing equation
∂µ ν + ∂ν µ = c(x)δµν .
Find them and compute their commutators (this Lie algebra is the Witt algebra).
(iv ) [painful] Check that the algebra of conformal Killing vectors in general dimension
is so(d + 1, 1).
(v ) Prove that in a unitary CFT, every local operator can be written as a sum of primaries
and descendants.
(vi ) Recall the definition of Hermitian conjugation for operators on the Euclidean plane
acting on the sphere Hilbert space
x
a`
†
a1 ···a`
−2∆ a1
† b1 ···b`
(O
(x)) = |x|
Ib1 · · · Ib` (O )
.
|x|2
Deduce that the conformal charges acting on this Hilbert space obey
(Q )† = QII .
(vii ) Prove the unitarity bound reported in the lectures for scalar operators,
d−2
)>0,
2
by computing the norm of the state |ψi = Pµ P µ |Oi.
∆(∆ −
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Pen and Paper/Mathematica
(viii ) [harder] Prove the bound reported in the lectures for traceless symmetric tensors of
spin `,
∆ − (l + d − 2) > 0 ,
by considering states of the form Pµ |Ωµµ2 ···µ` i.
(viii ) Determine whether a conserved, even-spin symmetric traceless tensor can have a
nonzero three-point function with two identical scalar operators.
(ix ) Derive the general form for the four-point function of non-identical scalar operators.
(x ) Write the conformal Casimir operator, C = 21 LAB LAB , in terms of the charges Mµν ,
Pµ , Kµ , D and check that you have the coefficients correct by verifying that it
commutes with all of the charges.
(xi ) [hard] Compute the differential operator that arises from inserting the conformal
Casimir in the four-point function of identical scalar operators. This was derived in
the paper Conformal Partial Waves and the Operator Product Expansion by Dolan
and Osborn.
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