defining a quantum field theory on the plane. This will prove to be useful when we want to look more deeply into the consequences of conformal invariance.
7.1 Radial Quantization
We will begin with a flat two dimensional Euclidean surface with coordinates labeled by σ0 for timelike positions and σ1 for spacelike positions, i.e. a point on our surface is
specified by a time and space position, (σ0, σ1). The metric on the surface is given by
ds2 = (dσ0)2+ (dσ1)2. (7.1)
Note that the left- and right-moving boson fields become Euclidean fields that have purely holomorphic or anti-holomorphic dependence on the coordinates. Also, note that in order to eliminate any infrared divergences we will compactify the space coordinate of our surface, i.e. we will take σ1 to be periodic, σ1 = σ1 + 2π, which gives us the
topology of an infinitely long (in both directions of σ0) cylinder for our surface. Thus,
we can think of our Euclidean surface as the product space R× S1, where S1 denotes
the circle.
We can define light-cone coordinates for our Euclidean surface, ζ, ζ = σ0 ± iσ1,
which are just Wick rotations of the light-cone coordinates previously used for the string worldsheet in the previous sections. In terms of these coordinates, the metric becomes
ds2 = dζdζ. (7.2)
Now, for reasons that will become clear
z Exp equal time slices equal radial slices
later‡, we will map the infinitely long cylin-
der to the complex plane, coordinatized by z, via the map exp : R× S1 → C defined
by
ζ 7→ z = exp(ζ) = e(σ0+iσ1),
ζ 7→ z = exp(ζ) = e(σ0−iσ1). (7.3)
‡By defining our CFT on the complex plane we will see that many of the components of the theory will split up into holomorphic and anti-holomorphic parts. Thus, one can use the properties of complex analysis to study the theory.
From the mapping we can see that the infinite past and future of the cylinder, σ0 =±∞,
are mapped to the points z = 0, for infinite past, and z = ∞, for infinite future, in the complex plane. Also, equal time slices of the cylinder, i.e. the surface defined by σ0 =constant and σ1 taking all values in [0, 2π), become circles of constant radius
exp(σ0) in the complex plane (see the previous figure). Time translations, σ0+ a where
a is a constant, are mapped to exp(a) exp(σ0 + iσ1), i.e. z 7→ exp(a)z, which are
the dilatations in the complex plane. Note that since the Hamiltonian generates time translations we can see that the dilatation generator on the complex plane corresponds to the Hamiltonian on the cylinder. And so, the Hilbert space defined on the cylinder is built up of constant time slices while the Hilbert space defined on the complex plane is built up of circles of constant radius. Finally, a word about nomenclature, this procedure of quantizing a theory on a manifold whose geometry is given by the complex plane is known as radial quantization. Also, it is useful to radially quantize two dimensional CFT’s since it allows for one to use complex analysis to analyze short distance operator expansions, conserved charges, etc. as we will see later.
7.2 Conserved Currents and Symmetry Generators
For what follows, we will treat z and z as independent coordinates. Thus, we are really mapping the cylinder (i.e. subset of R2) to C2. And so, one must remember throughout
that we are really sitting on the real slice R2 ⊂ C2 defined by setting z = z∗.
In general, symmetry generators can be constructed via the Noether method which states that if your d + 1 dimensional quantum theory has an exact symmetry then associated to this symmetry is a conserved current jµ. For example, if the theory is
invariant under an infinitesimal coordinate transformation, xµ 7→ xµ+ δxµ = xµ+ µ,
then the corresponding conserved current is given by
jµ= Tµνν, (7.4)
where Tµν is the stress-energy tensor. In particular, for translations along xα by a we
have that
µ= aδαµ, (7.5)
and so our current is given by jµ = aTµα. If our theory is translationally invariant
we would have that this current is conserved. While for dilatations, i.e. scaling the coordinates, we have that
µ= λxµ, (7.6)
and so the current corresponding to this transformation jµ is proportional to Tµνxν. If
our theory is invariant under dilatations then we have that ∂µj
that
∂µjµ= ∂µ(bTµνxν),
where b is the constant of proportionality, and so ∂µ(bTµνxν) = b∂µTµνxν
= bTµν∂µxν
= bTµνδνµ
= bTµµ.
Thus, in a conformally invariant classical field theory we have that the stress-energy tensor is traceless, T µ
µ = 0. Note that even when the conformal invariance survives in
a 2d quantum theory, the vanishing trace of the stress-energy tensor will only turn out to hold in flat space.
Furthermore, there also exists a conserved charge Q, defined by
Q = Z
∂M
ddxj0, (7.7)
where ∂M is the d dimensional manifold constructed by taking a fixed time slice of the d+1 dimensional spacetime manifold in which your theory is defined. The conservation of Q, i.e.
d
dτQ = 0,
follows straight from Stokes’ theorem. The conserved charge Q generates the symmetry, i.e. if one has a field A and a symmetry of your theory then under this symmetry the variation of the field A is given by the expression
δA = [Q, A].
We will now specialize to the case of two dimensions.
Before we begin to study the conserved charges on the complex z-plane, we need to first introduce some properties of our plane, namely we need to know the components of the metric and stress-energy tensor in complex coordinates z and z. So, to start we have that on the Euclidean surface (i.e. the cylinder) our metric is given, in light-cone coordinates, by (7.2)
To transform the previous expression for the metric into complex coordinates we note that according to (7.3) one has that ζ = ln(z) and ζ = ln(z) which gives us that
ds2 = 1
|z|2dzdz.
Now, the scaling factor of 1/|z|2 can be removed via a conformal transformation and
since our field theory is a CFT it will be invariant under this transformation. Therefore, we can, without loss of generality, take the metric on our complex plane to be given by
ds2 = dzdz. (7.8)
In order to find the components of the metric in the z, z coordinate system we can simply read off from equation (7.8) to get that gzz = gzz = 0 while gzz = gzz = 1/2.
Now that we have the components of the metric in the complex coordinates, we can find the components of the stress-energy tensor in terms of these components as well and we see that
Tzz = 1 4(T00− 2iT10− T11) , (7.9) Tzz = 1 4(T00+ 2iT10− T11) , (7.10) Tzz = Tzz = 1 4(T00+ T11) = 1 4T µ µ. (7.11)
Now, by translational invariance, we have that ∂νT
µν = 0 which implies that
∂zTzz + ∂zTzz = 0, (7.12)
∂zTzz + ∂zTzz = 0 . (7.13)
Also, imposing dilatation invariance gives us that the stress-energy tensor is traceless, Tµ
µ= 0, and so we see that T00+T11 = 0 which, from (7.11), implies that Tzz = Tzz = 0.
Combining this result with equations (7.12) - (7.13) gives us that
∂zTzz = 0, (7.14)
and
∂zTzz = 0. (7.15)
Equation (7.14) tells us that Tzz is a holomorphic function of z only, while equation
denote these functions by T (z)≡ Tzz(z) and T (z)≡ Tzz(z). Thus, the only nonvanish-
ing components of the stress-energy tensor, for a two-dimensional CFT, are given by T (z) and T (z), and so we see that the stress-energy tensor factorizes into holomorphic and antiholomorphic pieces. We will see this factorization for other quantities as we proceed further into analyzing symmetries of CFT’s living on the plane.
Now that we have the components of the metric and stress-energy tensor in terms of complex coordinates, we are in a position to study symmetries and their corresponding conserved currents for two dimensional CFT’s on a plane. So, consider the generator of a general coordinate transformation δxµ= µ or, in terms of complex coordinates,
δz = (z),
δz = (z) ,
where (z) is a holomorphic function and (z) is an anti-holomorphic function. The corresponding charge to this transformation is given by
Q = Z
∂M
j0dσ1, (7.16)
where again we integrate over a constant time slice of the cylinder or, in terms of complex coordinates, the corresponding charge is given by
Q = 1 2πi I C dz T (z)(z) + dz T (z)(z), (7.17)
where the contour C is over a circle in the complex plane whose radius corresponds with the value of σ0 for the time slice of the cylinder, i.e. if we pick a time slice at
the value σ0 = s, for some s ∈ [−∞, ∞], then the corresponding contour is a circle
of radius exp(s). Also, we will chose the positive orientation of the circle to be in the counter-clockwise sense.
The variation of a field Φ(w, w) with respect to the above transformation is given by the “equal-time” commutator of Φ(w, w) with the charge Q, which follows from (7.17), δ,Φ(w, w) ≡ [Q, Φ(w, w)] = 1 2πi h I dz T (z)(z), Φ(w, w)i+h I dz T (z)(z), Φ(w, w)i. (7.18)
What does this equal-time commu- z z z z
∼
∼
homotopic Figure 6:tator mean? First, note that the oper- ators are T (z), T (z), and Φ(w, w) and we can think of the commutators in (7.18) as, for example just considering the first commutator, letting Φ(w, w) act first and then acting with T (z), but this only well-defined when |w| < |z| which is equivalent to extending our contour C to enclose the point w in the complex plane (see figure 6), plus the contribution when we let T (z) act first then Φ(w, w), which is equivalent to shrinking the contour to exclude the point w, i.e. |z| < |w|.
If we define the radial ordering‡op-
erator R, of two operators A(z) and B(w), as
RhA(z)B(w)i= (
A(z)B(w) if |w| < |z| ,
B(w)A(z) if |z| < |w| , (7.19)
then we can rewrite δ,Φ(w, w) as
δ,Φ(w, w) = 1 2πi I |w|<|z|− I |z|<|w| dz (z)RhT (z)Φ(w, w)i+dz (z)RhT (z)Φ(w, w)i. (7.20)
‡Note that radially ordering operators on the complex plane corresponds to time ordering the operators on the cylinder. This is because we have that constant time slices map to circles of constant radius in the complex plane. So, we have that, for example, a correlation function of the fields A1(t1),· · ·An(tn) on the cylinder, given by
hk|ThA1(t1)· · · An(tn) i
|k0i,
becomes, after the conformal mapping, the correlation function of the fields A1(z1, z1)· · · An(zn, zn) on the complex plane given by
hk|RhA1(z1, z1)· · · An(zn, zn) i
|k0i,
whereT is the time ordering operator, k and k0 are arbitrary in and out states and R is the radial ordering operator.
This expression can be further evaluated by noting that the difference of the two con- tours, |w| < |z| and |z| < |w|, is homotopy equivalent to the one contour centered around the point w in the complex plane, see figure ??.
So, we have that
~
z z
z
Figure 7: The commuator
T (z), Φ(w, w) of the two operators, T (z) and Φ(w, w), is homotopic to a closed contour enveloping the point w. δ,Φ(w, w) = 1 2πi I C0 n × × dz (z) RhT (z)Φ(w, w)i + dz (z) RhT (z)Φ(w, w)io (7.21)
where the contourC0 is the contour enclosing the
point w, see figure ??. But we know what this should be equal to because we know how Φ trans- forms since it is composed of primary fields. Namely, we know that
δ,Φ(w, w) = h∂w(w)Φ(w, w) + (w)∂wΦ(w, w)
+ h∂w(w)Φ(w, w)
+ (w)∂wΦ(w, w) . (7.22)
Thus, setting (7.22) equal to (7.21) we see that in order for the charge Q, given by equation (7.17), to induce the correct infinitesimal conformal trans- formations, we infer that the short distance, i.e. when z 7→ w, singularities of T and T with Φ should be given by† RhT (z)Φ(w, w)i = h (z− w)2Φ(w, w) + 1 z− w∂wΦ(w, w) + regular terms , (7.23) RhT (z)Φ(w, w)i = h (z− w)2Φ(w, w) + 1 z− w∂wΦ(w, w) + regular terms¶,(7.24)
†By regular terms, in the following expressions, we mean terms that are non-singular, or equiva- lently, terms with zeroth order poles.
where (h, h) are called the conformal weights of the primary field Φ(w, w). To see this is indeed correct, note that from (7.21) we have that the variation of Φ(w, w), due to a general coordinate transform δz = (z), is given by
δΦ(w, w) = 1 2πi I C dz (z)RhT (z)Φ(w, w)i,
where C is a closed contour enclosing the point w. Plugging in for RhT (z)Φ(w, w)i, from (7.23), the above becomes
δΦ(w, w) = 1 2πi I C dz (z) hΦ(w, w) (z− w)2 + ∂wΦ(w, w) (z− w) .
Now, since (z) is a holomorphic function, i.e. it has no singularities, its Laurent expansion around the point w is given by
(z) = (w) + ∂w(w)(z− w) + · · · .
Plugging this in gives
δΦ(w, w) = 1 2πi I C dz (w) + ∂w(w)(z− w) + · · · hΦ(w, w) (z− w)2 + ∂wΦ(w, w) (z− w) = 1 2πi I C dz (w)hΦ(w, w) 1 (z− w)2 + (w)∂wΦ(w, w) 1 (z− w) +∂w(w)hΦ(w, w) 1 (z− w) + ∂w(w)∂wΦ(w, w) +· · · = (w)hΦ(w, w) 1 2πi I C dz 1 (z− w)2 + (w)∂wΦ(w, w) 1 2πi I C dz 1 (z− w) +∂w(w)hΦ(w, w) 1 2πi I C dz 1 (z− w)+ ∂w(w)∂wΦ(w, w) 1 2πi I C dz(1) +(· · ·) 1 2πi I C dz(1) .
This can be simplified via the residue theorem‡ to give
δΦ(w, w) = h∂w(w)Φ(w, w) + (w)∂wΦ(w, w),
which is what we should get, see (7.22).
We have just seen that the transformation property of primary fields leads to a short distance operator product expansion (OPE) for the holomorphic and anti-holomorphic stress-energy tensors, T and T , with the field Φ. Note that if the OPE of a field, with the stress-energy tensor, is not of this form, then the field called primary. For example, we will see later on that the field X(z, z) is not a primary field.
7.3 Operator Product Expansion (OPE)
The operator product expansion (OPE) tells one what happens as a collection of local operators approach each other, i.e. if one has two operators A(x) and B(y) and wants to see what happens as they approach each other, x7→ y, then one needs the OPE of A(x) and B(y). The basic idea behind the OPE is that it should be possible to write the product of local operators, at close points, as a linear combination of a complete set of operators at one of the two points. Thus, if Ai(z, z) is a local operator at (z, z) and
if Aj(w, w) is a local operator at (w, w) then the OPE of the product Ai(z, z)Aj(w, w)
is given by
Ai(z, z)Aj(w, w) =
X
k∈I
ckij(z− w, z − w)Ok(w, w), (7.25)
where the indexing set I can either be finite or countable. Also, the functions ck ij(z−
w, z− w) depend only on the differences between the points (z, z) and (w, w), due to translational invariance, and the set of operators{Ok} is complete. Note that the above
expression is an operator expression and thus only holds inside a general (time/radial- ordered) expectation value,
Figure 8: This is a figure of local op- erators defined on the complex plane, C. The OPE for Ai(z)Aj(w) has ra- dius of convergence given, pictorially, by the circle.
hAi(z, z)Aj(w, w)· ··i =
=X
k∈I
ckij(z− w, z − w)hOk(w, w)· ··i, (7.26)
where the dots represent other local operators de- fined at points which are much further away from (w, w) than|z − w|, i.e. Ai(z, z) and Aj(w, w) are
much closer to each other than to any of the other local operators. Also, the above expression for the OPE is, in a two dimensional CFT, a convergent sequence whose radius of convergence is equal to the distance of the next nearest operator, see fig- ure 8. Finally, in the limit z 7→ w we have that
some of the functions become singular. But, one should not get concerned because this is precisely the part of the OPE that matters to us as can be seen from the relations (7.23) and (7.24) and so, in general, we will write the OPE sum up to singular terms only since we will not care about the regular terms in the expansion, which in fact vanish when inside a contour integral.
So, to recap, we have seen that the transformation law for primary fields leads to a short distance OPE for the holomorphic and anti-holomorphic stress-energy tensors, T and T , with a primary field Φ. From now on we shall drop the R symbol and consider the OPE as a shorthand notation for radially ordered products. In general, we have that the OPE of a primary field Φ(w, w), of conformal weight (h, h), with T (z) and T (z) is given by T (z)Φ(w, w)∼ h (z− w)2Φ(w, w) + 1 z− w∂wΦ(w, w), (7.27) T (z)Φ(w, w)∼ h (z− w)2Φ(w, w) + 1 z− w∂wΦ(w, w) , (7.28) where ∼ means up to regular terms (i.e. non-singular terms). Note that we could, in fact, take this to be the definition of a primary field and thus any field whose OPE with T (z) and T (z) is not of this form is not a primary field. Also, the weights, h and h, are not as unfamiliar as they appear. They simply tell us how operators transform under rotations and scalings. But we already have names for these concepts from undergraduate days. The eigenvalue under rotation is usually called the spin, s, and is given in terms of the weights as
s = h− h. (7.29)
Whereas the scaling dimension, 4, of an operator is defined by
4 = h + h. (7.30)
In the next chapter we will calculate the OPE’s for some specific quantities in the free bosonic field theory. Then we will show that the corresponding charges for the conserved current arising from global conformal transformation are the Virasoro generators, Lm. Afterwords we will look at the link between physical states and high-
est weight representations of the Virasoro algebra. Finally, we will define the Ward identities for a conformal field theory.
7.4 Exercises Problem 1
The expansion of the free massless scalar X living on a Lorentzian cylinder is
X(τ, σ) = x + 4pτ + iX n6=0 1 n(αne 2inσ+ ˜α ne−2inσ)e−2inτ, (7.31)
where σ ∼= σ + π. This is obtained by setting ls = 2 (or α0 = 2) in the expansion
formula for the Xµ fields for closed string (BBS (2.40), (2.41)).
(a) Write down the expansion formula on a Euclidean cylinder. First do a Wick rotation τ → −iτ and then express the result in terms of the complex coordinates ζ = 2(τ − iσ), ¯ζ = 2(τ + iσ).
(b) Derive the following expansion formula for a complex z-plane, by defining z = eζ,
¯ z = eζ¯: X(z, ¯z) = x− ip log |z|2+ iX n6=0 1 n(αnz −n+ ˜α nz−n). (7.32)
(c) The commutation relations for x, p, αn, ˜αn are given by
[x, p] = i, [αm, αn] = [˜αm, ˜αn] = mδm+n,0, (7.33)
with all other commutators vanishing. We define the creation-annihilation normal ordering by
: xp : = : px : = xp,
: αmα−n: = : α−nαm: = α−nαm,