Download Geometrical Aspects of Conformal Quantum Field Theory

Geometrical Aspects of Conformal
Quantum Field Theory
Diploma Thesis by
Michael Klaput
February 2009
Advisor
Prof. Dr. Michael Ratz
“To those who do not know mathematics it is difficult to get across a real feeling
as to the beauty, the deepest beauty, of nature ... If you want to learn about nature,
to appreciate nature, it is necessary to understand the language that she speaks in."
Richard Feynman (1918 – 1988)
“Understanding is, after all, what science is all about – and science is a great deal
more than mindless computation."
Sir Roger Penrose (1931 – )
To my family
Abstract
This work studies conformal quantum field theory (CFT) with special focus on its geometrical
aspects. Conformal invariance is first introduced as a classical symmetry and later incorporated
into the framework of quantum field theory using the Euclidean path integral formalism. Quantum anomalies and algebraic structures in CFT are revealed by the analysis of operator product
expansions of fields. Furthermore, an operator picture of CFT is introduced in which the complete integrability of CFTs is studied. Finally, the relation between CFT and string theory is
explored.
Contents
1 Introduction
1.1 Physics beyond the standard model . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 String theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Conformal invariance
2.1 Conformal transformations . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Conformal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Defining equations . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Solving the equations . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Riemann sphere and Möbius transformations . . . . . . . . .
2.2.4 The algebra of conformal vector fields on the Riemann sphere
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3 Conformal Quantum Field Theory
3.1 Conformal Tensors and Primary Fields . . . . . . . . . . . . . . . . . . . . .
3.2 Conformal Quantum Field Theory – a physicists definition . . . . . . . . . .
3.3 Symmetries and the Energy–Momentum Tensor . . . . . . . . . . . . . . . .
3.3.1 Classical Symmetries and the Classical Energy–Momentum Tensor .
3.3.2 Quantum Symmetries and the Quantum Energy–Momentum Tensor
3.3.3 The Conformal Ward identity . . . . . . . . . . . . . . . . . . . . . .
3.4 The one–, two–, three– and four–point functions . . . . . . . . . . . . . . . .
3.5 Operator Product Expansion of Free Field Theories . . . . . . . . . . . . . .
3.5.1 The Free Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 The Free Boson Energy–Momentum Tensor . . . . . . . . . . . . . .
3.5.3 The Free Boson Quantum Energy–Momentum Tensor . . . . . . . .
3.5.4 The General OPE of a CFT . . . . . . . . . . . . . . . . . . . . . . .
3.5.5 The Physical Meaning of c . . . . . . . . . . . . . . . . . . . . . . . .
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4 Operator formalism
4.1 Chiral Splitting . . . . . . . . . . . . . . .
4.2 Radial Quantization . . . . . . . . . . . .
4.2.1 Radial Ordering . . . . . . . . . .
4.2.2 Normal Ordering . . . . . . . . . .
4.2.3 Scattering States . . . . . . . . . .
4.2.4 Mode Expansions . . . . . . . . . .
4.3 Central Charge and the Virasoro Algebra
4.4 Space of States . . . . . . . . . . . . . . .
4.5 The Operator Algebra . . . . . . . . . . .
4.6 Unitarity, Minimal Models and RCFT . .
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5 String Theory
5.1 Sum over Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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10
Chapter Contents
5.2
5.3
5.4
Gauge Fixing and Ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Superstring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Compactifications of the heterotic string . . . . . . . . . . . . . . . . . . . . . . .
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6 Conclusions and Outlook
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7 Acknowledgements
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A Differential Geometry
A.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Complex Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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B Path integrals and two–point functions
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1 Introduction
“The modern physicist is a quantum theorist on Monday, Wednesday, and Friday
and a student of gravitational relativity theory on Tuesday, Thursday, and Saturday.
On Sunday he is neither, but is praying to his God that someone, preferably himself,
will find the reconciliation between the two views."
Norbert Wiener (1894 – 1964)
The greatest achievements in 20th century theoretical physics were undoubtedly the formulation of quantum theory and general relativity. These manage to describe the smallest and largest
physical objects known to us.
Soon after the invention of quantum mechanics and general relativity it became apparent
that there is no evident way to unify these two in one consistent theory of quantum gravity.
Henceforward—stimulated by experimental progress—other fundamental questions moved into
the focus and less and less physicists kept on working on such a “theory of everything”. Instead,
the study of elementary particles became one of the most active area in fundamental physics.
This development culminated in what is now being called the standard model of elementary
particle physics. It succeeds in tracing back the fundamental laws governing physical phenomena
to a handful of elementary particles, interacting through three so–called fundamental forces.
These forces are the electromagnetic, weak and strong force.
The mathematical formalism of the standard model combines the concepts of quantum field
theory, symmetry and geometry. More precisely, the standard model is the unique1 renormalizable quantum gauge field theory with the gauge group
GSM = SU(3) × SU(2) × U(1) .
The elementary particles form representations of GSM and occur in three different flavors or
families. Therefore, the particles of different families carry identical quantum numbers and
differ only in their masses. All family members carry spin 1/2 and obey—according to the
spin–statistics connection—Fermi–Dirac statistics. The interaction of these particles is mediated
through spin one bosonic particles belonging to the adjoint representation of GSM . They obey
Bose–Einstein statistics and are called gauge bosons.
An interesting aspect of the standard model is the generation of mass. In the first place, all
particles have to be massless as any mass term in the field theory lagrangian would explicitly break
the gauge symmetry and render the theory inconsistent. Therefore, the masses are generated
dynamically through the interaction with a fundamental scalar field—the Higgs particle. It is
expected to be detected in collider experiments beginning in 2009/2010 at the Large Hadron
Collider at CERN. Particle physicists are very anxious about these measurements since the
detailed knowledge of the Higgs sector might reveal physics beyond the standard model.
1
Strictly spoken, this is not true. Still, one has to specify the particle content and 19 free parameters a priori, but
after doing this the framework is unique. Note also in this context that the particle content is not completely
arbitrary but highly restricted through the requirement of anomaly freedom (see below).
12
Chapter 1 – Introduction
Mathematically, the standard model is described as a field theory on principal fibre bundles.
A principal bundle is a geometric object that can be mapped locally to a manifold with a group
glued to each point of it—the gauge group. An example would be the Möbius–strip which
locally looks like R × [0, 1]. Naturally, every field described on a principal bundle has to form a
representation of the gauge group. Additionally, the global structure of glueing a group onto a
manifold defines a geometric structure called connection. It turns out that locally a connection
is described as a field in the adjoint representation. Therefore, the connection—defining the
geometrical structure of the principal bundle—corresponds to the gauge bosons mediating the
interactions of fundamental particles.
The process of quantization of a gauge field theory may cause the breakdown of classical
symmetries, which is called a quantum anomaly. In this context, the standard model has a
remarkable feature. Although anomalies occur in the the gauge group of the standard model,
the quantum numbers of the elementary particles take precisely the values needed to have these
anomalies cancel each other.
Until today, only few experimental results are known which cannot be explained by the standard model. The most important being the existence of dark matter and dark energy in our
universe2 . Nevertheless, it is far from being a complete description of nature. Quite the contrary, the search for physics beyond the standard model is one of the most active areas of research
in modern physics. The progress made in the last centuries has brought the question of quantum
gravity back again into the minds of many physicists. Consequently, they study gravitational
theory again on Tuesday, Thursday, and Saturday.
1.1 Physics beyond the standard model
“With four parameters I can fit an elephant, and with five I can make him wiggle his
trunk."
John von Neumann (1903 – 1957)
Despite the standard model’s great success in describing and predicting experimental results
many questions remain unanswered. Why is there a hierarchical structure in the masses of
elementary particles and what is the origin of mass? Why do the particles form so many different
representations of the gauge group? Why do we observe exactly three families with nearly
identical properties? What is the origin of the fundamental forces? How is it possible to unify
Gravity with the other three forces? What is dark matter and dark energy?
Many of the open questions are related to the fact that the standard model incorporates 19
free parameters that have to be determined by experiment and are not predicted from the theory
itself. Although 19 is a small number compared to the vast variety of described phenomena, it
still represents our deep ignorance on fundamental aspects of nature.
Many attempts have been made in theoretical physics to improve the framework of the standard
model in order to shed light on the open problems. We will only mention some of these which
are of direct relevance to our work at hand.
Supersymmetry
The dynamical generation of masses through the Higgs mechanism gives rise to one of the main
problems of the standard model: the hierarchy problem. As the Higgs mechanism in the standard
2
We are not considering the observation of neutrino oscillations a contradiction to the standard model, because it
is possible to incorporate these observations into the standard model without having to alter its mathematical
framework.
1.2 String theory
13
model is used to give mass to the gauge bosons of the weak interaction, it should naturally possess
a mass near the weak breaking scale. It can be shown that a Higgs mass beyond ≈ 1 TeV renders
the symmetry breaking inconsistent. The hierarchy problem arises when one considers corrections
to this mass due to quantum effects. This corrections will be quadratically divergent with respect
to the cut–off scale of the standard model. Due to the fact that the standard model does not
incorporate gravity, it is expected to break down at latest on a scale where effects of a quantum
theory of gravity play an important role. This scale is the Planck scale and is expected to be
about 1016 TeV and therefore causes the Higgs mass to be many orders of magnitude too large.
If one does not allow for massive fine–tuning one has to find an explanation for this fact.
A possible solution is the introduction of supersymmetric extensions of the standard model.
In these, a new type of symmetry is introduced which relates bosons and fermions. Hence every
known elementary particle gets assigned a superpartner of the same mass. Every boson has
a fermionic superpartner and vice versa. As a consequence, the calculation of the quantum
corrections to the Higgs mass now contains a negative contribution from the superpartners—
which cancel the previous one and hence stabilize the mass of the Higgs. Of special interest
is the minimal supersymmetric standard model (MSSM), the smallest possible supersymmetric
extension of the standard model.
Since we have not seen any superpartners so far, supersymmetry has to be broken below
a certain energy scale. To solve the hierarchy problem, this scale has to be roughly 1 TeV.
Therefore—given that supersymmetry is indeed realized by nature—it is very likely that upcoming collider experiments at the LHC will be able to detect superpartners.
Supersymmetry might also provide a candidate for dark matter: the lightest supersymmetric
particle (LSP).
However, supersymmetry also introduces new problems the biggest of which is probably to
find the exact mechanism of supersymmetry breaking. Other questions are related to the implementation of supersymmetry as a gauge symmetry. This local supersymmetry is also known as
supergravity because a locally supersymmetric theory necessarily incorporates gravity (at least
in a perturbatively).
Grand Unification
Another interesting idea beyond the standard model is the concept of Grand Unified Theories
(GUTs). Here, one tries to unify all fundamental forces hence integrating the standard model
gauge group into a bigger one. In fact it is possible to describe all standard model particles as
representations of a single group. One possible choice is
SO(10) ⊃ GSM .
Furthermore, all particles of one family (including right–handed neutrinos) form a complete
representation of this group.
Another necessary condition for a successful unification of all forces is the equality of all
gauge couplings on some specific energy scale MGUT , the GUT scale. However, if one does a
standard renormalization group analysis of the gauge coupling constants in the standard model,
one encounters that they do not meet at any scale. Fortunately, the situation improves when one
takes supersymmetry into account after which the coupling constants meet. Usually, the GUT
scale is expected to be of order 1012 TeV.
1.2 String theory
“The progress of science requires the growth of
understanding in both directions, downward from the whole to the parts and upward
14
Chapter 1 – Introduction
from the parts to the whole. A reductionist philosophy, arbitrarily proclaiming that
the growth of understanding must go only in one direction, makes no scientific sense.
Indeed, dogmatic philosophical beliefs of any kind have no place in science."
Freeman Dyson (1923—)
In the early seventies, when the mysterious nature of nuclear forces was in the focus of scientific
attention, string theory was invented to solve this mystery. Its most remarkable feature is that
its basic constituents are not point particles—in contrast to all other theories before—but rather
one–dimensional extended objects, called strings.
But early enthusiasm quickly settled down when it was discovered that a consistent quantum
theory of strings is only possible in 26 space–time dimensions and that it does not contain any
fermionic states. Therefore, this early formulation of string theory is known as bosonic string
theory nowadays. Moreover, the spectrum contains a spin 2 particle which makes no sense in
the context of nuclear forces. These properties disqualified string theory as a realistic theory of
nuclear forces.
In the 80s string theory was reborn with the discovery of superstring theory, a supersymmetric
extension of bosonic string theory. Its consistency requires 10 space–time dimensions and it
appears in five distinct formulations: Type I, Type IIa, Type IIb, Heterotic E8 ×E8 and Heterotic
SO(32)/Z2 .
Superstring theory is believed to be able to unify all of nature’s interactions including gravity.
Its consistency fixes the theory completely, leaving no free parameters nor necessary input.
However, there is no known mechanism which chooses the string’s vacuum configuration. This is
known as the string landscape problem since one is left with a huge landscape of possible vacua.
1.3 Outline
“We must know—we will know! "
David Hilbert (1862 - 1943)
Our work was motivated by recent progress in model building from orbifold compactifications
of the heterotic string [L+ 07] [L+ 08] [LNRS+ 08]. It was shown that there are a few hundred
models—the mini–landscape—exhibiting the complete MSSM spectrum and absence of exotic
particles at the same time. This makes the mini–landscape a promising “ballpark” where to look
for the string vacuum which might describe our reality.
It would be of great interest to further investigate the mini–landscape and search for Yukawa
hierarchies and models without proton decay. In order to do so, it is necessary to learn more
about the couplings in these models.
Since couplings between strings are calculated by the use of conformal quantum field theory
(CFT) methods, this work is concentrated on the development of these methods. We will put
a special focus on geometrical aspects of CFT since they are especially important in orbifold
CFTs.
The work is organized as follows: At first, we are going to establish conformal invariance as
a classical symmetry in chapter 2. We will study its group structure, as well as its algebra
carefully. Of special interest will be the case of two space–time dimensions. Here, we are going
to encounter an infinite dimensional symmetry algebra. Furthermore, we will see how complex
geometry naturally appears in the context of conformal invariance.
1.3 Outline
15
After having treated classical conformal invariance, we will formulate a quantum theory obeying this symmetry in chapter 3. This is going to be done using path integral methods of Euclidean quantum field theory. Special attention has to be paid to the fact that the conformal
group breaks down upon Wick rotation. This will make it necessary to develop conformal invariance infinitesimally, introducing the concept of primary fields. We will see how the complex
structure of two–dimensional manifolds leads to a natural splitting of the degrees of freedom into
holomorphic and anti–holomorphic components.
On this basis, consequences of conformal invariance for correlators are carried out. We uncover
the special role of the energy–momentum tensor as generator of conformal symmetry. This is
reflected in the conformal Ward identity. From our analysis we deduce how conformal invariance
fixes the one–, two– and three–point functions of arbitrary primary fields.
The system of a free, massless scalar field in two dimensions is studied. It reveals the possibility
of quantum anomalies in the conformal invariance. The concepts of central charge and operator
product expansions are introduced. The latter are shown to play a vital role in CFT.
We conclude our study of CFT in chapter 4, where we introduce operator methods which
will prove useful for concrete calculations and generalizations of CFT. In the scheme of radial
quantization we uncover the celebrated Virasoro algebra from mode expansions of the energy–
momentum tensor. Representations of this algebra are constructed and the intricate structure
of conformal families is investigated. We study the integrability of the theory and treat a large
class of CFTs which can be solved completely from conformal invariance alone.
Finally, in chapter 5 we see how the formalism of CFT appears in string theory. We discuss
the path integral formulation of bosonic string theory and its gauge fixing. Superstring theory
is introduced. We comment on orbifold compactifications of the heterotic string.
2 Conformal invariance
“The Unreasonable Effectiveness of Mathematics in the Natural Sciences"
Eugene Paul Wigner (1902 – 1995)
In the following, we will establish the concept of conformal symmetry in geometrical language.
Some basic knowledge about differential geometry is assumed, on the level of an introduction to
general relativity. A brief summary of useful facts can be found in appendix A.
After defining conformal transformations, we proceed to determine their algebraic and group
structure in various dimensions. In the further analysis we will concentrate on the two–dimensional
case, which will turn out to be special in various ways.
2.1 Conformal transformations
We introduce conformal transformations on general pseudo–Riemannian manifolds. One of our
main goals is to establish the distinctiveness of the two–dimensional case.
Definition 2.1: Let (M, g) be a pseudo–Riemannian manifold with metric g. A diffeomorphism
M −→ M
f :
(2.1)
p 7−→ f (p)
is called conformal transformation if the metric tensor g transforms as
f ∗ gf (p) = eσ(p) gp
with some σ : M → R .
(2.2)
Remarks.
• Equation (2.2) is more convenient to physicists if formulated in local coordinates
′
′
∂f µ ∂f ν
gµ′ ν ′ (f (x)) = eσ(x) gµν (x) .
∂xµ ∂xν
(2.3)
• If we recall that the angle between two tangent vectors u = uµ ∂µ and v = v ν ∂ν in a
manifold is given by
cos θuv := p
g(u, v)
g(u, u) g(v, v)
=p
uµ gµν v ν
uσ gστ uτ v λ gλκ v κ
,
(2.4)
we see at once, that it remains constant under the action of a conformal transformation.
Therefore we conclude that conformal transformations leave the angles between tangent
vectors constant. It can also be shown that the reverse is true, i.e. every diffeomorphism
leaving the angles between tangent vectors constant is a conformal transformation.
18
Chapter 2 – Conformal invariance
• More important for physical applications is the following fact: A diffeomorphism is a conformal transformation if and only if it preserves the light–cone structure. Equivalently,
one could say more loosely that conformal transformations are those transformations preserving the causal structure of space–time. This important fact underlies the theory of
Carter–Penrose diagrams.
• Conformal invariance is not to be confused with Weyl transformations/(re)scalings. These
are all mappings ω which transform the metric according to
ω : gp 7−→ e2 σ(p) gp .
(2.5)
It is apparent that there is a close relationship between Weyl scalings and conformal transformations. However, there are subtleties and one should carefully distinguish the two.
Consider for instance an action S[g, φr ] depending on the metric tensor and some (scalar)
fields φr . The action of a Weyl scaling is
ω : S[g, φr ] 7−→ S[e2 σ g, φr ] ,
whereas a conformal transformation acts as
f : S[g, φr ] 7−→ S[f ∗ g, f ∗ φr ] .
The important difference is that the Weyl scaling leaves the fields invariant, which means
that it will not be a symmetry of our resulting quantum field theory (which will not
incorporate the metric tensor as a quantum field). However, there are Weyl scalings which
are generated by diffeomorphisms. Comparing (2.5) with (2.2) we see that these are exactly
the conformal transformations defined above. We will see in chapter 5 that after fixing a
gauge for the metric—the conformal gauge—we are left with conformal invariance as the
symmetry of our theory.
In physics, the group structure of symmetries is of great use. This is also true for conformal
transformations:
Corollary 2.2: The set of conformal transformations on a pseudo–Riemannian manifold (M, g) with
signature (r, s) is a group, denoted by
CM (r, s) .
(2.6)
We will usually work with its connected component containing the identity, which we call the
conformal group and write
COM (r, s) .
(2.7)
The group product is given by the composition of transformations.
Proof: As the set of diffeomorphisms already is a group (with the composition of maps as product),
it remains to show that the composition of conformal transformations is a conformal transformation as
well.
Let f , h be two conformal transformations. The action of f ◦ h on the metric at (f ◦ h)(p) ∈ M is
h
i
−1
(2.8)
(f ◦ h)∗ gf ◦h(p) = h∗ ◦ f ∗ gf ◦h(p) = h∗ eσf (h(p)) gh(p) = eσh (h ◦ h(p)) eσf (p) gp = eσ(p) gp ,
with σ(p) := σh (p) + σf (p). Hence, f ◦ h is a conformal transformation as well.
In our work the study of two–dimensional manifolds is of main interest and in that case one
has the important
2.2 Conformal algebra
19
Theorem 2.3: Let (M, g) a two–dimensional pseudo–Riemannian manifold. For every p ∈ M there
exists a chart in which the metric takes the form
±
gµν = eσ(x) δµν
.
(2.9)
±
Here δµν
is not the usual Kronecker–delta, but one with the appropriate number of minus signs to match the
signature of the metric.
Proof:
see [Sch97], p. 81.
Remark. We shall refer to this special chart as the conformal gauge.
Written out in local coordinates, the statement is equivalent to the following: Let gµ′ ν ′ the
′
metric of a two–dimensional manifold in a certain chart and coordinate basis xµ . Then there
exists another coordinate basis xµ such that
′
′
∂xµ ∂xν
±
.
gµ′ ν ′ (x′ (x)) = eσ(x) δµν
∂xµ ∂xν
(2.10)
A manifold which admits a metric for which the above equation holds, is generally called
conformally flat—i.e. flat modulo some conformal transformation. Therefore, the statement of
the above theorem is that every two–dimensional pseudo–Riemannian manifold is conformally
flat. This fact will be very useful when considering string theory, which is a quantum field theory
on a generally curved two–dimensional manifold. In this case, we are able to map the CFT on
the curved space to an identical CFT with flat background. However, one has to assure that no
quantum anomalies show up in the conformal invariance of the theory. Hence, given we manage
to avoid anomalies, we can circumvent any problems that might occur because of the numerous
difficulties one encounters when trying to formulate a quantum field theory on curved space–time.
2.2 Conformal algebra
2.2.1 Defining equations
We investigate the algebra of vector fields that generate the conformal transformations of the
previous section. During the discussion we will encounter an interesting dependence on the
space–time dimension.
We analyze the action of a diffeomorphism generated by a vector field on the metric tensor
and use the defining equations of conformal transformations to get restrictions on this vector
field.
Let fε be a flow of diffeomorphisms of the space–time manifold M, in local coordinates ϕ
ϕ ◦ fε ◦ ϕ−1 :
xµ 7−→ f µ (x, ε) ,
which is generated by the vector field X = Xµ ∂µ , i.e. in local coordinates
d µ
f (x, ε)
= Xµ (x) .
dε
ε=0
(2.11)
(2.12)
Every vector field generating a conformal transformation is called a conformal Killing vector
(field) (CKV) of M.
We now calculate the variation of the metric tensor with respect to the flow fǫ to first order
in ε
f∗ g − g
.
(2.13)
lim ε
ε→0
ε
20
Chapter 2 – Conformal invariance
Note that the right hand side is nothing else than the Lie derivative £X of the metric tensor
with respect to X:
f∗ g − g
,
(2.14)
£X g ≡ lim ε
ε→0
ε
which reads in local coordinates
(£X g)µν = ∇µ Xν + ∇ν Xµ ,
(2.15)
£X gµν = ̺(x) gµν ,
(2.16)
according to formula (A.15) of the appendix.
On the other hand, the defining property of a conformal transformation is that the metric
changes only by a scale factor eσ(x,ǫ) . Hence, the variation to first order—the Lie derivative—is
with an appropriate function ̺(x). It is determined by contracting µ and ν:
2
̺(x) = ∇µ Xµ ,
d
with d := dim(M). Together with the general expression, it follows that
(2.17)
Ξµν := ∇µ Xν + ∇ν Xµ − ̺(x) gµν = 0 .
(2.18)
∇µ Ξνρ + ∇ν Ξρµ − ∇ρ Ξµν = 0
(2.19)
This equation is known as conformal Killing equation.
From this we deduce that
and by using that ∇ρ gµν = 0,
=⇒
2 ∇µ ∇ν Xρ = ∇µ ̺(x) gνρ + ∇ν ̺(x) gρµ − ∇ρ ̺(x) gµν .
(2.20)
Contraction of the indices µ and ν gives
2 ∇µ ∇µ Xρ = (2 − d) ∇ρ ̺(x) .
(2.21)
(d − 1) ∇ν ∇ν ̺(x) = 0 .
(2.22)
Applying ∇ρ to equation (2.21) and using equation (2.17), we finally get
2.2.2 Solving the equations
The conformal algebra is so far determined through the equations
2 ∇µ ∇ν Xρ − ∇µ ̺(x) gνρ − ∇ν ̺(x) gρµ + ∇ρ ̺(x) gµν = 0
2 ∇µ ∇µ Xρ − (2 − d) ∇ρ ̺(x) = 0
where ̺(x) =
2
d
∇ µ Xµ .
(d − 1) ∇ν ∇ν ̺(x) = 0
One dimension
In one dimension we see at once that the third equation is automatically satisfied. Because the
covariant derivative has only one component ∇, the second equation reads
2 ∇∇ X − ∇ 2 ∇X = 0 ,
(2.23)
2 ∇∇ X − ∇ 2 ∇X g − ∇ 2 ∇X g + ∇ 2 ∇X g = 0 ,
(2.24)
which is also automatically satisfied. Finally, the first equation is
and again automatically satisfied. We therefore conclude, that every diffeomorphism of one
dimensional manifolds is conformal. This result is rather obvious, as there are only two possible
angles between two vectors in one dimension: 0 and π. These cannot continuously change and
therefore remain constant under the action of any diffeomorphism.
2.2 Conformal algebra
21
More than two dimensions
As we are mainly interested in the case of two dimensions, we will only quote the results in this
section. Solving the equations of the conformal algebra in more than two dimensions, leads to
the following base of generating vector fields
X(1)
µ = aµ ,
ν
X(2)
µ = bµν x ,
ν
X(3)
µ = λx ,
ν
ν
X(4)
µ = 2 cν x xµ − x xν cµ ,
with λ ∈ R, aµ , cµ constant vectors in M and bµν = −bνµ. The vector fields can be integrated
(1)
(2)
to form the conformal group which consists of translations (Xµ ), rotations (Xµ ), dilatations
(3)
(4)
(Xµ ) and special conformal transformations (Xµ ). In a manifold with signature (r, s) the
resulting group is isomorphic to the Lorentz group1
CM (r, s) ∼
= O(r + 1, s + 1)/Z2 .
(2.25)
For the connected component containing the identity transformation this means
COM (r, s) ∼
= SO(r + 1, s + 1) ,
(2.26)
where SO(r + 1, s + 1) denotes the Lorentz group’s connected component containing the identity.
Note that according to this analysis we would expect the conformal group in a two dimensional
Euclidean space (i.e. signature (2, 0)) to be the Lorentz group in four dimensions O(3, 1)/Z2
(modulo Z2 ) and its connected component containing the identity to be the proper orthocronous
Lorentz group SO(3, 1).
Two dimensions
For d = 2 we will encounter a quite different structure than in higher dimensions. To see this,
we reconsider the defining equation of conformal invariance for a diffeomorphism f
′
′
!
f ∗ g ≡ gµ′ ν ′ (f (x)) df µ ⊗ df ν = eσ(x) gµν (x) dxµ ⊗ dxν .
(2.27)
As we are interested in formulating physical theories, we will assume one space and one time
dimension. Hence, our metric has the signature (1, 1).
The relation of dxµ and df µ is
′
∂f µ
µ′
dxµ ,
(2.28)
df =
∂xµ
hence, equation (2.27) reads
′
gµ′ ν ′ (f (x))
′
∂f µ ∂f ν
!
dxµ ⊗ dxν = eσ(x) gµν (x) dxµ ⊗ dxν .
∂xµ ∂xν
(2.29)
To find solutions of the above requirement, we use theorem 2.3 to map the metric g to a
conformally flat one, which simplifies our discussion. The above equation is then written as2
′
η
µ′ ν ′
′
∂f µ ∂f ν
!
dxµ ⊗ dxν = eσ(x) ηµν dxµ ⊗ dxν ,
∂xµ ∂xν
with the flat Minkowski metric
ηµν =
1
−1 0
0 1
.
(2.30)
(2.31)
One has to be a bit careful in the case of a curved manifold M where it is not obvious how the Lorentz group
acts on M. But since any manifold is by definition locally isomorphic to a flat space the statement remains
true locally. See [Sch97] for a more careful treatment.
2
Note that the conformal factor eλ(x) coming from the application of theorem 2.3 cancels.
22
Chapter 2 – Conformal invariance
This will not restrict our analysis of possible conformal transformations since their definition
(2.2) is coordinate independent. For convenience, we introduce light–cone coordinates 3
x+ := x + t ,
x− := x − t .
(2.32)
The metric in light–cone coordinates is now
(+)
ηµν
=
1
2
0
1
2
0
(2.33)
,
and therefore the left–hand side of (2.30) reads in lightcone coordinates
1
∂f + ∂f − +
dx ⊗ dx+ +
+
+
∂x ∂x
2
∂f + ∂f − ∂f + ∂f −
+ − +
∂x+ ∂x−
∂x ∂x
dx+ ⊗ dx− + dx− ⊗ dx+ + . . .
∂f + ∂f −
. . . + − − dx− ⊗ dx− .
∂x ∂x
(2.34)
If we compare this with the right hand side of (2.30) in lightcone coordinates, we obtain the
relations
∂f + ∂f −
=0,
∂x+ ∂x+
∂f + ∂f −
=0,
∂x− ∂x−
1
2
∂f + ∂f − ∂f + ∂f −
+ − +
∂x+ ∂x−
∂x ∂x
= eσ(x) .
From the third, we deduce that every component of f has to possess at least one nonvanishing
derivative with respect to x+ or x− . Otherwise, the left hand side would be identically zero
which contradicts the strict positiveness of eσ(x) .
Together with the first two relations, this implies that we have either
∂f +
=0
∂x−
and
∂f −
=0
∂x+
(2.35)
∂f +
=0
∂x+
and
∂f −
=0.
∂x−
(2.36)
or
The second case does not include the identity transformation f + = x+ and f − = x− . Hence,
these two possibilities represent two different connected components of the conformal group. The
component including the identity is the first one.
In conclusion, every conformal transformation takes the form (in light–cone coordinates)
f + = f + (x+ )
f − = f − (x− )
(2.37)
f + = f + (x− )
f − = f − (x+ ) .
(2.38)
or
This expressions will look very familiar, after we have performed a Wick–rotation to complex
time as we will do in the next section.
3
This change of coordinates is conformal, as it simply represents a rotation about π2 . Therefore, corollary 2.2
ensures that our discussion is unaltered since every composition of conformal transformations is a conformal
transformation.
2.2 Conformal algebra
23
2.2.3 Riemann sphere and Möbius transformations
In the previous section, we were able to restrict our analysis on conformally flat manifolds by
making use of theorem 2.3. Our ultimate goal is to do quantum field theory with conformal
invariance. As is known from quantum field theory in general, the procedure of quantization is
best achieved in the path integral formalism. To have it well–defined, it is common to perform
a Wick–rotation from real to complex time τ
t 7−→ τ ≡ i t ,
(2.39)
what results in the action transforming as
i SMinkowski 7−→ − SEuclidean .
(2.40)
The light–cone coordinates are now coordinates on the complex plane
x+ 7−→ ξ = x + i t
x− 7−→ ξ̄ = x − i t ,
(2.41)
with metric
1
(dξ ⊗ dξ̄ + dξ̄ ⊗ dξ)
(2.42)
2
which has signature (2, 0). The condition on f to define a conformal transformation becomes
g=
f + = f + (ξ) =: ζ(ξ)
f − = f − (ξ̄) =: ζ̄(ξ̄)
(2.43)
f + = f + (ξ̄) =: ζ(ξ̄)
f − = f − (ξ) =: ζ̄(ξ) ,
(2.44)
or
where f has to be a diffeomorphism, i.e. invertible. We notice at once that exactly every invertible
holomorphic (anti–holomorphic) mapping of the complex plane is a conformal transformation.
To be more precise, we have to allow for every invertible meromorphic mapping, since our analysis
might have missed isolated singularities4 .
As we know from algebraic geometry, the natural playground for meromorphic functions is not
the complex plane. It is far more convenient to view them on the Riemann sphere5
CP1 ∼
= C ∪ {∞} ,
(2.45)
where they become holomorphic (anti–holomorphic) mappings
CP1 −→ CP1 .
(2.46)
The conformally flat metric of the Riemann sphere is
g=
2
¯ 2 (dξ ⊗ dξ̄ + dξ̄ ⊗ dξ) .
(1 + ξ ξ)
(2.47)
Let us use the conformal mapping (the minus sign in the exponential is a convention)
ξ 7−→ z := e−i ξ ,
(2.48)
1
(dz ⊗ dz̄ + dz̄ ⊗ dz) .
2
(2.49)
to make the metric locally flat
g=
4
Recall that theorem 2.3 is only valid locally, i.e. in single charts. In fact, it holds that every holomorphic
function on a compact complex manifold is a constant.
5
Recall that {∞} := {ξ ∈ C | |ξ| = ∞}.
24
Chapter 2 – Conformal invariance
To get a better understanding of the steps taken so far, let us repeat the above discussion from
a physicists perspective.
At first, we have Wick–rotated the Minkowski space–time to the infinite complex plane in
order to have a well–defined path integral. Now, since we would like to set up a scattering theory
on this manifold—including asymptotic states—we include ∞ into our manifold. This leads to
the one–point compactification C ∪ {∞} ∼
= CP1 and we end up on the Riemann sphere. Since
we do not know how to quantize a quantum field theory in curved space, we use the conformal
symmetry of our theory to map the Riemann sphere locally onto a flat one. This is done by
ξ 7−→ z := e−i ξ = et e−ix .
(2.50)
Now, the space degree of freedom has been compactified on a circle. Furthermore, t = −∞
corresponds to z = 0 and t = ∞ corresponds to z = ∞. The curves of constant time are the circles
with the origin at their center. For that reason, the canonical quantization in these coordinates
is called radial quantization since the direction of time is pointed radially. Although, this is
the common procedure to introduce conformal quantum field theory, we will avoid canonical
quantization and make use of the more powerful path integral quantization in the first part of
chapter 3. However, it will be interesting to return to operator methods and we will treat radial
quantization in detail in the second part of chapter 3.
Let us now determine the set of conformal transformations on the Riemann sphere. Therefore,
it is useful to view CP1 as a (real) two–dimensional Euclidean sphere S 2 carrying the usual
complex structure of C. The south–pole is equivalent to the point 0 on the plane, while the
north–pole represents the point ∞. We use two coordinate patches to cover the sphere:
CP1 \ {∞} 7−→ C
CP1 \ {0} 7−→ C
ϕw :
(2.51)
ϕz :
p
7−→ z
p
7−→ w
with transition functions on the overlap
CP1 \ {0, ∞} 7−→ C
ϕw ◦ ϕ−1
:
z
z
7−→ w =
1
z
,
ϕz ◦ ϕ−1
w :
CP1 \ {0, ∞} 7−→ C
z
7−→
z=
1
w
.
(2.52)
In the following, we would like to obtain all invertible holomorphic (anti–holomorphic) functions
ζ :
CP1 −→ CP1 .
(2.53)
This means, that the function
(ϕ ◦ ζ ◦ ϕ−1 )(z) :
C −→ C
(2.54)
has to be meromorphic (anti–meromorphic) for ϕ = ϕz and ϕ = ϕw . Let us start in coordinate
patch z and concentrate on meromorphic mappings. In the next few steps, we will show that
every meromorphic function has to be an algebraic function, i.e. the quotient of two polynomials.
Our considerations pose no proof of mathematical rigor but intend to give some insights of how
the group of meromorphic mappings of CP1 emerges. A rigorous proof can be found in the
standard literature on Riemann surfaces.
As the functions have to be single–valued (i.e. to be invertible), logarithms and fractional
powers of z are not allowed to occur since these would introduce branches.
Now, consider functions of the type
e̺(z) .
Assuming this function is meromorphic, we can allow for ̺(z) to have zeros but not to have poles
since these points would be essential singularities of e̺(z) . Now take the Laurent expansion of
2.2 Conformal algebra
25
̺(z). We choose to expand around z = 0 since the series converges everywhere in
because of the absence of poles:
X
an z n .
̺(z) =
CP1 \ {∞}
N
n∈
Taking this function in the w patch gives
̺(w) =
X
N
an w−n ,
n∈
which has a pole at w = 0. Therefore, we cannot allow for poles nor for zeros in ̺, which forces
it to be a constant. Note that from our discussion follows the fact that every function on the
Riemann sphere has the same number of poles and zeros because a pole in the one patch is a
zero in the other one. Therefore, no exponentials—and hence no sines, cosines, etc.—can occur
as conformal mappings.
Thus, the most general meromorphic function will be the quotient of two complex polynomials
(we drop the explicit reference to coordinate patches in favor of brevity):
ζ(z) =
P (z)
.
Q(z)
(2.55)
Clearly, they have to be of same order since the function has to have an equal number of zeros
and poles.
To be invertible, the function has to possess a nonvanishing derivative in every point of its
coordinate patch. Take the z patch. Here, the derivative reads
∂z ζ(z) =
P ′ Q − P Q′
Q2
(2.56)
and the non–vanishing implies P ′ Q − P Q′ 6= 0. But according to the fundamental theorem of
algebra, every (non–constant) polynomial of z has at least one zero. Therefore, P ′ Q − P Q′ has
to be independent of z. It is possible to show6 that this requires P and Q to be linear functions
in z. Let P (z) = a z + b and Q(z) = c z + d then
P ′ Q − P Q′ = a d − b c .
(2.57)
We conclude that all invertible meromorphic functions on the Riemann sphere are given by
ζ(z) =
az +b
cz + d
,
a d − b c 6= 0 .
(2.58)
Note that the multiplication of all constants with the same factor results in no change in ζ.
Therefore we can specify a, b, c and d only up to an overall sign ambiguity. Furthermore, we
may√take a d − b c = 1 without loss of generality, because we could always divide all constants
by a d − b c which would give a d − b c = 1 in terms of this new constants. Now we can specify
every transformation by a 2 × 2 matrix
a b
∈ SL(2, C)/Z2 .
(2.59)
c d
The Z2 accounts for the mentioned sign ambiguity, as A and −A represent the same transformation ζ. These transformations are the well–known Möbius transformations. They form
6
The proof can be done as follows: Take two polynomials of arbitrary order n for which P ′ Q−P Q′ is independent
of z. Then add a term c z n+1 and calculate P ′ Q − P Q′ . It follows that independence of z is equivalent to
c = 0. Clearly, P ′ Q − P Q′ is independent of z in the case n = 1. Hence, we deduce by complete induction
that P ′ Q − P Q′ depends on z for all cases n > 1 and the proposition is proved. 26
Chapter 2 – Conformal invariance
the so–called Möbius group which is isomorphic to SL(2, C)/Z2 , as we have shown above. The
analysis of the anti–holomorphic mappings would have revealed another Möbius group (acting
on z̄), as can be seen easily by replacing all z’s, w’s with z̄’s, w̄’s in the derivation. Hence, the
conformal group is
CM = SL(2, C)/Z2 × SL(2, C)/Z2 ,
and the identity component is
COM = SL(2, C)/Z2 .
Surprisingly, all this is what we would have expected when reviewing the analysis of the higher
dimensional case. Recall that we obtained for a manifold with signature (2,0) for the connected
component of the conformal group which contains the identity
COM = SO(2 + 1, 1) = SO(3, 1) ∼
= SL(2, C)/Z2 ,
(2.60)
where the last isomorphism is the Caley–Klein transform, well known from the representation
theory of the Lorentz group in 4 dimensions.
This leads us to the following question: have we—despite our initial claim—really obtained
nothing new compared to the higher dimensional case? As we will soon understand, this is not
the case.
Remark (Complex plane vs. Riemann sphere).
In the last section, we changed in our discussion from the complex plane (or R2 ) onto the
Riemann sphere (or S2 ). We argued that meromorphic functions are more naturally viewed as
being defined on the Riemann sphere. We were supported by the fact that this procedure has a
natural motivation from a physicists point of view. At this point, we would like to make up a
more precise justification of this step.
One can show [Sch97] that every conformal transformation on a pseudo–Riemannian manifold
M has a unique continuation on a minimal conformal compactification MC of M. For the complex
plane it turns out that
CC = CP1 .
(2.61)
Therefore, the analysis of the conformal group on the complex plane can be carried out without
loss of generality on the Riemann sphere—its minimal conformal compactification. A deeper
reason for this is that in general every conformal transformation in arbitrary dimensions can
only be globally defined with the exception of a set of (Lebesgue) measure zero—i.e. a set of
points in case of the complex plane. Therefore the addition of a measure zero set (i.e. the point
{∞}) to form CP1 ∼
= C ∪ {∞} does not effect the analysis of the conformal group.
Remark (Number of Poles is Number of Zeros). There is a very imaginative way to explain the
fact that every function on the Riemann sphere must have the same number of poles and zeros.
Consider the Riemann sphere and some function f (z) having only one simple pole somewhere
on the sphere—we choose the south pole z = 0 for definiteness. Now take an integral with small
contour encircling the south pole (in local coordinates):
I
dz f (z)
C0
Thinking locally (in one coordinate patch), one would expect the integral to yield 2πi Res(f (z), 0)
from Cauchy’s theorem. However, this is inconsistent since the contour C0 may be deformed to
a point continuously (we are on a sphere) and hence the integral yields 0.
The resolution of this paradox is that the integral is always evaluated in local coordinates.
Recall that there are several patches necessary to cover the whole sphere and a single one can
only cover the sphere minus one point. So we have at least two patches: one covering the north
and one the south pole.
2.2 Conformal algebra
27
Now, imagine the contour to be a rubber string around the sphere and the singularity at the
south pole is represented by an (infinitely long) spike. To remove the string from the sphere
continuously (i.e. no cutting) one will obviously have to move it across the north pole and hence
switch the coordinate patch. So for the integral to be consistent (the integral should be a
coordinate independent statement) something has to prevent us from pulling the contour over
the north pole. Switching the patches is done by z 7→ z −1 and therefore exchanging 0 and ∞.
After this change, we need a “spike” at the north pole to stop our rubber string. So the function
has to possess another singularity (pole) at the north pole. But a pole in terms of z −1 is a zero
in terms of z. So f needs one pole and one zero.
More generally, this forces any function with n poles to have n zeros and vice–versa. This can
be seen with the rubber argument too although it is a bit harder to imagine. Take a function with
two poles for instance and consider two small contours C1 , C2 encircling each of the singularities.
Hence, we expect the integral to yield the sum of the residues Res1 + Res2 . But as before
we can deform one of the contours to have it go around the other singularity. However, this
second contour now has the opposite orientation of the first one and hence the integral evaluates
to zero: Res1 − Res1 = 0. To resolve this contradiction, there had to be a zero somewhere
which prevented us from deforming the contour C2 . The same argument holds for the other
contour C1 and therefore we conclude that f has two poles and two zeros. Similar (although
more complicated) arguments apply to a higher number of poles.
Hence, we have seen that the fact that the topology of CP1 (to be more precise: the global
properties of closed curves with respect to continuous deformations) forces our function to have
the same number of poles and zeros.
2.2.4 The algebra of conformal vector fields on the Riemann sphere
The experience with symmetries in quantum field theory shows, that their local symmetry properties (i.e. algebras) are often more important than global ones (i.e groups). In this section, we
will therefore explore the local structure of conformal transformations on the Riemann sphere.
We have seen above, that all meromorphic mappings of the Riemann sphere correspond to
conformal transformations. In local coordinates, in some neighborhood of z it is
(z, z̄) 7−→ (ζ(z), ζ̄(z̄)) = (z + ε(z), z̄ + ε̄(z̄))
(2.62)
for some meromorphic (anti–meromorphic) function ε(z) (ε̄(z̄)). Therefore, the action on a
(0,0)–tensor field φ (i.e. a scalar field) is
£ǫ φ(z, z̄) = (ε(z) ∂z + ε̄(z̄) ∂z̄ ) φ(z, z̄) .
(2.63)
Let us now Laurent expand ε and ε̄ in z:
ε(z) =
∞
X
cn z
n+1
,
n=−k
ε̄(z̄) =
∞
X
c̄n z̄ n+1 .
(2.64)
n=−k̄
With this, the action on a (0,0)–tensor becomes
£ǫ φ(z, z̄) =
∞
X
n=−k̃
cn z n+1 ∂z + c̄n z̄ n+1 ∂z̄ φ(z, z̄) ,
(2.65)
where k̃ = max(k, k̄) and the undefined cn ’s or c̄n ’s are considered zero. At this point we see
the crucial difference to the higher dimensional case. While there was only a finite number of
generators in the former case, there are now (countably) infinitely many of them. We define the
generators as
(2.66)
ℓn := −z n+1 ∂z ,
ℓ̄n := −z̄ n+1 ∂z̄
,n∈Z
28
Chapter 2 – Conformal invariance
and calculate their commutation relations
[ℓm , ℓn ] = (m − n) ℓm+n ,
ℓ̄m , ℓ̄n = (m − n) ℓ̄m+n ,
ℓn , ℓ̄m = 0 .
(2.67)
This algebra corresponds to two copies of the so–called Witt algebra W [PS86]. It is the complexified Lie algebra of orientation preserving diffeomorphisms on the circle Diff + (S 1 ), i.e.
W = C ⊗ diff+ (S 1 ) .
(2.68)
Note that the Witt algebra contains sl(2, C) as a subalgebra. The generators of sl(2, C) are
{ℓ−1 , ℓ0 , ℓ1 } or equivalently ε(z) ∈ {1, z, z 2 }. In fact it turns out that this is the only closed
subalgebra and the only one that can be integrated to form a group, which is exactly what we
expect from the results of our analysis in the preceding section.
It is an important fact that we encounter two independent algebras for the holomorphic and
the anti–holomorphic parts of conformal transformations. In the context of string theory these
two algebras are often also referred to as chiral and anti–chiral or left– and right–moving algebras.
Remark (The Witt Algebra and the Conformal Group).
As we have shown in section 2.2.3, the group corresponding to the Witt algebra is finite
dimensional. Nevertheless, we argued in the current section that the infinite–dimensional local
(i.e. algebraic) structure is more important to physics than the finite–dimensional global (i.e.
group) one. This point is going to be a crucial step in our work, as we will use the algebra
to obtain—in principle—all correlation functions of a CFT. But why should we use the whole
algebra, if only a finite part of it actually appears to be a full symmetry? Especially when doing
string theory it is a bit disturbing to forget about the group without any doubts, as mathematics
is our main guideline.
Therefore it is very relieving that there exists also a motivation for this crucial step from a
more mathematical viewpoint. [Sch97]
It turns out in a careful analysis that on a compact real two–dimensional manifold with
Minkowskian metric, the conformal algebra can indeed be completely integrated to form an
infinite dimensional group. In particular the conformal group on a (1,1) Minkowski sphere is
given by
(2.69)
COS2 (1, 1) = Diff + (S 1 ) × Diff + (S 1 ) .
Hence, we realize that the main reason for the breakdown of the infinite–dimensional conformal
group to a finite–dimensional one, originates in the Wick-rotation we made and is not a general
flaw of our approach. The precise meaning of a Wick–rotation in quantum field theory has
been an open problem for a long time and we won’t make any attempts to resolve it here.
Henceforward, we will believe in the physical significance of a Wick rotated quantum field theory
and ignore the caused mismatch between the local and global conformal symmetry in our theory.
We would also like to note that it is a widely held misbelief that the conformal group on the
complex plane is infinite dimensional. See for example [GO89], p. 333
“ The conformal group in two–dimensional Euclidean space is infinite dimensional
and has an algebra consisting of two commuting copies of the Virasoro algebra. ”
Similar statements can be found in [BPZ84] (p. 335) or [FQS84a] (p. 420). This misbelief can
be traced back to the fact that “physicists mostly think and calculate infinitesimally, while they
write and talk globally” [Sch97].
3 Conformal Quantum Field Theory
“It is often stated that of all the theories proposed in this century, the
silliest is quantum theory. In fact, some say that the only thing that quantum theory
has going for it is that it is unquestionably correct."
Michio Kaku (1947 – )
Conformal quantum field theory (CFT) is the mathematical framework used to study continuous physical quantum systems that obey conformal invariance. Typical examples are two–
dimensional critical statistical systems as the 2d–Ising model or the Ashkin–Teller model. Our
major goal is to describe the dynamics on the two–dimensional world–sheet of a vibrating string,
the topic of chapter 5.
In the last section we established conformal invariance as a space–time symmetry of manifolds.
We now proceed to formulate a quantum field theory which obeys this symmetry. Here, our
method of choice for quantization is the path–integral formalism. Nevertheless, it will be usefull
to return to operator methods in chapter 4 since they make powerful algebraic and group theoretical methods applicable. Moreover, there are many conformal quantum field theories which
have been defined through their operator algebra and for which no Langrangian formulation is
known.
Conformal quantum field theory was introduced in the work [BPZ84]. Aspects connected to
the analytic geometry of Riemann surfaces have been studied in [FS87]. A generalization to
higher genus Riemann surfaces is found in [EO87].
3.1 Conformal Tensors and Primary Fields
In quantum field theory, all fields are representations of the Poincaré group P(r, s) or equivalently
tensors with respect to P(r, s). Geometrically, the fields are sections on the associated bundles of
the principal bundle over the space–time M with fibre P(r, s). A straightforward generalization
of this notion to the conformal group in two (Euclidean) dimensions is not going to work since we
already noticed that the group after Wick–rotation does not represent the whole symmetry of our
theory. Hence, we will generalize the above notion of fields to make use of the full infinitesimal
symmetry.
As was established in chapter 2, the natural playground for conformal symmetry in two dimensions after Wick–rotation is the Riemann sphere CP1 . Let us carefully repeat the introduction of
the Riemann sphere. We consider it an S 2 with the standard complex structure on C. According
to theorem 2.3, we choose local coordinates on the sphere S 2 for which the metric is conformally
flat, i.e.
g = eσ(ξ) dξ 0 ⊗ dξ 1 + dξ 1 ⊗ dξ 0 .
(3.1)
We now define the complex structure by introducing complex coordinates through
z := ξ 1 + iξ 0 ,
z̄ := ξ 1 − iξ 0 .
(3.2)
30
Chapter 3 – Conformal Quantum Field Theory
In these, the tangent space T+ S 2 ⊕ T− S 2 is spanned by
¯ ≡ { 1 (∂1 − i ∂0 ) , 1 (∂1 + i ∂0 )}
{∂z , ∂z̄ } ≡ {∂, ∂}
2
2
(3.3)
and the cotangent space T+∗ S 2 ⊕ T−∗ S 2 by
{dz , dz̄} ≡ {dx1 + i dx0 , dx1 − i dx0 } .
(3.4)
The metric tensor is written
g=
1 σ(z,z̄)
e
(dz ⊗ dz̄ + dz̄ ⊗ dz) .
2
(3.5)
Now, let us take a closer look on tensors on the Riemann sphere, which are our candidates
for physical fields. Since the metric defines an isomorphism between the tangent and cotangent
spaces, we will restrict ourselves to cotensors on the cotangent space T+∗ S 2 ⊕ T−∗ S 2 . Note that
both T+∗ and T−∗ are (real) one–dimensional vector spaces, orthogonal to each other and that this
decomposition is unique, i.e. independent of coordinates. It is determined by the choice of the
complex structure solely.
Let us investigate the action of a general S 2 diffeomorphism on the basis of the tangent space.
Let1
(z, z̄) 7→ (w, w̄) = (f (z, z̄), f¯(z, z̄))
(3.6)
be a diffeomorphism. Then the bases of T+∗ and T−∗ transform as
dw =
∂f
∂f
dz +
dz̄
∂z
∂ z̄
dw̄ =
∂ f¯
∂ f¯
dz +
dz̄ .
∂z
∂ z̄
(3.7)
Hence, in order to preserve the complex structure (or equivalently to preserve the splitting into
T+∗ and T−∗ ), a diffeomorphism has to be either holomorphic or anti–holomorphic:
or
(z, z̄) 7→ (w, w̄) = (f (z), f¯(z̄))
(3.8)
(z, z̄) 7→ (w, w̄) = (f (z̄), f¯(z)) .
(3.9)
We see that they are exactly the conformal mappings discussed in chapter 2. The second possibility does not strictly preserve the complex structure. Let, for example (w, w̄) := (z̄, z). Then
both structures are said to be conjugate to each other. However, looking at (3.3) and (3.4), we
see that T+ ∼
= T− . Hence, the conjugate complex structures are isomorphic. This isomorphism is
equivalent to complex conjugation on the Riemann sphere—hence the term conjugate complex
structure.
In general, any cotensor of CP1 is a tensor product of T ∗ elements. This product may now
be decomposed according to the complex structure into tensor products of T+∗ and T−∗ elements.
For example tensors of weight 2 decompose as
T ∗ ⊗ T ∗ = (T+∗ ⊕ T−∗ ) ⊗ (T+∗ ⊕ T−∗ ) = (T+∗ ⊗ T+∗ ) ⊕ (T+∗ ⊗ T−∗ ) ⊕ (T−∗ ⊗ T−∗ ) .
A general tensor of weight ∆ is decomposed as
M
∗
∗
∗
T(∆)
=
T(h,
with T(h,
T+∗ ⊗ . . . ⊗ T−∗ ⊗ . . . ,
h̄)
h̄) = |
{z } | {z }
h+h̄=∆
1
h times
(3.10)
(3.11)
h̄ times
We will often introduce symbols with and without a bar. This notation emphasizes the splitting according to
the complex structure, which does not necessarily mean that symbols with and without bars are related by
complex conjugation. For example f¯ 6= f ∗ below.
3.1 Conformal Tensors and Primary Fields
31
∗
Written out in components, this means that an element Φ(h,h̄) of T(h,
is of the form
h̄)
Φ(h,h̄) ≡ φ z . . . z
dz ⊗ . . .
| {z } | {z }
h
h
⊗ ψ z̄ . . . z̄
dz̄ ⊗ . . .
| {z } | {z }
h̄
h̄
= φ z . . . z ψ z̄ . . . z̄
| {z }
h
dz ⊗ . . .
| {z } | {z }
h̄
h
Its components transform with respect to a holomorphic mapping on
(z, z̄) 7→ (w, w̄) = (f (z), f¯(z̄)) as
φ w . . . w ψ w̄ . . . w̄ =
| {z }
h
| {z }
h̄
∂f (z)
∂z
⊗
dz̄ ⊗ . . .
| {z }
h̄
.
(3.12)
CP1 (in local coordinates)
−h ¯ −h̄
∂ f (z̄)
φ z . . . z ψ z̄ . . . z̄ .
| {z } | {z }
∂ z̄
h
(3.13)
h̄
Note that every tensor has exactly one component, due to the fact that there is exactly one
basis element in both the tangent space T+∗ and its complex structure equivalent T−∗ . We will,
therefore, omit the indices in our notation and write henceforward
φ z...z ψ z̄...z̄ ≡ Φ(z, z̄) .
The tensor transformation rule now reads
−h̄
∂f (z) −h ∂ f¯(z̄)
Φ(z, z̄) .
Φ(w, w̄) =
∂z
∂ z̄
(3.14)
(3.15)
In principle, this is the desired defining equation for the components of conformal tensors. However, as we realized in the preceding chapter, the conformal group breaks down to a finite–
dimensional group after Wick–rotation—the Möbius group. Fortunately, for the purposes of
quantum field theory it will be sufficient to use local properties with respect to conformal transformations, i.e. the (infinite–dimensional) Lie–algebra. We, therefore, need to express the global
transformation property (3.15) in local terms.
Since we are dealing with tensors, the infinitesimal action of a diffeomorphism is given by the
Lie derivative with respect to the generating vector field of the diffeomorphism. Let
(fs (z), f¯s (z̄)) = (z + s ε(z), z̄ + s ε̄(z̄)) + O(s2 )
(3.16)
be the flow of diffeomorphisms generated by the vector field (ε(z)∂z , ε̄(z̄)∂z̄ ). We now use (3.15)
to calculate the Lie derivative of a (h, h̄) tensor:


1 
1 ∗
h̄
(∂w fs (w))h ∂w̄ f¯s (w̄) Φ(w, w̄) dw
⊗ . . . ⊗ dw̄ ⊗ . . . − Φ =
£ǫ Φ ≡ lim (fs Φ − Φ) = lim
| {z } | {z }
s→0 s
s→0 s
h
h̄
1
h
h̄
= lim
1 + s ∂ε + O(s2 )
1 + s ∂¯ε̄ + O(s2 )
1 + s ε∂ + s ε̄∂¯ + O(s2 ) Φ(z, z̄) − . . .
s→0 s
. . . Φ(z, z̄) dz
⊗ . . . ⊗ dz̄ ⊗ . . .
| {z }
| {z }
h
h̄
= ε∂ + ε̄∂¯ + h (∂ε) + h̄ (∂¯ε̄) Φ(z, z̄) dz
⊗ . . . ⊗ dz̄ ⊗ . . . .
| {z }
| {z }
h
h̄
This is now of the desired local form and we are almost ready to give a definition of conformal
fields. But before doing so we still need to establish a more precise language for the infinitesimal
transformation properties of fields. Reviewing our above discussion, we see that it was not crucial
to have CP1 as base manifold but to have a two–dimensional manifold with complex structure.
Therefore, we will drop the restriction to CP1 and allow for a general Riemann surface of genus
g as base manifold.
32
Chapter 3 – Conformal Quantum Field Theory
Definition 3.1: Let Φ be a (h, h̄) tensor on a Riemann surface Σg and Φ(z, z̄) its component. We
call δε,ε̄ Φ(z, z̄) the first variation of Φ(z, z̄). It is defined by
£ǫ Φ =: δε,ε̄ Φ(z, z̄) dz
⊗ . . . ⊗ dz̄ ⊗ . . . .
| {z }
| {z }
h
(3.17)
h̄
Lemma 3.2: The first variation is a derivation, i.e. it satisfies ∀α , β ∈ R and arbitrary tensor fields
T , S (with components t, s)
(i)
(ii)
Proof:
(T and S of same rank)
δ(α t + βs) = α δt + β δs
(Leibnitz rule)
δ(t s) = δ(t) s + t δ(s)
(i) As the Lie derivative is linear, it holds that
(3.18)
£(α T + βS) = α £T + β £S .
Writing T = t |dz ⊗
. . . ⊗ dz̄ ⊗ . . . , S = s dz ⊗ . . . ⊗ dz̄ ⊗ . . ., the left hand side is
{z } | {z }
| {z } | {z }
h1
h1
h̄1
h̄1
δ(α t + α s) dz ⊗ . . . ,
(3.19)
α £(t dz ⊗ . . .) + β £(s dz ⊗ . . .) = (α δt + β δs)dz ⊗ . . . .
(3.20)
while the right hand side is
By comparing both and equating the components we obtain the proposition.
(ii) Major part of the proof is the Leibnitz rule for the Lie derivative (A.14)
£ǫ (T ⊗ S) = (£ǫ T ) ⊗ S + T ⊗ (£ǫ S) .
(3.21)
T ⊗ S = t s |dz ⊗
. . . ⊗ dz̄ ⊗ . . . .
{z } | {z }
(3.22)
We have by definition
h1 +h2
h̄1 +h̄2
Then the first variation of this product is given through
£(T ⊗ S) ≡ δ(t s) dz
⊗ . . . ⊗ dz̄ ⊗ . . . .
| {z } | {z }
h1 +h2
(3.23)
h̄1 +h̄2
But according to the Leibniz rule we have
δ(t s) dz
⊗ . . . ⊗ dz̄ ⊗ . . . = £(T ⊗ S) = £(T ) ⊗ S + T ⊗ £(S) ≡ [δ(t) s + t δ(s)]
| {z } | {z }
h1 +h2
h̄1 +h̄2
dz ⊗ . . . ⊗ dz̄ ⊗ . . .
| {z } | {z }
h1 +h2
, (3.24)
h̄1 +h̄2
from which we deduce that the first variation fulfils the Leibnitz rule as well.
It is now straightforward to infer the following:
Corollary 3.3: The first variation for a product of fields is given by
δε,ε̄ {Φi1 (z1 , z̄1 ) . . . Φin (zn , z̄n )} =
n
X
k=1
Φi1 (z1 , z̄1 ) . . . {δε,ε̄ Φik (zk , z̄k )} . . . Φin (zn , z̄n ) .
(3.25)
Finally, we establish the notion of conformal fields.
3.2 Conformal Quantum Field Theory – a physicists definition
33
Definition 3.4: Let Φ be a tensor on a Riemann surface Σg . Its component Φ(z, z̄) is called a
primary conformal field of weight (h, h̄), if its first variation with respect to any conformal transformation, generated by (ε, ε̄), is given by
δε,ε̄ Φ(z, z̄) = ε∂ + ε̄∂¯ + h (∂ε) + h̄ (∂¯ε̄) Φ(z, z̄) .
(3.26)
We call h + h̄ the scaling dimension of the field Φ and h − h̄ its conformal spin.
Definition 3.5: A tensor component for which the above relations only hold for conformal transformations in sl(2, C)/Z2 is called a quasi–primary conformal field.
Remark. The restriction to sl(2, C)/Z2 in the formulae above is equivalent to the choice ε(z) ∈
{1, z, z 2 }, as was noted in 2.2.4.
3.2 Conformal Quantum Field Theory – a physicists definition
Having established the connection between infinitesimal conformal invariance and the notion of
fields, we are now able to formulate a corresponding quantum field theory. We will take up
the position that a quantum field theory is defined through the knowledge of all its correlation
functions. Furthermore, we assume that the correlation functions can be obtained by using
(Euclidean) path integral techniques.
Definition 3.6: Let {Φr }r=1 be a collection of fields on the pseudo Riemann manifold M (with one
time–like direction), L({Φr }) their field theory Lagrangian and
Z
p
dd x | det(g)| L({Φr })
S[{Φr }] ≡
(3.27)
M
the corresponding action. The partition function of ({Φr }, L) is defined as
Z
Z := DΦ ei S[{Φr }] .
(3.28)
We call
Z
1
hΦi1 (x1 ) . . . Φin (xn )i :=
DΦ Φi1 (x1 ) . . . Φin (xn ) ei S[{Φr }]
(3.29)
Z
a Wightman function, correlation function, correlator or n–point function of the field theory ({Φr }, L).
The points xi are called insertion points or insertions of the fields.
R
RQ
DΦr .
In the above expressions, the path integral ranges over all fields, i.e. DΦ :=
r
Remarks.
• Note that we do not specify what exactly a field is. We only assume that it is some function
on the manifold M. It would be more natural to define a field to represent the symmetry
of our theory. Then it would be a section on an appropriate bundle associated to the
principal bundle of the symmetry group of our theory. However, we will not pursue this
approach for two reasons. Firstly, we would like to keep close to the standard physics
literature in order to make our reasoning easily accessible for particle physicists. Secondly,
as we are mainly interested in two-dimensional conformal symmetry, we need to focus on
infinitesimal properties to make use of the infinite dimensional conformal algebra.
• Concerning our notion of fields, we always implicitly assume their correlators to fulfill
Wightman’s axioms, though we will not check this explicitly, since we are not going to
make much use of these properties. The two most important facts are that correlators
are distributions (hence, their equations are only valid in a distributional sense) and the
concept of local fields, i.e. the validity of certain causality relations.
34
Chapter 3 – Conformal Quantum Field Theory
• Note that our definition does not specify the fields to take values in the real numbers. They
are free to take values in the Grassmanian numbers in order to incorporate fermions into
our theory.
• Regarding a field simply as a function with appropriate boundary conditions, note that
its derivative is not necessarily a field itself. Usually we will assume that for every field,
its derivatives are indeed fields as well and therefore fulfil the same boundary conditions.
However, they may have very different symmetry properties. For example, they differ in
their mass dimensions and will therefore have different scaling properties.
• Correlation functions are analytic functions in their arguments, as long as the insertion
points are pairwise distinct. However, they diverge in general when two points coincide,
which reflects their distributional nature.
Definition 3.7: Let {Φr }r=1 be a set of fields on the manifold M and L({Φr }) their field theory
Lagrangian. We call the collection
{hΦi1 (x1 ) . . . ΦiL (xL )i | L ∈ N , ij ∈ [1, . . . , M ] ∀ j , xk ∈ M ∀ k}L
(3.30)
a Quantum Field Theory (QFT).
It will be necessary to work in Euclidean QFT, i.e. QFT on a Riemann manifold with Euclidean metric. Starting with a QFT defined on a manifold with one time–like direction, the
corresponding Euclidean QFT is defined as follows
Definition 3.8: Let a QFT Q be given. Its Euclidean QFT QE is obtained via
Q 7−→ QE
where
iS =
Z
M
d
d x
√
g L(x) 7−→ −S = −
(3.31)
Z
dd ξ
ME
√
gE L(ξ) .
(3.32)
ξ are coordinates of the Euclidean Manifold ME . We call the mapping from Q to QE Wick rotation.
Remark. The bottom–line of the above definition is, that correlators in Euclidean QFT are
calculated by using −S instead of iS (making the path–integrals better defined) and everything
has to be expressed in Euclidean coordinates. Usually, it is assumed that the correlators obtained
in a (Minkowskian) QFT can be obtained from the Euclidean correlators by analytic continuation.
Next, we will include conformal invariance into our concept of Quantum Field Theory. This
is most easily done by forcing the fields to obey conformal invariance as was established in the
preceding section. From now on we focus solely on two–dimensional systems to make full use of
the conformal algebra.
Definition 3.9: Let {Φr }r=1 be a set of primary conformal fields over the Riemann surface Σg and
L({Φr }) their field theory lagrangian. We call the collection
{hΦi1 (z1 , z̄1 ) . . . ΦiL (zL , z̄L )i | L ∈ N , ij ∈ [1, . . . , M ] ∀ j , (zk , z̄k ) ∈ Σg ∀ k}L
(3.33)
a Conformal Quantum Field Theory (CFT).
Remarks.
• Note that this definition is only sensible for two–dimensional field theories (on Riemann
surfaces) according to our definition of primary fields. Therefore, every CFT in the above
sense is by definition living on a two–dimensional manifold.
3.3 Symmetries and the Energy–Momentum Tensor
35
• The insertions (zk , z̄k ) are points in M. Though, we will usually take them in local coordinates which we denote by (zk , z̄k ) as well.
• We have defined a CFT to be Euclidean right from the start. However, we shall motivate
its importance for QFTs on two–dimensional Minkowskian space–time.
Although we have introduced CFT in an axiomatic style, it has to be pointed out that our
approach is far from being rigorous. Already the functional integral is not well–defined beyond
physical rigour, not to mention the various other problems encountered when trying to rely on a
minimal set of axioms. But since we are mainly interested in calculational aspects of quantum
field theory, the above definition will be suitable for our purposes and we can avoid to be drawn
into the depths of axiomatic (conformal) quantum field theory. For true axiomatic approaches
see [Seg87], [Hua92], [GG00] and the reviews [Gaw96], [Gab00].
3.3 Symmetries and the Energy–Momentum Tensor
The so–called energy–momentum tensor plays an exceptional role already in classical and quantum field theory. As we will realize soon, it is of even greater importance in conformal quantum
field theory.
3.3.1 Classical Symmetries and the Classical Energy–Momentum Tensor
Historically, the energy–momentum tensor is defined by making use of Noether’s theorem. The
major drawback of this approach is that the energy–momentum tensor defined this way is only
unique up to a total derivative. This prevents an interpretation as the source for gravitational
forces in general relativity and furthermore makes a straightforward generalization to quantum
field theory difficult.
Hence, we choose a different way of defining the energy–momentum tensor, motivated by
general relativity.
Definition 3.10: Let ({Φr }, L) be a field theory with corresponding action S. The unique symmetric
tensor T defined through
Z
1
√
dd x g Tµν δǫ gµν ,
(3.34)
δǫ S =: −
2
p
√
is called the (true) energy–momentum tensor of the field theory ({Φr }, L). ( g ≡ | det(g)|)
Remarks.
• Note that we have defined the energy–momentum tensor to be symmetric, i.e. Tµν = Tνµ .
This is necessary for our definition to be equivalent to
2 δS
,
Tµν = − √
g δgµν
(3.35)
where δgδµν denotes the functional derivative. As we shall see in a moment, this is crucial for
T to take the role of energy–momentum in Einstein gravity, which justifies our definition.
• It may seem a bit unnatural to consider the variation with respect to the inverse metric.
One reason for this is that the resulting energy–momentum tensor is more naturally related
to Einstein’s equations (see the following remark). Another reason is that in a rigorous
Lagrangian formulation the natural gravitational degrees of freedom are not found to be
the components of the metric itself, but rather the so–called “Vielbeins” eα ≡ eαµ dxµ .
36
Chapter 3 – Conformal Quantum Field Theory
They account for the freedom to choose an orthogonal (Minkowskian) coordinate system
locally:
gµν = eαµ (x) eβν (x) ηαβ
with the flat (Minkowskian) metric ηαβ . Therefore, the action is varied with respect to the
eα , which produces the upper indices naturally. Details can be found in [Thi97].
For our purposes, varying with respect to the inverse metric can be considered a convention
rather than a fundamental statement.
• In the context of general relativity it has to be pointed out that the action written above
reflects only the matter part, i.e. does not contain the Einstein–Hilbert action
Z
1
√
dd x g R ,
Sgrav =
4
8πG/c
with the Ricci scalar R and the gravitational constant G. Consequently, varying the sum
of both actions leads to the Einstein equations with T as source:
Rµν −
1
8πG
gµν R = 4 Tµν .
2
c
This is the physical motivation for defining the energy-momentum tensor in this way.
• The corresponding definition of the energy–momentum tensor from Noether’s theorem is
given by
∂L
∂ν φr − L η µν ,
(3.36)
(Tc )µν =
∂∂µ φr
and is called the canonical energy–momentum tensor. It has the apparent drawbacks of
lacking symmetry in its indices (what disqualifies it as the source for gravitational forces)
and having no genuine gauge invariance. Furthermore, this definition is only valid on a flat
(Minkowskian) space–time.
It can be easily seen from the definition, that T is covariantly conserved (∇T = 0) in any
diffeomorphism invariant field theory (see below for a proof). But of greater importance for us
is the connection between conformal invariance and the energy–momentum tensor.
Proposition 3.11: Any field theory with a traceless energy–momentum tensor (Tµµ = 0) is invariant
under conformal transformations.
Proof:
From equation (2.18) we know that for any conformal transformation generated by ǫ it is
δǫ gµν ≡ ∇µ ǫν + ∇ν ǫµ =
2
∇σ ǫσ gµν .
d
(3.37)
The variation of the inverse metric is deduced from the invariance of the identity matrix:
0 = δǫ (δ µα ) = δǫ (gµν gαν ) = gαν δǫ gµν + gµν δǫ gαν
⇒
2
2
gαβ g δ gµν = −gαβ gµν δǫ gαβ = −gαβ gµν ∇σ ǫσ gαν = − ∇σ ǫσ gµβ .
| {z αν} ǫ
d
d
(3.38)
(3.39)
δ βν
Therefore, invariance of a field theory action under conformal transformations is equivalent to
Z
Z
1
1
√
√
!
0 = δǫ S ≡ −
(3.40)
dd x g Tµν δǫ gµν =
dd x g Tµµ ∇σ ǫσ .
2
d
Consequently, this is automatically satisfied for any field theory with T µµ = 0.
3.3 Symmetries and the Energy–Momentum Tensor
37
Remark. It has to be stressed that the converse of the above theorem is not true in general,
as can be seen at once from the proof: Given δǫ S = 0, the function ∇σ ǫσ is not arbitrary
(but restricted due to 3.37) and hence one cannot draw the conclusion that Tµµ has to vanish.
Nevertheless, this will never pose a restriction for the cases of interest in this work.
For further use we state the well–known Noether theorem without proof. We focus on space–
time symmetries for these are our main concern.
Theorem 3.12: Noether’s theorem for space–time symmetries. Every continuous space–
time symmetry xµ 7→ xµ + ω a δa xµ + O(ω 2 ) (summation over a) of the action of a physical system
has a corresponding conservation law
∂µ jaµ (x) = 0 ,
(3.41)
where (for all solutions of the equations of motion φr )
∂L
µ
µ
∂ν φr − L η ν δa xν ≡ (Tc )µν δa xν .
ja (x) =
∂∂µ φr
(3.42)
d
Q = 0 are
The corresponding conserved quantities dt
Z
Z
d−1
0
Q a := d x ja ≡ dd−1 x (Tc )0 ν δa xν .
(3.43)
Remark. Although we do not prove the theorem here, we will need an interesting side result
of its derivation later: The action transforms as (the ∗ is just to avoid confusion with the first
variation defined earlier)
S 7−→ S + ω a δa∗ S + O(ω 2 ) ,
(3.44)
where it is
δa∗ S
=−
Z
d
d x ∂µ
∂L
µ
ν
∂ν φr − L η ν δa x
.
∂∂µ φr
Proposition 3.13: Let ({Φr }, L) be a field theory and Tc its canonical
The following set of symmetries and properties of Tc are equivalent


Translations
∂µ (Tc )µν


µν
⇐⇒
Lorentz transformations
(Tc ) − (Tc )νµ


Dilatations
(Tc )µµ
Proof:
energy momentum tensor.
=0
=0
=0
.
(3.46)
(i) An arbitrary translation
xµ 7→ xµ + ω aµ
µ
(3.45)
(3.47)
µ
(i.e. δx = a with an arbitrary vector a) is according to Noether’s theorem equivalent to the conservation
law
(3.48)
∂µ ((Tc )µν aν ) = 0 .
Since this is true for arbitrary a it follows that
∂µ (Tc )µν = 0 .
(3.49)
xµ 7→ Λµν xν = xµ + ω Xµν xν + O(ω 2 )
(3.50)
(ii) For Lorentz invariance, we have
with Xµν a generator of the Lorentz group, i.e. Xνσ = −Xσν . We now have
δxν = Xνσ xσ ,
(3.51)
38
Chapter 3 – Conformal Quantum Field Theory
and the conservation law reads
(∂µ (Tc )µν )Xνσ xσ + (Tc )µν Xνσ ηµσ = 0 .
(3.52)
The first term vanishes due to translation invariance. Further, using the antisymmetry of Xνσ , we obtain
[(Tc )µν − (Tc )νµ ] Xµν = 0 .
(3.53)
From this it follows immediately that (Tc )µν − (Tc )νµ = 0.
(iii) Finally, for dilatations we have
xµ 7→ eω xµ = xµ + ωxµ + O(ω 2 )
(3.54)
δxν = xν .
(3.55)
(∂µ (Tc )µν ) xν + (Tc )µν δµν = 0 .
(3.56)
and therefore
Now the conservation law reads
Again, the first term vanishes due to translation invariance and the second term gives
(Tc )µµ = 0 .
(3.57)
An interesting question is what the precise relationship between the canonical and true energy–
momentum tensors is. Let us look on the conservation law implied by the definition of the
energy–momentum tensor. Assuming the action to be invariant under a general diffeomorphism
generated by ǫ, we have
Z
1
√
0 = δǫ S = −
dd x g Tµν δǫ gµν .
(3.58)
2
From the appendix we know that
δǫ gµν ≡ (£X g)µν = ∇µ ǫν + ∇ν ǫµ .
(3.59)
Then the symmetry condition on S reads
Z
Z
Z
√
1
d √
µ ν
ν µ
d √
µ ν
d x g Tµν (∇ ǫ + ∇ ǫ ) = − d x g Tµν (∇ ǫ ) = dd x g ǫν ∇µ Tµν ,
0=−
2
(3.60)
In the first step we made use of the symmetry of T and in the second step we integrated partially
and used that the covariant derivative of g is zero. Since this has to be true for arbitrary ǫ, we
conclude
∇µ Tµν = 0 .
(3.61)
On a flat background this reads
∂ µ Tµν = 0 .
(3.62)
Furthermore, if we restrict our analysis to dilatations ǫµ = ω xµ and evaluate on a flat background
(i.e. ∇µ = ∂ µ ), it follows from (3.59) that
δǫ gµν |g=η = ω (η µν + η νµ ) .
(3.63)
For the further reasoning, we need to include a small technicality, which could be avoided in our
other calculations. The precise formulation of the symmetry condition is that
Z
√
1
′
(3.64)
dd x g Tµν δǫ gµν = 0
δǫ S(Ω ) ≡ −
2 Ω′
3.3 Symmetries and the Energy–Momentum Tensor
for all Ω′ ⊆ Ω, where Ω is the domain of our theory. Hence, together with (3.63) it is
Z
dd x Tµµ
0 = −ω
39
(3.65)
Ω′
for all Ω′ ⊆ Ω. Accordingly, the integrand has to vanish and it follows
Tµµ = 0.
(3.66)
As the true energy–momentum tensor is symmetric by its definition, we conclude that in a
diffeomorphism invariant theory it is divergence free, traceless and symmetric if evaluated on a
flat background. Conversely, if we start with a field theory on a flat manifold which is translation,
rotation and dilatation invariant, then we know from Noether’s theorem (proposition 3.13) that
the canonical energy momentum tensor is divergence free, traceless and symmetric as well.
In fact, it turns out that both tensors coincide on flat background in any conformally invariant
theory modulo some total derivative term [Wal84]. However, we might encounter the breakdown
of the expected tensor identities (symmetric, traceless, divergence free). In this case, it will be
necessary to add a total derivative term to the canonical energy–momentum tensor in order to
restore these. The resulting tensor will always coincide with the true energy–momentum tensor
(in flat background)[LL80].
From this we conclude one more statement
Proposition 3.14: In flat background, invariance with respect to translations, rotations and dilatations implies classical conformal invariance.
Remark. Standard examples for the above statement are pure Yang–Mills theory in four dimensions or the theory of a free scalar field in two dimensions.
3.3.2 Quantum Symmetries and the Quantum Energy–Momentum Tensor
In a quantum field theory the energy–momentum tensor will only be sensible inside a correlator
hTµν . . .i. There is a very elegant way to incorporate this fact into its (quantum) definition.
First, we define
Definition 3.15: Let ({Φr }, L) be a field theory. We call W the connected vacuum functional or
connected partition function of ({Φr }, L) if
Z[g] = ei W[g] .
(3.67)
Here we explicitly indicated the dependence on the metric tensor by [g].
Now, what happens under the action of a diffeomorphism? For the partition function it follows
that (we always omit O(kδgk2 ) terms, if not stated otherwise)
Z
Z
i (S[{Φ}]+δS[{Φ}])
Z[g + δg] ≡ (DΦ)(g+δg) e
= (DΦ)(g+δg) (1 + i δS[{Φ}]) ei S[{Φ}] . (3.68)
We now use the classical definition of the energy–momentum tensor and assume additionally
that it accounts for any changes in the functional measure as well. Then it follows:
Z
Z
i
µν
d √
Z[g + δg] = (DΦ)g 1 −
ei S[{Φ}]
d x g Tµν δg
2
Z
(3.69)
Z
i
µν
i S[{Φ}]
d √
(DΦ)g Tµν e
= Z[g] −
d x g δg
.
2
40
Chapter 3 – Conformal Quantum Field Theory
As Tµν is some function of the fields, it follows by definition that
Z
(DΦ)g Tµν ei S[{Φ}] ≡ Z[g] hTµν i
and hence
Z[g + δg] =
i
1−
2
Z
d
d x
√
g δg
µν
hTµν i Z[g] .
(3.70)
(3.71)
In terms of the connected partition function it is
Z[g + δg] = ei (W+δW) = (1 + i δW) ei W = (1 + i δW) Z[g] .
(3.72)
Comparing this with the previous expression for Z[g + δg], we can define the correlator of the
energy–momentum tensor in terms of the connected partition function.
Definition 3.16: Let T be a symmetric tensor–valued function of fields {Φr }.
quantum energy– momentum tensor of the field theory ({Φr }, L) if it satisfies
1
δǫ W = −
2
Z
dd x
√
g δǫ gµν hTµν i .
T is called
(3.73)
Remarks.
• It is important that we have assumed the energy–momentum tensor to incorporate any
changes of the path integral measure as well. This means, that if the measure fails to be
invariant under some symmetry transformation of the theory (i.e. a classical symmetry of
the action), then the quantum energy–momentum tensor will lose its tensor property with
respect to this transformation.
This breakdown of classical symmetries upon quantization (i.e. path integral formulation)
is called quantum anomaly. It is very dangerous in view of possible physical applications
since anomalies will spoil the physical consistency of the theory. An improtant anomaly
we are going to consider is the so–called conformal anomaly which causes the breakdown
of classical conformal invariance in a theory.
• Note that in analogy to the classical energy–momentum tensor it is
2
δW
= − √ hTµν i .
µν
δg
g
(3.74)
3.3.3 The Conformal Ward identity
We will now investigate the role of symmetries in conformal quantum field theories. As we
will see, there is a quantum version of Noether’s theorem, expressed in the Ward identities for
correlation functions. Again, our focus is on (continuous) space–time symmetries only. Further,
we assume a flat metric on our manifold which is necessary for using Noether’s theorem.
In our approach, the basic elements of quantum field theories are the correlation functions
obtained from path integral methods. Therefore, it is of great importance to determine their
transformation behavior with respect to space–time transformations.
We have seen in 3.1 that the infinitesimal action of a space–time transformation (i.e. a diffeomorphism of the base manifold) on the theory’s fields is given by the first variation
δǫ Φi (z, z̄) .
(3.75)
3.3 Symmetries and the Energy–Momentum Tensor
41
According to our remark to Noether’s theorem 3.12, the action transforms as
S 7−→ S + ωa δ∗a S ,
with
δ∗a S
=−
Z
d
d x
∂µ [Tµν
a ν
δ x ]≡−
Z
(3.76)
dd x ∂µ jaµ (x) .
(3.77)
We can now calculate the transformation of a correlator by using the path integral method. The
correlator
Z
Z hΦi1 (z1 , z̄1 ) . . . Φin (zn , z̄n )i = DΦ Φi1 (z1 , z̄1 ) . . . Φin (zn , z̄n ) ei S[Φr ]
(3.78)
is mapped onto
Z
a
DΦ′ [Φi1 (z1 , z̄1 ) . . . Φin (zn , z̄n ) + ωa δǫa {Φi1 (z1 , z̄1 ) . . . Φin (zn , z̄n )}] ei S[Φr ]+i ωa δ∗ S[Φr ] .
(3.79)
We further simplify the integrand by dropping O(ω 2 ) terms, which gives
Z
DΦ′ [X + ωa δǫa X + i ωa X δ∗a S] ei S[Φr ] = Z hX i′ + ωa Z hδǫa X i′ + i ωa Z hX δ∗a Si′
(3.80)
with X := Φi1 (z1 , z̄1 ) . . . Φin (zn , z̄n ) and the ′ indicates that the correlator is calculated using the
transformed integration measure. So we have obtained the following relation for the transformed
correlator to first order (take the left–hand side as a definition for that)
h∆ω X i = hX i′ + ωa hδǫa X i′ + i ωa hX δ∗a Si′ .
(3.81)
Now take a look on the inside of the second correlator on the right–hand side. For a product
of fields we know from corollary 3.3 that
δǫ {Φi1 (x1 ) . . . Φin (xn )} =
n
X
k=1
Φi1 (x1 ) . . . {δǫ Φik (xk )} . . . Φin (xn ) .
(3.82)
Recall that δǫ will act like a linear partial differential operator on fields. Take translations
xµ 7→ xµ + ω aµ for instance. Then, it holds that
δǫ Φ(x) = aµ (x) ∂µ Φ(x) .
(3.83)
If we now require the transformation to be a symmetry of our theory, it holds that
h∆ω X i = hX i .
(3.84)
This means that for symmetry transformations, equation (3.81) provides us with a linear partial
differential equation between correlation functions. This equation is called Ward–identity in
quantum field theory.
However, there is still a subtlety. In the path integral formalism we have to be careful about the
functional measure. Therefore, we include this subtlety into our concept of quantum symmetry:
Definition 3.17: Let Q a QFT in the sense of definition 3.7. We call the diffeomorphism f :
xµ 7→ xµ + ωa δa xµ + O(ω 2 ) a quantum (space–time) symmetry of Q iff
f
Z 7−→ Z + O(ω 2 ) ,
and
f
f
DΦ 7−→ DΦ
hΦi1 (z1 , z̄1 ) . . . Φin (zn , z̄n )i 7−→ hΦi1 (z1 , z̄1 ) . . . Φin (zn , z̄n )i + O(ω 2 )
(3.85)
(3.86)
42
Chapter 3 – Conformal Quantum Field Theory
Remarks.
• It may seem strange to define the correlation functions to be invariant as well as the
functional measure, since the former contain the latter. However, this is justified by the fact
that this definition ensures that every quantum symmetry is a symmetry of the underlying
classical field theory (not vice–versa!). To see this, look at the invariance of Z (using the
invariance of DΦ)
Z
Z
f
DΦ exp(i S) 7−→ DΦ exp i S + i ωa δ∗a S + O(ω 2 ) .
(3.87)
Hence, we have necessarily δ∗a S = 0 and consequently every quantum symmetry is a classical
symmetry.
• We have defined a symmetry from the start infinitesimally. This is justified from the fact,
that this definition can be shown to be equivalent to invariance with respect to the whole
transformation. See [FMS97], chapter 2 for details.
We will now prove the quantum equivalent to Noether’s theorem. It consists of two major
parts: Firstly, it relates the variation of correlators to the classical Noether current. Secondly, it
gives the quantum conserved charge.
Lemma 3.18: Let Q a QFT in the sense of definition 3.7. Let xµ 7→ xµ + ωa δa xµ a quantum
space–time symmetry of Q. Then the following statements are true:
(i) The linear partial differential equation—called the Ward–identity—
Z
∂
a
δǫ hΦi1 (x1 ) . . . Φin (xn )i = i
dd x µ hTµν (x) δa xν Φi1 (x1 ) . . . Φin (xn )i
∂x
(3.88)
is fulfilled for an arbitrary insertion of fields.
(ii) Q admits the following conservation law:
n
X
∂
µ
hj (x) Φi1 (x1 ) . . . Φin (xn )i = −i
δ(x − xk ) hΦi1 (x1 ) . . . {δǫa Φik (xk )} . . . Φin (xn )i ,
∂xµ a
k=1
(3.89)
with the Noether current jaµ (x) = Tµν (x) δa xν and for an arbitrary insertion of fields.
Proof:
(i) We assume xµ 7→ xµ + ωa δ a xµ to be a quantum symmetry of Q. Hence,
h∆ω X i = hX i
(3.90)
for an arbitrary insertion of fields X := Φi1 (x1 ) . . . Φin (xn ).
With this, it follows from (3.81) that
′
′
′
hX i = hX i + ωa hδǫa X i + i ωa hX δ∗a Si .
(3.91)
Since the functional measure remains invariant under the action of a quantum symmetry, we have
−i ωa hX δ∗a Si = ωa hδǫa X i .
Using (3.77) this gives
i ωa
for all ωa . Hence,
Z
dd x ∂µ (Tµν δ a xν ) X
hδǫa X i = i
Z
= ωa hδǫa X i
dd x ∂µ hTµν δ a xν X i
(3.92)
(3.93)
(3.94)
3.3 Symmetries and the Energy–Momentum Tensor
43
which is the first part of the proposition.
(ii) For the second part, we will write down a local version of (3.94). In order to do so, we have
to rewrite the left–hand side as an integral. This can be done by inserting delta distributions in an
appropriate way. First, let us use corrolary 3.3 to write the left–hand side explicitly:
hδǫa
Xi =
n
X
k=1
hΦi1 (x1 ) . . . {δǫa Φik (xk )} . . . Φin (xn )i .
(3.95)
Inserting delta distributions for every xk and integrating yields
Z
n
n
X
X
δ(x − xk ) hΦi1 (x1 ) . . . {δǫa Φik (xk )} . . . Φin (xn )i .
hΦi1 (x1 ) . . . {δǫa Φik (xk )} . . . Φin (xn )i = dd x
k=1
k=1
(3.96)
Inserting (3.96) into (3.94) gives
Z
Z
n
X
dd x
dd x ∂µ hTµν δ a xν X i .
δ(x − xk ) hΦi1 (x1 ) . . . {δǫa Φik (xk )} . . . Φin (xn )i = i
(3.97)
k=1
With the usual reasoning of Noether’s theorem we conclude
n
X
k=1
δ(x − xk ) hΦi1 (x1 ) . . . {δǫa Φik (xk )} . . . Φin (xn )i = i ∂µ hTµν δ a xν X i .
(3.98)
This is the second proposition.
Remark. Note that we will be a bit loose with the term Ward–identity, using this term for
(3.89) as well as (3.88). However, this should not give rise to any confusion.
We will now rewrite the Ward–identity (3.88) for a Euclidean QFT on a Riemann surface.
Proposition 3.19: The Ward identity in a Euclidean QFT on a Riemann surface takes the form
I
I
1
1
dz̄ h−2πjz̄ (z, z̄) X i .
(3.99)
dz h−2πjz (z, z̄) X i +
δε,ε̄ hX i =
2πi C
2πi C ∗
Q
Φil (zl , z̄l ) is an arbitrary insertion of (not necessary primary) fields. C is a curve
where X :=
l
encircling all insertions of X anti–clockwise and C ∗ its complex conjugate.
Proof: The proof is performed by writing down the Ward–identity (3.88) in complex coordinates and
making use of Stokes’ theorem (A.17) :
Z
I
I
2
dz̄ vz̄ .
(3.100)
i
d z (∂z vz̄ + ∂z̄ vz ) = − dz vz −
ω
∂ω
From (3.88) we have
δǫa hX i = i
Z
(∂ω)∗
d2 x ∂ µ jµa (x) X
(3.101)
with X = Φi1 (x1 ) . . . Φin (xn ) and jµa (x) := Tµν (x) δ a xν . Performing a Wick rotation, this expression
becomes (recall equation (3.77)):
Z
a
(3.102)
δǫ hX i = − d2 x ∂ µ jµa (x) X .
We now choose (flat) complex coordinates and obtain2
Z
1
a
δε,ε̄
hX i = −
d2 z gzz̄ ∂z̄ hjz (z, z̄) X i + gz̄z ∂z hjz̄ (z, z̄) X i
2
Z
= − d2 z (∂z̄ hjz (z, z̄) X i + ∂z hjz̄ (z, z̄) X i) .
2
Recall that d2 x
Wick rotation
7−→
i d2 xE =
i
2
d2 z and gz z̄ = gz̄z = 2 .
(3.103)
44
Chapter 3 – Conformal Quantum Field Theory
Using (3.100) with vz = hjz (z, z̄) X i and vz̄ = hjz̄ (z, z̄) X i, we obtain
I
I
1
a
− dz hjz (x) X i − dz̄ hjz̄ (x) X i
δǫ hX i =
i
I
I
1
1
=
dz h−2πjz X i +
dz̄ h−2πjz̄ X i .
2πi C
2πi C
(3.104)
Remark. Note that we have not required the theory to be quantum conformally invariant. In
the above proposition, the theory is (quantum) invariant with respect to the transformation
(z, z̄) 7→ (z + ε(z, z̄), z̄ + ε̄(z, z̄)) which is not necessarily a conformal transformation (or only a
subalgebra).
It is possible to prove the following lemma:
Lemma 3.20: Let a two–dimensional quantum field theory on flat background be given which is
quantum symmetric under translations, Lorentz transformations and dilatations. Then it holds for its
quantum energy–momentum tensor that
(i)
∂¯ hTzz (z, z̄)X ({zl , z̄l })i = 0
∀zl , z̄l 6= z, z̄ .
(3.105)
(ii)
∂ hTz̄ z̄ (z, z̄)X ({zl , z̄l })i = 0
∀zl , z̄l 6= z, z̄ .
(3.106)
hTz z̄ (z, z̄)X ({zl , z̄l })i = 0
∀zl , z̄l 6= z, z̄ .
(3.107)
(iii)
where X :=
Q
l
Φil (zl , z̄l ) is an arbitrary insertion of (not necessary primary) fields.
Proof: A proof of a similar statement was originally given in [Pol70] and can be found in modern
language in [FMS97] at the end of chapter 4.
Remark. The condition of flat background still allows for gravitational anomalies to appear,
which result in
µ
T µ (z, z̄)X = hTz z̄ (z, z̄)X i ∝ R(z, z̄)
(3.108)
with the Ricci scalar R, which vanishes on flat background. We will see this later in the concrete
example of one free scalar field.
We are now ready to give the condition for conformal invariance of a QFT on a Riemann
surface.
Proposition 3.21: Let Tµν be the quantum energy momentum tensor of a QFT on a Riemann
surface. The QFT is conformally invariant if and only if
(0)
Q
l
(3.109)
(i)
∂¯ hTzz (z, z̄)X ({zl , z̄l })i = 0
∀zl , z̄l 6= z, z̄ .
(3.110)
(ii)
∂ hTz̄ z̄ (z, z̄)X ({zl , z̄l })i = 0
∀zl , z̄l 6= z, z̄ .
(3.111)
hTz z̄ (z, z̄)X ({zl , z̄l })i = 0
∀zl , z̄l 6= z, z̄ .
(3.112)
(iii)
where X :=
DΦ 7−→ DΦ
Φil (zl , z̄l ) is an arbitrary insertion of (not necessary primary) fields.
3.3 Symmetries and the Energy–Momentum Tensor
45
Proof: “⇒” Let a conformally invariant quantum field theory be given. Then (0) is true by definition.
Further, the theory has to be invariant with respect to translations, rotations and dilatations since they
form a subalgebra of the conformal algebra. Hence, (i), (ii) and (iii) are true because of lemma 3.20.
“⇐” We start from equation (3.103)
Z
a
δε,ε̄
hXi = − d2 z (∂z̄ hjz (z, z̄) Xi + ∂z hjz̄ (z, z̄) Xi) .
(3.113)
For a general transformation (z, z̄) 7→ (z, z̄) + (ǫz (z, z̄), ǫz̄ (z, z̄)), jz and jz̄ are given by
jz = Tzz ǫz (z, z̄) + Tzz̄ ǫz̄ (z, z̄)
(3.114)
jz̄ = Tz̄z̄ ǫz̄ (z, z̄) + Tz̄z ǫz (z, z̄) .
(3.115)
and
For conformal transformations (z, z̄) 7→ (z + ε(z), z̄ + ε̄(z̄)) it is
∂z̄ jz = ε∂¯ Tzz + (∂¯ε̄) + ε̄ ∂¯ Tzz̄
∂z jz̄ = ε̄ ∂ Tz̄z̄ + [(∂ ε) + ε ∂ ] Tzz̄
(3.116)
where we used that T is symmetric by definition. This gives for the variation of a correlator
Z
Z
a
δε,ε̄
hXi = − d2 z ε ∂¯ Tzz + ∂¯ ε̄ + ε̄ ∂¯ Tzz̄ X − d2 z h(ε̄ ∂ Tz̄z̄ + [∂ ε + ε ∂ ] Tzz̄ ) Xi . (3.117)
Given equations (i), (ii) and (iii) of the proposition are true, we deduce
a
δε,ε̄
hXi = 0 .
(3.118)
Together with (0) this implies invariance of the theory with respect to conformal transformations.
Above proposition and lemma together imply the important conclusion:
Theorem 3.22: Any two–dimensional Quantum Field Theory which is invariant under translations,
rotations and dilatations is conformally invariant.
Remark. Note that of course the converse is also true, since translations, rotations and dilatations are conformal transformations. However, it is remarkable that this is sufficient for the
theory to be invariant with respect to the infinite dimensional conformal algebra.
The following will be of great importance in many calculations:
Theorem 3.23: The Ward identity in a conformally invariant QFT—the conformal Ward identity —
takes the form
I
I
1
1
δε,ε̄ hX i =
dz ε(z) hT(z) X i +
dz̄ ε̄(z̄) T̄(z̄) X .
(3.119)
2πi C
2πi C ∗
Q
Φil (zl , z̄l ) is an arbitrary insertion of (not necessary primary) fields and
where X :=
l
T(z) := −2π Tzz
T̄(z̄) := −2π Tz̄ z̄ .
(3.120)
C is a curve encircling all insertions of X anti–clockwise and C ∗ its complex conjugate.
Proof:
The general Ward identity on a Riemann surface is given in proposition 3.19:
I
I
1
1
δε,ε̄ hX i =
dz h−2πjz (z, z̄) X i +
dz̄ h−2πjz̄ (z, z̄) X i .
2πi C
2πi C ∗
(3.121)
For conformal transformations it is
h−2πjz (z, z̄) X i = h−2πTzz ε(z)X i + h−2πTzz̄ ε̄(z̄)X i
(3.122)
h−2πjz̄ (z, z̄) X i = h−2πTz̄z̄ ε̄(z̄)X i + h−2πTzz̄ ε(z)X i .
(3.123)
46
Chapter 3 – Conformal Quantum Field Theory
Because of proposition 3.21, conformal invariance implies
h−2πjz (z, z̄) X i = h−2πTzz ε(z)X i
and
h−2πjz̄ (z, z̄) X i = h−2πTz̄z̄ ε̄(z̄)X i .
(3.124)
Hence,
δε,ε̄ hX i =
1
2πi
I
C
dz h−2πTzz ε(z)X i +
1
2πi
I
C∗
dz̄ h−2πTz̄z̄ ε̄(z̄)X i ,
which is the proposition.
(3.125)
Remark. Note that because of the contour integrals, any terms on the right–hand side holomorphic in z will vanish due to Cauchy’s theorem. Hence, we deduce that the transformation
properties of the correlators are determined by their pole structure solely.
Corollary 3.24: Let hX i be an arbitrary correlator of a product of primary fields. Its invariance
with respect to conformal transformations is equivalent to
X
δε hXi∗ =
(ε(zk ) ∂k + hik ∂ε(zk )) hXi∗ = 0
(3.126)
k
X
δε̄ X̄ ∗ =
ε̄(z̄k ) ∂¯k + h̄ik ∂¯ε̄(z̄k ) X̄ ∗ = 0
(3.127)
k
where
hX i = X̄ ∗ X̄ ∗ .
(3.128)
The star emphasizes the fact that the correlators with stars are not obtained via the theory’s path
integral.
Proof:
Our basic fields are primary fields, for which the first variation with respect to conformal
transformations is by definition 3.4
δε,ε̄ Φ(z, z̄) = ε∂ + ε̄∂¯ + h (∂ε) + h̄ (∂¯ε̄) Φ(z, z̄) .
(3.129)
For a product of fields corollary 3.3 gives
δε,ε̄ {Φi1 (z1 , z̄1 ) . . . Φin (zn , z̄n )} =
n
X
k=1
Φi1 (z1 , z̄1 ) . . . {δε,ε̄ Φik (zk , z̄k )} . . . Φin (zn , z̄n ) .
Hence, the condition of vanishing variation of a correlator is
X
δε,ε̄ hX i =
ε(zk ) ∂k + hik ∂ε(zk ) + ε̄(z̄k ) ∂¯k + h̄ik ∂¯ε̄(z̄k ) hX i = 0 .
(3.130)
(3.131)
k
We have used h∂µ f (x)i = ∂µ hf (x)i, since the path integral is independent of the insertion point x. The
above equation has the form
(3.132)
Dz + D̄z̄ f ({zk }, {z̄k }) = 0 ,
with the linear differential operator Dz acting on the zk only and the operator D̄z̄ acting on the z̄k only.
By comparing (3.132) with (3.131) the operators are determined to be:
X
X
Dz =
[ε(zk ) ∂k + hik ∂ε(zk )]
D̄z̄ =
(3.133)
ε̄(z̄k ) ∂¯k + h̄ik ∂¯ε̄(z̄k ) .
k
k
Making the separation Ansatz
gives
f ({zk }, {z̄k }) = φ({zk }) χ({z̄k })
(3.134)
χ({z̄k }) Dz φ({zk }) + φ({zk }) D̄z̄ χ({z̄k }) = 0 .
(3.135)
Since φ and χ are functions of independent variables, this is equivalent to
Dz φ({zk }) = 0
and
D̄z̄ χ({z̄k }) = 0 .
With φ ≡ hXi∗ , χ ≡ X̄ ∗ and using (3.133), this is the proposition.
(3.136)
3.4 The one–, two–, three– and four–point functions
47
Remark. Note that the above notation h. . .i∗ is temptative. It suggests that the CFT we started
with can be decomposed into two independent quantum field theories. However, we have not
given any reason for this to be true. We will see in the operator formalism that there are strong
indications for this to be true and all known CFTs indeed admit this decomposition. We will
later refer to this as chiral splitting. At this point, we have only shown that the correlators
of primary fields can be obtained from two independent differential equations, which will be of
immediate use in the next section.
3.4 The one–, two–, three– and four–point functions
A direct consequence of the conformal Ward identity is that global conformal invariance – i.e.
SL(2, C)/Z2 invariance – already fixes the one–, two– and three–point functions up to a constant.
Furthermore, the four–point functions are fixed up to a function depending on certain cross–ratios
of the field insertions. We sum up this results in the following theorem, which we will proceed
to prove in the rest of this subsection.
Theorem 3.25: Let {Φr }r=1 be a collection
of quasi–primary
fields of weights (hr , h̄r ). Further,
P
P
let zij := zi − zj , z̄ij := z̄i − z̄j , h := k hik and h̄ := k h̄ik . Then it holds that
hΦi (z, z̄)i = Ci δhi ,0 δh̄i ,0 ,
hΦi1 (z1 , z̄1 )Φi2 (z2 , z̄2 )i =
Ci1 i2 δhi1 hi2 δh̄i
h̄
1 i2
,
(3.137)
(zkl )him −hik −hil (z̄kl )h̄im −h̄ik −h̄il .
(3.138)
(z12 )h (z̄12 )h̄
with some constants Ci and Cij .
hΦi1 (z1 , z̄1 )Φi2 (z2 , z̄2 )Φi3 (z3 , z̄3 )i = Ci1 i2 i3
Y
(m,k,l)
Where the product runs over the three tuples (m, k, l) ∈ {(1, 2, 3), (2, 1, 3), (3, 1, 2)} and Ci1 i2 i3 is
some constant
Finally, we have
Y
hΦi1 (z1 , z̄1 )Φi2 (z2 , z̄2 )Φi3 (z3 , z̄3 )Φi4 (z4 , z̄4 )i = f (x, x̄)
(zkl )h/3−hik −hil (z̄kl )h̄/3−h̄ik −h̄il ,
k<l
(3.139)
with an undetermined function that depends only on the harmonic cross–ratios x :=
12 z̄34
x̄ := z̄z̄13
z̄24 .
z12 z34
z13 z24
and
Proof: We will use the Ward identities for quasi–primary fields. They are a deduced from corollary
3.24 by taking ε(z) ∈ {1, z, z 2}.
X
X ∂k hXi = 0 ,
(3.140)
∂¯k X̄ = 0
k
X
k
X
k
(zk ∂k + hik ) hXi = 0 ,
(zk )2 ∂k + 2 hik zk hXi = 0 ,
k
X
k
X
k
z̄k ∂¯k + h̄ik
X̄ = 0
(z̄k )2 ∂¯k + 2 h̄ik z̄k
X̄ = 0 .
(3.141)
(3.142)
We will refer to them as identities one, two and three respectively. It can be seen at once that the analysis
of the holomorphic part is similar to the analysis of the anti–holomorphic part. Therefore, we will only
carry out the first one.
(a) One point function. We have X = φ(z) with holomorphic conformal dimension h. From the
first identity it follows
∂ hφ(z)i = 0 .
(3.143)
48
Chapter 3 – Conformal Quantum Field Theory
Consequently, hφ(z)i may not depend on z and has to be a constant. This yields together with the second
identity that
h hφ(z)i = 0 ,
(3.144)
and therefore hφ(z)i can only be non–zero if h = 0.
(b) Two point function. Writing hXi ≡ hφi1 (z1 ) φi2 (z2 )i =: W (z1 , z2 ), the first identity reads
(3.145)
(∂1 + ∂2 ) W = 0 .
Changing to variables z± := z1 ± z2 , this equation becomes
(3.146)
∂+ W = 0 .
Consequently, the two point function only depends on the difference of the insertions, i.e. W = W (z1 −z2 ).
We now use this information to simplify the second identity, which becomes
z1 ∂− W − z2 ∂− W + hi1 W + hi2 W = 0 .
(3.147)
This is a separable differential equation for W (z− ) and it follows that
ZW
1
= −(hi1 + hi2 )
dW̃
W̃
Zz−
dz̃−
1
+ const. .
z̃−
(3.148)
Integrating both sides we obtain, with h := hi1 + hi2 and C = const.,
W (z1 − z2 ) = C (z1 − z2 )−h .
(3.149)
Plugging this into the third identity gives
C
(−h) (z2 )2
(hi1 z1 + hi2 z2 )
(−h) (z1 )2
−
+2
(z1 − z2 ) (z1 − z2 )h
(z1 − z2 ) (z1 − z2 )h
(z1 − z2 )h
=0.
(3.150)
∀ z1 , z2 .
(3.151)
After factoring out (z1 − z2 )−(h+1) it follows that
C [hi1 − hi2 ] (z1 )2 + [hi1 − hi2 ] (z2 )2 − [hi1 − hi2 ] z1 z2 = 0
Therefore, either hi1 = hi2 or C = 0 (and hence W = 0). This proves equation (3.137).
(c) Three point function. The system of first order linear partial differential equations (3.140),
(3.141), (3.142) has a unique solution (up to one integration constant). We have three differential equations for a function of three independent variables. It can be verified that 3.138 is a solution of the system
and hence the proposition follows from uniqueness (up to the integration constant).
(d) four point function. In this case, we have three linear partial differential equations for a function
of four variables. This system is solved by (3.139) for any function f (x) of the cross ratios, as can be
checked by direct computation. Hence, every solution of the system (3.140), (3.141), (3.142) takes claimed
form, because of uniqueness of the solution.
3.5 Operator Product Expansion of Free Field Theories
We are now going to introduce the main tool of CFT, namely the Operator Product Expansion
(OPE). Differing from most of the introductory texts on CFT, we are going to do this in the
path integral formalism not in the operator formalism. Although, to remain easily accessible, we
will let us guide by the specific theory of one free boson.
3.5 Operator Product Expansion of Free Field Theories
49
3.5.1 The Free Boson
Let’s consider the QFT of a free, massless scalar field ϕ. We will usually refer to this special
QFT as “the free boson”. Its path integral is defined through the action
Z
p
ig
d2 x |g| (∂µ ϕ)(∂ µ ϕ) ,
(3.152)
iS =
2
with some dimensionless normalization constant g, to be specified later. We now perform a Wick
rotation to Euclidean space (and hence Euclidean metric g) after which the action gets
Z
p
g
S 7−→ −
d2 ξ |g| gµν (∂µ ϕ)(∂ν ϕ) ,
(3.153)
2
As explained before, we will work on the Riemann sphere CP1 , on which the action becomes
Z
Z
g
1
S=
d2 z ∂z̄ ϕ ∂z ϕ .
(3.154)
d2 z eσ(z,z̄) gz z̄ ∂z ϕ ∂z̄ ϕ + gz̄z ∂z ϕ ∂z̄ ϕ = g
2
2
p
Here we have used that |g| = 12 eσ(z,z̄) from (3.5) and that the inverse metric is given by
g−1 = 2 e−σ(z,z̄) (∂z ⊗ ∂z̄ + ∂z̄ ⊗ ∂z ) .
(3.155)
Note that the dependence on the conformal factor σ has dropped. This justifies to set e−σ = 1
in the following.
The action is obviously invariant with respect to translations and rotations. If it should be
conformally invariant we need invariance with respect to dilatations as well. Therefore, it follows
from dimensional analysis that the scaling dimension ∆ = h + h̄ of ϕ would have to be zero.
Since ϕ carries no spin s = h − h̄ either, we would deduce that its conformal weight has to
be (h, h̄) = (0, 0) – assuming it is a primary field. But this forces the theory’s one–, two– and
three–point functions to be constants (which can be seen immediately from the general formulae
in theorem 3.25) and the theory is trivial. Hence, we conclude that ϕ cannot be a primary field.
This fact can also be understood by examining the theory’s two–point function. After partially
integrating the action, the partition function can be written
Z
Z
1
2
d z ϕ(z, z̄) [−g (∂z ∂z̄ + ∂z̄ ∂z )] ϕ(z, z̄) .
(3.156)
Z = Dφ exp −
2
Using theorem B.1 from the appendix, the two–point function fulfils
(∂z ∂z̄ + ∂z̄ ∂z ) G(z, z̄, w, w̄) = −
1 (2)
δ (z − w, z̄ − w̄)
g
(3.157)
in a distributional sense3 .
Using the representation of the delta distribution in complex coordinates from the appendix
(A.21)
1
1
1
1
2
(∂z ∂z̄ + ∂z̄ ∂z ) ln |z − w| = −
+ ∂z̄
∂z
= δ(2) (z − w, z̄ − w̄) , (3.158)
−
4πi
4πi
z̄ − w̄
z−w
we conclude that
G(z, z̄, w, w̄) = hϕ(z, z̄) ϕ(w, w̄)i = −
1
[ln(z − w) + ln(z̄ − w̄)] + const. .
4πg
(3.159)
Because of the logarithms, this expression has no definite transformation behavior with respect
to conformal transformations (take scalings for instance). Therefore, ϕ cannot be a primary field.
Nevertheless, the free boson is indeed a CFT as we will see in a moment from an analysis of
the energy–momentum tensor.
3
That is, the equation is only valid when integrating both sides against suitable test functions. Note further,
that as a consequence of this ∂z̄ ∂z G(z, z̄, w, w̄) 6= ∂z ∂z̄ G(z, z̄, w, w̄) in general.
50
Chapter 3 – Conformal Quantum Field Theory
3.5.2 The Free Boson Energy–Momentum Tensor
At first, take a look on the equations of motion
∂µ
∂L
∂L
∂L
= ∂z
+ ∂z̄
= g (∂z ∂z̄ + ∂z̄ ∂z ) ϕ(z, z̄) = 0 .
∂∂µ ϕ
∂∂z ϕ
∂∂z̄ ϕ
(3.160)
It is equivalent to the system of equations
∂z ∂z̄ ϕ(z, z̄) = 0
∂z̄ ∂z ϕ(z, z̄) = 0 .
(3.161)
Therefore, we see that
∂z ϕ = ∂z ϕ(z) ,
(3.162)
∂z̄ ϕ = ∂z̄ ϕ(z̄) ,
(3.163)
is a holomorphic function of z and
is an anti–holomorphic function of z.
The canonical energy–momentum tensor is
(Tc )µν
∂L
1
σ
=
∂ν φr − L gµν = g ∂µ ϕ ∂ν ϕ − ∂σ ϕ ∂ ϕ gµν .
∂∂ µ φr
2
(3.164)
As can be seen easily, this tensor is symmetric and traceless4 . Using the equation of motion, it
also turns out to be divergence free. Therefore, it has to agree with the true energy–momentum
tensor of our theory since we are working on a flat background. Hence, we drop the index c in
the following.
In complex coordinates the tensor is
∂z ϕ ∂z ϕ
0
Tzz Tz z̄
=g
.
(3.165)
(Tµν ) =
0
∂z̄ ϕ∂z̄ ϕ
Tz̄z Tz̄z̄
Note that the vanishing of the trace now reads
Tµµ = gµν Tµν = gz z̄ Tz z̄ + gz̄z Tz̄z = 4 Tz z̄ = 0 .
(3.166)
We conclude that conformal invariance is equivalent to the vanishing of Tz z̄ in our coordinates.
The simplification of this condition in complex coordinates is is no surprise. We have seen in
section 3.1 that conformal invariance and the complex structure (i.e. the splitting of the tangential
spaces into dz and dz̄) are intimately connected.
Therefore, we are left with two nonvanishing components of the energy–momentum tensor
Tzz = Tzz (z)
Tz̄ z̄ = Tz̄z̄ (z̄) ,
(3.167)
which are holomorphic and anti–holomorphic functions of z, as a consequence of the equations
of motion (3.161). In the following, we shall use T, T̄ defined by
T(z) := −2π Tzz (z)
T̄(z̄) := −2π Tz̄z̄ (z̄) ,
(3.168)
and call them the holomorphic and anti–holomorphic components of the energy–momentum
tensor. The factors of −2π have been included to avoid them in later expressions. Note that this
is consistent with our earlier definition in the conformal Ward identity 3.23.
4
Note that this is only true in two–dimensions, although the action may be dilatation invariant in arbitrary
number of dimensions.
3.5 Operator Product Expansion of Free Field Theories
51
3.5.3 The Free Boson Quantum Energy–Momentum Tensor
Like with all composite fields in a QFT, we will encounter difficulties with the energy–momentum
tensor upon quantization. Their origin is the occurrence of fields at coincident space–time points.
As a solution we will use the method of normal ordering. In case of the energy–momentum tensor,
this means
D
E
hTzz . . .i = hg : ∂z ϕ ∂z ϕ : . . .i = g lim [(∂z ϕ ∂w ϕ) − h∂z ϕ ∂w ϕi] . . . .
(3.169)
w→z
With . . . we indicated any possible product of fields with insertions not at z. At this point,
the above definition of normal ordering will suit our needs, although it only works in free field
theories. Therefore, we will generalize it when dealing with the operator formalism of CFT in
the following chapter.
Note that we do not have to care for time–ordering issues, since we are working in the path–
integral formalism.
We will now introduce one of the main concepts of CFT, the operator product expansion (OPE).
In general, the expansion of a product of two fields A, B inside a correlator is given by [Wil69]
+
* N
X [AB]n (y)
... ,
(3.170)
hA(x) B(y) . . .i =
|x − y|n
n=−∞
where [AB]n (w) are fields non–singular at w = z.
As we will see later, only the singular terms in this expansion will be of physical importance.
Hence, we define the OPE of two fields to be
Definition 3.26: Let A, B be two fields. We call
A(x) B(y) ∼
N
X
[AB]n (y)
n=1
|x − y|n
,
(3.171)
their operator product expansion (OPE). The (non–singular) fields [AB]n are called OPE coefficients.
Remarks.
• The OPE makes only sense inside a correlator, what we shall always keep in mind.
• In a general QFT, the OPE of two arbitrary fields does not have to be of any use (since
we omitted all regular terms), nor to be well–defined (i.e. convergent) either. Usually, the
OPE is assumed to be an asymptotic expansion which can be used as a calculational tool.
But—as we shall see in the course of this work—it is in fact the most powerful structure
of a CFT.
• The OPE of the primary fields with themselves and with the energy–momentum tensor (i.e.
its (anti)holomorphic components) plays an exceptional role in CFT. In mathematics this
OPE appears in connection to vertex operator algebras [Bor86] [Bor92] [FHL93] [Zhu96]
[FLM88] [Kac97]. For a large class of CFTs this OPE is everything necessary to solve the
corresponding QFT completely—i.e. to calculate every possible correlator unperturbatively.
We shall come back to this point in more detail later.
The simplest example is the OPE of two fundamental fields (i.e. path integration variables) or
their derivatives. In the case of the free boson, differentiating (3.159) yields
h∂z ϕ(z) ∂w̄ ϕ(w̄)i = 0 .
(3.172)
52
Chapter 3 – Conformal Quantum Field Theory
Hence, we conclude the OPE
∂z ϕ(z) ∂w̄ ϕ(w̄) ∼ 0 ,
(3.173)
since the contraction (i.e. two–point function) of ∂z ϕ and ∂w̄ ϕ never introduces singularities in
the limit w → z. 5
Let us now try to identify the primary fields of the free boson theory. Simplifying our notation
by writing
¯
∂ϕ(z) := ∂z ϕ(z) and ∂ϕ(z̄)
:= ∂z̄ ϕ(z̄) ,
(3.174)
we obtain from (3.159)
h∂ϕ(z)∂ϕ(w)i = −
and
1
1
4πg (z − w)2
1
¯
¯ w̄) = − 1
∂ϕ(z̄)
∂ϕ(
.
4πg (z̄ − w̄)2
(3.175)
(3.176)
From these, we deduce the following OPEs
∂ϕ(z)∂ϕ(w) ∼ −
and
1
1
4πg (z − w)2
1
¯
¯ w̄) ∼ − 1
∂ϕ(z̄)
∂ϕ(
.
4πg (z̄ − w̄)2
(3.177)
(3.178)
We recognize the two–point functions of a primary field of weight (1, 0) and (0, 1) (see 3.137).
¯ are indeed primary fields, we need to investigate their transforHowever, to check if ∂ϕ and ∂ϕ
mation properties. They can be calculated through the conformal Ward–Identity
I
I
1
1
dz ε(z) hT(z) ∂ϕ(w)i +
dz̄ ε̄(z̄) T̄(z̄) ∂ϕ(w)
(3.179)
δε,ε̄ h∂ϕ(w)i =
2πi w
2πi w̄
and
¯ w̄) = 1
δε,ε̄ ∂ϕ(
2πi
I
w
¯ w̄) + 1
dz ε(z) T(z) ∂ϕ(
2πi
I
w̄
¯ w̄) .
dz̄ ε̄(z̄) T̄(z̄) ∂ϕ(
(3.180)
¯ with T and T̄. They can
To calculate the integrals above, we will need the OPEs of ∂ϕ and ∂ϕ
6
be obtained from Wick’s theorem . We have (two fields connected by a line are contracted, i.e.
replaced by their two–point function)
T(z) ∂ϕ(w) ≡ −2πg : ∂ϕ(z)∂ϕ(z) : ∂ϕ(w)
∼ −2πg : ∂ϕ(z)∂ϕ(z) : ∂ϕ(w) − 2πg : ∂ϕ(z)∂ϕ(z) : ∂ϕ(w)
1
−1
∼ −4πg ∂ϕ(z)
4πg (z − w)2
∂ϕ(w) + ∂∂ϕ(w) (z − w) + O (z − w)2
∂ϕ(z)
∼
∼
(z − w)2
(z − w)2
∂ϕ(w)
∂∂ϕ(w)
∼
+
.
2
(z − w)
z−w
5
6
(3.181)
Strictly spoken, we ought to take the limit (w, w̄) → (z, z̄).
Wick’s theorem states that the normal ordered product of normal ordered operators is obtained from summing
over all possible contractions between the operators. Take for instance four bosonic fields:
X
X
hAi Bj i hAk Bl i .
hAi Bj i : Ak Bl : +
: A1 A2 : : B1 B2 :=
i,j6=k,l
i,j6=k,l
In the case of fermionic fields, one has to be careful with minus signs introduced when contracting fields.
3.5 Operator Product Expansion of Free Field Theories
53
In the fourth line we have Taylor expanded ∂φ around z = w and omitted non–singular terms,
which will drop out of the integral in the conformal Ward–Identity. Analogously, we obtain
¯ w̄) ∼
T̄(z̄) ∂ϕ(
¯ w̄)
¯ w̄)
∂¯∂ϕ(
∂ϕ(
.
+
(z̄ − w̄)2
z̄ − w̄
(3.182)
Because of (3.173) it is
T̄(z̄) ∂ϕ(w) ∼ 0
and
¯ w̄) ∼ 0 .
T(z) ∂ϕ(
(3.183)
Now the conformal Ward–identity
I
1
∂ϕ(w)
∂∂ϕ(w)
δε,ε̄ h∂ϕ(w)i =
dz ε(z)
+
2πi w
(z − w)2
z−w
I
∂ϕ(w)
1
∂∂ϕ(w)
2
=
dz ε(w) + ∂ε(w) (z − w̄) + O((z − w̄) )
+
2πi w
(z − w)2
z−w
= [∂ε(w) + ε(w) ∂ ] h∂ϕ(w)i
(3.184)
where we Laurent expanded ε(z) at z = w and used Cauchy’s theorem. In complete analogy
¯ w̄) = ∂¯ε̄(w̄) + ε̄(w̄) ∂¯ ∂ϕ(
¯ w̄) .
(3.185)
δε,ε̄ ∂ϕ(
The above equations will also hold for correlators involving additionally an arbitrary number
of fields as long as their insertions are all different from w (w̄), since they play no role in the
complex integrations. Hence, we may write
δε,ε̄ ∂ϕ(w) = [∂ε(w) + ε(w) ∂] ∂ϕ(w) ,
and
¯ w̄) = ∂¯ε̄(w̄) + ε̄(w̄) ∂¯ ∂ϕ(
¯ w̄) ,
δε,ε̄ ∂ϕ(
(3.186)
(3.187)
being true inside any correlator without w (w̄) insertions. Comparing this with our definition
¯ a
(3.26) of primary fields, we see that ∂ϕ is a primary field of weight (h, h̄) = (1, 0) and ∂ϕ
primary field of weight (h, h̄) = (0, 1).
Next, we want to investigate if the energy–momentum tensor itself is a primary field. As Tzz
¯ we expect them to be primary fields of
and Tz̄ z̄ are products of the primary fields ∂ϕ and ∂ϕ,
higher weight classically. In particular—since the fields appear quadratic—we expect them to
be of conformal weight (2, 0) and (0, 2) respectively. However, as is known from QFT in general,
the procedure of quantization may cause the breakdown of classical symmetries.
To obtain the energy–momentum tensor’s transformation properties, we will again make use
of the conformal Ward–identity. Therefore, we need the OPEs of T and T̄ with themselves. The
OPE
T(z) T̄(w̄) ∼ 0
(3.188)
¯ w̄) ∼ 0.
is rather obvious, since ∂φ(z) ∂φ(
For T with itself we have
T(z) T(w) ≡ 4π 2 g2 : ∂ϕ(z) ∂ϕ(z) : : ∂ϕ(w) ∂ϕ(w) :
"
#
∼ 4π 2 g2 4 : ∂ϕ(z) ∂ϕ(z) : : ∂ϕ(w) ∂ϕ(w) : + 2 : ∂ϕ(z) ∂ϕ(z) : : ∂ϕ(w) ∂ϕ(w) :
,
(3.189)
54
Chapter 3 – Conformal Quantum Field Theory
where we have not written out all the equivalent contractions explicitly. Evaluating the contractions using (3.175) we get
1
1
−1 : ∂ϕ(z) ∂ϕ(w) :
+2
T(z) T(w) ∼ 4π g 4
4πg
(z − w)2
(4πg)2 (z − w)4
: ∂∂ϕ(w) ∂ϕ(w) : (z − w)
1/2
: ∂ϕ(w) ∂ϕ(w) :
− 4πg
+
∼ −4πg
(z − w)2
(z − w)2
(z − w)4
2 T(w)
∂T(w)
1/2
+
+
.
∼
4
2
(z − w)
(z − w)
z−w
2 2
(3.190)
Again, the according OPE for T̄ follows in complete analogy
T̄(z̄) T̄(w̄) ∼
1/2
2 T̄(w̄)
∂¯T̄(w̄)
.
+
+
(z̄ − w̄)4 (z̄ − w̄)2
z̄ − w̄
(3.191)
Making use of the conformal Ward–identity now yields the transformation law
I
1/2
1
2 T(w)
∂T(w)
dz ε(z) hT(z) T(w)i =
dz ε(z)
+
+
2πi C
(z − w)4 (z − w)2
z−w
C
1 3
∂ ε(w)
= [∂ε(w) + ε(w) ∂ ] hT(w)i +
12
(3.192)
1
δε,ε̄ hT(w)i =
2πi
I
and
δε,ε̄
1
T̄(w̄) =
2πi
I
C∗
1 ¯3
dz̄ ε̄(z̄) T̄(z̄) T̄(w̄) = ∂¯ε̄(w̄) + ε̄(w̄) ∂¯ T̄(w̄) +
∂ ε̄(w̄) . (3.193)
12
Again, these statements also hold in presence of an arbitrary number of other fields inside the
correlator, as long as their insertions are different from w (w̄).
Comparing (3.192) and (3.193) with the definition for primary fields we see that the energy–
momentum tensor splits into one (h, h̄) = (2, 0) primary–field T and one (h, h̄) = (0, 2) primary–
field T̄, except for the anomalous term
1 3
∂ ε(w) ,
12
(3.194)
which is being caused by the anomalous OPE term
1/2
.
(z − w)4
(3.195)
Note that for transformations in the subgroup ε ∈ {1, z, z 2 } (i.e. in sl(2, C)) the anomalous
term vanishes and T, T̄ are still quasi–primary fields. Hence, the energy–momentum tensor
still transforms tensorially with respect to translations, rotations and dilatations. Therefore, it
remains to be a sensible quantum field of definite spin and mass dimension.
In conclusion, we have seen that the free boson in two–dimensions is an example of a CFT.
¯ as primary fields of weight (1, 0) and (0, 1). Nevertheless, we observed
We identified ∂ϕ and ∂ϕ
the breakdown of conformal symmetry after quantization in the OPE of the energy–momentum
tensor with itself. As we will see in the following section, this is not a special property of the
free boson, but rather characteristic for CFTs in general.
3.5 Operator Product Expansion of Free Field Theories
55
3.5.4 The General OPE of a CFT
The conformal Ward–identity 3.23 relates the transformation properties of fields to their OPE
(the singular part of the general operator product expansion) with the energy–momentum tensor.
We have seen this in the last section for the free boson. Let us now determine the most general
OPE of a primary field Φ of weight (h, h̄) with the energy–momentum tensor.
The conformal Ward–identity reads
I
I
1
1
dz ε(z) hT(z) Φ(w, w̄)i +
dz̄ ε̄(z̄) T̄(z̄) Φ(w, w̄) .
(3.196)
δε,ε̄ hΦ(w, w̄)i =
2πi w
2πi w̄
From the definition of a primary field of weight (h, h̄) in 3.4 the left–hand side is
δε,ε̄ hΦ(w, w̄)i = ε∂ + ε̄∂¯ + h (∂ε) + h̄ (∂¯ε̄) hΦ(w, w̄)i = .
(3.197)
Recall that for a function f (z) holomorphic in z
1
2πi
I
f (z)
1
dz
=
2
(z − w)
2πi
w
I
dz
w
f (w) + ∂f (w) (z − w) + O((z − w)2 )
= ∂f (w) .
(z − w)2
(3.198)
Hence,
1
2πi
I
h Φ(w, w̄) ∂Φ(w, w̄)
dz ε(z)
+
= [h (∂ε)Φ(w, w̄) + ε ∂Φ(w, w̄)]
(z − w)2
z−w
w
(3.199)
and analogously
1
2πi
I
¯
h̄ Φ(w, w̄) ∂Φ(w,
w̄)
dz̄ ε̄(z̄)
+
= h̄ (∂¯ε̄)Φ(w, w̄) + ε̄ ∂¯ Φ(w, w̄) .
2
(z̄ − w̄)
z̄ − w̄
w̄
(3.200)
We conclude
Proposition 3.27: The OPE of a primary field Φ(w, w̄) of weight (h, h̄) with the energy–momentum
tensor is given by
h Φ(w, w̄) ∂Φ(w, w̄)
T(z) Φ(w, w̄) ∼
(3.201)
+
(z − w)2
z−w
T̄(z̄) Φ(w, w̄) ∼
¯
h̄ Φ(w, w̄) ∂Φ(w,
w̄)
+
.
2
(z̄ − w̄)
z̄ − w̄
(3.202)
In the following, we will motivate the most general OPE of the energy–momentum tensor with
itself and investigate the meaning of anomalous terms occurring therein.
Let us begin with a naive approach, taking the classical point of view as a starting point. Let
a conformally invariant field theory on a two–dimensional manifold M be given. Its (Euclidean)
energy–momentum tensor on T+∗ M ⊕ T−∗ M is
T = Tzz dz ⊗ dz + Tz̄z̄ dz̄ ⊗ dz̄ .
(3.203)
We expect it to transform as a primary field of weight (2, 0) ⊕ (0, 2) and therefore,
δε,ε̄ Tzz = [2 (∂ε) + ε∂] Tzz
δε,ε̄ T̄z̄z̄ = 2 (∂¯ε̄) + ε̄∂¯ T̄z̄ z̄ .
(3.204)
(3.205)
56
Chapter 3 – Conformal Quantum Field Theory
After quantization, we obtain these transformation properties for correlators involving Tzz and
Tz̄z̄ from the conformal Ward–Identity 3.23:
I
I
1
1
dz̄ ε̄(z̄) T̄(z̄) T(w)
(3.206)
dz ε(z) hT(z) T(w)i +
δε,ε̄ hT(w)i =
2πi C
2πi C ∗
and
1
δε,ε̄ T̄(w̄) =
2πi
I
C
1
dz ε(z) T(z) T̄(w̄) +
2πi
I
C∗
dz̄ ε̄(z̄) T̄(z̄) T̄(w̄) .
(3.207)
Accordingly, the OPEs of the energy–momentum tensor should be of the form
T(z) T(w) ∼
2 T(w)
∂T(w)
+
,
2
(z − w)
z−w
(3.208)
T̄(z̄) T̄(w̄) ∼
2 T̄(w̄)
∂ T̄(w̄)
+
(z̄ − w̄)2
z̄ − w̄
(3.209)
and
T(z) T̄(w̄) ∼ 0 .
(3.210)
But as we have seen in case of the rather simple example of one free boson, quantization may
cause the transformation of the energy–momentum tensor to be anomalous. Let us therefore
examine what additional terms are possible to occur in (3.208) and (3.209).
Motivated by the example of the free boson, we require the energy–momentum tensor components to remain sensible quantum fields even after the breakdown of full conformal symmetry.
Hence, the additional terms have to respect spin and mass dimension of the energy–momentum
tensor. We take spin into account by requiring the terms to obey Bose–Einstein statistics, since
the quantum energy–momentum tensor is bosonic from its definition in equation (3.73). Alltogether, any additional term in (3.208) and (3.209) has to scale with mass dimension 2 + 2 = 4
and be symmetric with respect to the exchange z ↔ w (or z̄ ↔ w̄ respectively). This leaves two
one parameter families of sensible terms, i.e.
and
c/2
(z − w)4
∀c∈C,
(3.211)
c̄/2
(z̄ − w̄)4
∀ c̄ ∈ C .
(3.212)
We included a conventional factor of 21 . Both complex numbers c and c̄ are called central charge.
Their values are not fixed through symmetry considerations, they are characteristic parameters
of the theory. In case of the free boson it is c = c̄ = 1, as can be read off from (3.192) and
(3.193).
We conclude this chapter with a summarizing definition
Definition 3.28: Let T(z), T̄(z̄) be the holomorphic and anti–holomorphic components of the
quantum energy–momentum tensor of a CFT. If their OPEs fulfil
T(z) T(w) ∼
2 T(w)
∂T(w)
c/2
+
+
4
2
(z − w)
(z − w)
z−w
(3.213)
T̄(z̄) T̄(w̄) ∼
2 T̄(w̄)
∂ T̄(w̄)
c̄/2
+
+
,
4
2
(z̄ − w̄)
(z̄ − w̄)
z̄ − w̄
(3.214)
then (c, c̄) ∈ C2 is called the central charge of the CFT.
3.5 Operator Product Expansion of Free Field Theories
57
Remarks.
• c and c̄ are in principle unrelated. However, it has been shown in [AGW83] that the
coupling of any CFT with two dimensional gravity (or equivalently: the fully quantized
formulation of CFT on curved backgrounds) requires
c = c̄
for consistency.
• Although the considerations given in this section make the above form of the OPEs plausible, they cannot be considered a proof. However, from a physicists point of view, it is
hard to imagine any other possible singular terms in the OPE as explained above. Another point is that any other term—even if it is consistent with Bose symmetry and mass
dimension—will spoil an understanding of the anomalous OPE from a geometrical point
of view, where quantum anomalies can be understood as the non–trivial transformation
behavior of the path integral measure (an approach known as Fujikawa’s method) [Gaw96].
• Fortunately, the above OPEs are true for all known CFTs. In axiomatic approaches, the
above equations (or their equivalent formulation in terms of the transformation properties
of the path integral measure), are usually assumed to hold as one of the axioms. See
[Gaw96] for example.
3.5.5 The Physical Meaning of c
What is the consequence of the anomalous term in the OPEs (3.213), (3.214) for physical applications? We have seen that the OPE determines an operator’s transformation behavior with
respect to conformal transformations. For the energy–momentum tensor this means (we concentrate on T(z) since the discussion of T̄(z̄) is analogous)
I
I
c/2
1
2 T(z)
∂T(z)
1
dw ε(w) T(w)T(z) =
dw ε(w)
+
+
δε,ε̄ T(z) =
2πi z
2πi z
(w − z)4 (w − z)2
w−z
c 3
= ∂ ε(z) + 2T(z) ∂ε(z) + ε(z) ∂T(z) .
12
(3.215)
It can be shown that this can be integrated to give the global transformation behavior [FMS97]
z7→w
T(z) 7−→
∂w(z)
∂z
−2
T(z) +
c
{z, w} ,
12
(3.216)
where the last term is
{f, w} :=
f ′′′ (w) 3
−
f ′ (w)
2
f ′′ (w)
f ′ (w)
2
, with f ′ (w) :=
∂w
∂z
.
(3.217)
It is known as the Schwarzian derivative [BE53]. Analogously, we obtain for T̄
z̄7→w̄
T̄(z̄) 7−→
∂ w̄(z̄)
∂ z̄
−2
T̄(z̄) +
c̄
{z̄, w̄} .
12
(3.218)
Comparing equations (3.216), (3.218) with (3.15), we see that the anomalous term in the OPE
leads to a non–tensorial behavior of the quantum energy–momentum tensor. This anomaly in
the transformation behavior is proportional to (c, c̄) and vanishes for transformations in SL(2, C).
This can be seen from the fact that ε(z) ∈ {1, z, z 2 } (respectively ε̄(z̄) ∈ {1, z̄, z̄ 2 }) for SL(2, C)
58
Chapter 3 – Conformal Quantum Field Theory
diffeomorphisms generated by ε(z) (ε̄(z̄)). Then the anomalous term in (3.215) (and its anti–
holomorphic counterpart) vanishes since it is proportional to the third derivative of ε(z) (ε̄(z)).
It was shown in proposition 3.11 that any theory with a traceless energy–momentum tensor is
conformally invariant. Hence, we expect this property of the energy–momentum tensor to break
down upon quantization if c, c̄ 6= 0. More precisely, the Ward identity for the vanishing of the
energy–momentum tensor trace has to be violated. Indeed it can be shown that (c = c̄) [AGW83]
c
R(x) ,
Tµµ (x) =
24π
(3.219)
where R denotes the Ricci scalar of our theory’s manifold. In higher dimensions (3.219) is usually
known as the trace anomaly.
Let us have a closer look on the conformal anomaly in complex coordinates. By definition we
choose a conformally flat metric given by equation (3.5). Then, the Ricci scalar is given by
¯ ,
R = −4 e−σ(z,z̄) ∂ ∂σ
(3.220)
and the equation for the conformal anomaly reduces to
hTz z̄ (z, z̄)i =
c
¯
∂ ∂σ(z,
z̄) .
24π
(3.221)
The correlator on the left is calculated by using its definition 3.73
2 δ
ln Z .
hTz z̄ (z, z̄)i ≡ − √
g δgz z̄
Using the form of the metric to simplify the functional derivative7 , we obtain
Z
δ
δ
−S[{Φr }]
hTz z̄ (z, z̄)i ≡ −2i
DΦ e
ln Z = −2i
ln
.
δσ(z, z̄)
δσ(z, z̄)
(3.222)
(3.223)
The action S has to be independent of the conformal factor σ since we require the theory to
be conformally invariant classically. Hence, we see that the breakdown of conformal invariance
originates from the fact that the path integral measure depends on the conformal factor.
In conclusion, we have seen that a non–zero central charge leads to a non–tensorial transformation behavior of T and T̄ with respect to a conformal transformation. However, both fields
remain sensible quantum fields since their translation, rotation and scaling behavior is unaffected
by the anomaly. We allow for such anomalies, since the central charge c will turn out to be one
of the parameters specifying a CFT and forcing c = 0 from the start would eliminate many
interesting CFTs. However, in physical applications we will add up CFTs so that their central
charges sum to zero and the resulting theory is anomaly free.
7
One has to be a bit careful when performing this step. At first glance it looks like one could show by a similar
calculation that every correlator including Tzz and Tz̄ z̄ vanishes because gzz = gz̄ z̄ = 0. The paradox is
solved by observing that the crucial point which makes our argument work is not solely that gz z̄ = 2 e−σ but
furthermore that every variation δgz z̄ has a one–to–one correspondence to a variation δσ(z, z̄). This is not
true for variations δgzz and δgz̄ z̄ , which need the introduction of the Beltrami differential [FS87].
4 Operator formalism
In this chapter we will employ the operator formulation of conformal quantum field theory.
Although the path integral formalism allowed us to gain deeper insights into the geometrical
structure of CFT, the operator formalism is more convenient for calculational purposes.
For a working operator formalism it is crucial to distinguish between time and space directions.
The reason for this is that only vacuum expectations of time–ordered operators are meaningful.
However, this distinction was made arbitrary due to the Wick rotation we employed. But recalling
the original physical meaning of our coordinates, we are going to implement a scheme allowing
to set up a sensible operator formalism—the radial quantization.
4.1 Chiral Splitting
Before we proceed with the implementation of the operator formalism, we would like to stress
an important aspect of CFT which will be of great importance. In the theory of the free boson,
we encountered a splitting into degrees of freedom depending solely on z (∂ϕ) and degrees of
¯
freedom depending solely on z̄ (∂ϕ).
Recalling our discussion in section 3.1 we might get the
impression that there is a deeper reason for this, connected with the complex structure of our
base manifold. However, we do not want to go into details here.
For now, it will be sufficient to employ the following assumption:
The degrees of freedom of a CFT split into chiral parts, each belonging to one part
of the symmetry algebra
coΣg × coΣg .
4.2 Radial Quantization
Recalling the discussion in 2.2.3, we originally started on flat Minkowski space–time M2 . For a
better imagination, assume the space direction to be compactified on a circle of radius L. Then
the space–time is an infinite cylinder with time t reaching from −∞ to +∞ along the cylinder and
space direction x around the cylinder. We now map this Minkowski cylinder onto the complex
plane (more precisely on the Riemann sphere) by using the conformal transformation
z := et e−ix .
(4.1)
Now the space direction points along circles with center z = 0. The time direction is pointed
radially, with t = −∞ at z = 0 and t = +∞ at z = ∞. Hence, we define time–ordering on the
Riemann sphere as radial ordering.
4.2.1 Radial Ordering
But before doing so, let us first (loosely) define what is meant in our sense by an operator
formalism.
60
Chapter 4 – Operator formalism
Definition 4.1: Let C := {hΦi1 (z1 , z̄1 ) . . . ΦiL (zL , z̄L )i}L be a CFT in the sense of definition 3.9.
Let HC be a Hilbert space with inner product h·|·iHC . We call the collection of linear operators on
HC
b = {Φ
bi | Φ
b i : HC → HC}
C
(4.2)
an operator formalism of C, if there exists a time–ordering prescription T such that
hΦi1 (z1 , z̄1 ) . . . ΦiL (zL , z̄L )i = h0|T
for some |0i ∈ HC, which we call vacuum of C.
n
o
b i (z1 , z̄1 ) . . . Φ
b i (zL , z̄L ) |0iH
Φ
1
L
C
(4.3)
Remark. We really want to stress that this definition is not to be taken too seriously. We always
assume that such an operator formalism exists and is well–defined. This will lead to no problems
in the applications discussed here.
In free–field theories—which will be the main application in this work—the operator formalism
of a CFT can always be obtained through canonical quantization. This means expansion into
positive/negative frequency modes and interpretation of the coefficients as creation/annihilation
operators. Since time is pointed in radial direction, the necessary time–ordering prescription is
given by the radial–ordering:
b be the operator formalism of a free–field CFT. Then the time–ordering T
Proposition 4.2: Let C
is given by the radial–ordering on the complex plane:
b i (z, z̄) Φ
b j (w, w̄)} =
R{Φ
(
b i (z, z̄) Φ
b j (w, w̄)
Φ
b j (w, w̄) Φ
b i (z, z̄)
ηΦ
|z| > |w|
|w| > |z|
,
(4.4)
with η = +1 for commuting fields (bosons) and η = −1 for anti–commuting fields (fermions).
b is defined. Since we already noticed that
Remark. We didn’t specify the manifold on which C
the crucial property of a two–dimensional manifold is solely its complex structure, we allow for
a general Riemann surface of genus g again.
The OPE of fields translates directly into the operator formalism
Proposition 4.3: Let Φ, Ψ be two fields of a CFT with OPE
∞
X
[ΦΨ]n (w, w̄)
Φ(z, z̄) Ψ(w, w̄) ∼
|z − w|n
n=1
b Ψ
b
then it holds for their corresponding operators Φ,
∞ [
n
o X
[ΦΨ]n (w, w̄)
b z̄) Ψ(w,
b
R Φ(z,
w̄) ∼
,
n
|z
−
w|
n=1
(4.5)
(4.6)
[ (w, w̄) are the operators corresponding to the fields [ΦΨ]n (w, w̄).
where [ΦΨ]
n
Proof:
Let
Φ(z, z̄) Ψ(w, w̄) ∼
∞
X
[ΦΨ]n (w, w̄)
|z − w|n
n=1
(4.7)
be given. Then by definition of the operator formalism it is
∞
n
o
X
h[ΦΨ]n (w, w̄) . . .i
b z̄)Ψ(w,
b
h0|R Φ(z,
w̄) . . . |0iHC = hΦ(z, z̄) Ψ(w, w̄) . . .i ∼
|z − w|n
n=1
(4.8)
4.2 Radial Quantization
61
where . . . denotes an arbitrary product of fields with insertions away from (z, z̄), (w, w̄). Using the
definition of the operator formalism again the right–hand side is
∞
∞
[ (w, w̄) . . . |0i
X
h[ΦΨ]n (w, w̄) . . .i X h0|[ΦΨ]
n
=
n
|z
−
w|
|z
− w|n
n=1
n=1
(4.9)
Hence,
∞ [
n
o X
[ΦΨ]n (w, w̄)
b z̄) Ψ(w,
b
R Φ(z,
w̄) ∼
,
|z − w|n
n=1
(4.10)
is true inside all correlators and the proposition is proven.
Remark. Note that the crucial difference to the path integral formalism is the radial ordering
on the left–hand side of the OPE.
From now on, we will be working in the operator picture of CFT if not stated otherwise.
To avoid clumsy notation we are going to use the same symbols as before, which now denote
operators in HC.
4.2.2 Normal Ordering
To give meaning to composite operators, we need a prescription of normal ordering. Recall
we defined the normal ordering of the energy–momentum tensor (focusing on the holomorphic
component) for the free boson to be
: T(z) : = lim R {∂ϕ(z)∂ϕ(w)} − h∂ϕ(z)∂ϕ(w)i .
(4.11)
w→z
Looking at the OPE of R {∂ϕ(z)∂ϕ(w)}
R {∂ϕ(z)∂ϕ(w)} ∼ −
1
1
,
4πg (z − w)2
(4.12)
we see that the definition of normal ordering just subtracts the (only) singular term and renders
the product (∂ϕ∂ϕ)(z) well–defined. Lets see what happens if we try to normal order the product
TT in the same way:
?
: T(z)T(z) : = lim R {T(z)T(w)} − hT(z)T(w)i .
(4.13)
w→z
It is crucial not to interchange the limit with the correlator. By making use of the OPE
R {T(z)T(w)} ∼
2 T(w)
∂T(w)
1/2
+
+
(z − w)4 (z − w)2
z−w
we calculate the naive normal ordered product to be
lim
w→z
1/2 − 1/2 2 T(w) − 2 hT (w)i ∂T(w) − h∂T(w)i
+
+
+ regular
(z − w)4
(z − w)2
z−w
.
(4.14)
Hence, the naive normal ordering (4.13) removed only the leading of the three divergent terms
and failed to make the operator well–defined. Therefore, we will modify our definition of normal
ordering in such a way as to remove these singular terms as well. To do this, we define the
normal ordered product of two general operators to be the constant part of the full OPE:
62
Chapter 4 – Operator formalism
Definition 4.4: Let A, B be two (local) fields in a QFT with general operator product expansion
(including all terms)
N
X
[AB]n (y)
.
(4.15)
A(x) B(y) =
|x − y|n
n=−∞
Their normal ordered product (at the point x) : A(x) B(x) : is defined as
: A(x) B(x) : ≡ [AB]0 (x) .
(4.16)
Since this definition is not so practicable for direct calculations, we shall prove the following
Proposition 4.5: Let A(z, z̄), B(z, z̄) be two (local) fields in a CFT (on a genus g Riemann
surface). Then their normal ordered product is given by
I
I
1
1
1
dw
A(z, z̄) B(w, w̄) + dw̄
A(z, z̄) B(w, w̄) . (4.17)
: A(z, z̄) B(z, z̄) : =
4πi
w−z
w̄ − z̄
z̄
z
where the index z denotes a curve encircling the point z anti–clockwise and the index z̄ denotes a
curve encircling the point z̄ anti–clockwise.
Proof: The proof is straightforward: Substitute (4.15) into (4.17) and show that the definition (4.16)
is fulfilled. We have in a CFT (omitting the radial ordering symbol for convenience)
A(z, z̄) B(w, w̄) =
N
X
[AB]n (w, w̄)
.
(z
−
w)n (z̄ − w̄)n
n=−∞
(4.18)
Therefore, it is
1
4πi
I
dw
z
1
1
A(z, z̄) B(w, w̄) =
w−z
4πi
I
dw
z
N
X
1
1
[AB]n (w, w̄)
= [AB]0 (z, z̄) ,
w − z n=−∞ (z − w)n (z̄ − w̄)n
2
where we used Cauchy’s theorem in the last step. Further, the integral over z̄ is
1
4πi
I
z̄
dw̄
1
1
A(z, z̄) B(w, w̄) =
w̄ − z̄
4πi
I
dw̄
z̄
N
X
1
1
[AB]n (w, w̄)
= [AB]0 (z, z̄) .
w̄ − z̄ n=−∞ (z − w)n (z̄ − w̄)n
2
Putting together the expressions yields
1
1
[AB]0 (z, z̄) + [AB]0 (z, z̄) = [AB]0 (z, z̄)
2
2
which proves the proposition.
With the hypothesis of chiral splitting from section 4.1 we can simplify the notion of normal
ordering.
Lemma 4.6: Let A(z, z̄), B(z, z̄) be two (local) fields in a CFT (on a genus g Riemann surface)
obeying the chiral splitting assumption, i.e. they can be written
A(z, z̄) = A(z) ⊗ Ā(z̄)
and
B(z, z̄) = B(z) ⊗ B̄(z̄) .
(4.19)
Then their normal ordered product is given by
: A(z, z̄) B(z, z̄) : = : A(z) B(z) : ⊗ : Ā(z̄) B̄(z̄) : ,
with
1
: A(z) B(z) : ≡
2πi
I
z
dw
1
A(z) B(w)
w−z
(4.20)
(4.21)
4.2 Radial Quantization
63
1
: Ā(z̄) B̄(z̄) : ≡
2πi
I
dw̄
z̄
1
Ā(z̄) B̄(w̄) .
w̄ − z̄
(4.22)
Again, the index z denotes a curve encircling the point z anti–clockwise and the index z̄ denotes a
curve encircling the point z̄ anti–clockwise.
Proof:
The splitting of the normal ordering follows from the OPE
A(z, z̄) B(w, w̄) =
N
X
[AB]n (w, w̄)
= A(z) B(z) ⊗ Ā(z̄) B̄(z̄) .
|z − w|n
n=−∞
(4.23)
Since [AB]n (w, w̄) are by definition local fields themselves, they have to split chirally as well:
A(z, z̄) B(w, w̄) =
N
N
X
X
[AB]n (w, w̄)
[AB]n (w) ⊗ [ĀB̄]n (w̄)
=
n
|z − w|
|z − w|n
n=−∞
n=−∞
(4.24)
and therefore
: A(z, z̄) B(z, z̄) : ≡ [AB]0 (z, z̄) = [AB]0 (z) ⊗ [ĀB̄]0 (z̄) .
(4.25)
We have
: A(z) ⊗ B(z) : ≡
I
1
2πi
dw
z
I
1
1
A(z) B(w) =
w−z
2πi
dw
z
N
X
1
[AB]n (w)
= [AB]0 (z) ,
w − z n=−∞ (z − w)n
by use of Cauchy’s theorem. Analogously
1
: Ā(z̄) B̄(z̄) : ≡
2πi
I
1
1
Ā(z̄) B̄(w̄) =
dw̄
w̄
−
z̄
2πi
z̄
I
N
X
1
[ĀB̄]n (w̄)
= [ĀB̄]0 (z̄) .
dw̄
w̄
−
z̄
(z̄ − w̄)n
z̄
n=−∞
Hence, (4.25) gives
: A(z, z̄) B(z, z̄) : = : A(z) ⊗ B(z) : ⊗ : Ā(z̄) B̄(z̄) :
and the lemma is proven.
(4.26)
4.2.3 Scattering States
We have seen that on the complex plane t = −∞ corresponds to the origin (z, z̄) = 0 and t = +∞
corresponds to (z, z̄) → ∞. Therefore, we identify the incoming state created by the (primary)
field operator Φ(z, z̄) to be
|Φin
r i := lim Φr (z, z̄) |0i ≡ Φr (0, 0) |0i .
z,z̄→0
(4.27)
?
in
What about the outgoing states? The naive hΦout
s |Φr i = h0|Φs (∞, ∞) Φr (0, 0)|0i is not satisfying. According to the general form of a two–point function of primary fields (3.137) we have
δrs
2
h
z,z̄→∞ z r z̄ 2h̄r
h0|Φs (∞, ∞) Φr (0, 0)|0i = lim
(4.28)
and the product of states would not define a hermitian scalar product on HC. So in order to
have
in
δrs = hΦout
lim h0|Φr (z, z̄)† Φs (w, w̄)|0i
(4.29)
r |Φs i ≡
z,z̄,w,w̄→0
we define the out–states to be the hermitian conjugate of the in–states
1 1
†
†
−2hr −2h̄r
¯ .
h0|Φr ( , ) = lim ξ 2hr ξ¯2h̄r h0|Φr (ξ, ξ)
z̄
|Φin
r i = lim h0|[Φr (z, z̄)] ≡ lim z
z,z̄→0
z,z̄→0
z z̄
ξ,ξ̄→∞
(4.30)
64
Chapter 4 – Operator formalism
4.2.4 Mode Expansions
We now employ the core of canonical quantization, the mode expansion. In quantum field theory,
the fields are expanded in negative and positive frequency parts according to
X
(4.31)
cn en i (x+t) + c̄n e−n i (x−t)
Φ=
Z
n∈
which reads after Wick–rotating t 7→ t′ = it
X
X
′
′
cn en (t +ix) + c̄n en (t −ix) =
(cn z n + c̄n z̄ n ) .
7−→
Z
Z
(4.32)
n∈
n∈
This motivates the following proposition:
Corollary 4.7: Let a CFT admitting the chiral splitting assumption be given. Every primary
conformal field of weight (h, h̄), Φ(z, z̄) = Φ(z) ⊗ Φ̄(z̄) can be expanded around (z, z̄) = (w, w̄) as
follows:
X
X
z −n−h Φn (w) ⊗
z̄ −m−h̄ Φ̄m (w̄) ,
Φ(z, z̄) =
(4.33)
Z
Z
m∈
n∈
with
Φn (w) =
1
2πi
I
w
dz (z − w)n+h−1 Φ(z)
and
Φ̄m (w̄) =
1
2πi
I
w̄
dz̄ (z̄ − w̄)m+h̄−1 Φ̄(z̄) .
(4.34)
We call this expansion mode expansion. Usually w = 0 and we define
Φn := Φn (0)
Proof:
and
Φ̄n = Φ̄n (0) .
Every complex function may be Laurent expanded:
X
bn (z − w)n .
f (z) =
n∈Z
The Laurent coefficients bn are known to be (see for example [WW78])
I
1
bn :=
dz (z − w)−n−1 f (z) .
2πi w
(4.35)
(4.36)
(4.37)
Every correlator containing a primary field Φ(z, z̄) will be a product of complex functions
h0| . . . Φ(z, z̄) . . . |0i = h0| . . . Φ(z) ⊗ Φ̄(z̄) . . . |0i = h0| . . . Φ(z) . . . |0i∗ h0| . . . Φ̄(z̄) . . . |0i∗
(4.38)
because of the chiral splitting assumption. Laurent expanding both gives with (4.37)
I
X
1
(z − w)−n−h h0| . . .
dz (z − w)n+h−1 Φ(z) . . . |0i∗
h0| . . . Φ(z) . . . |0i∗ h0| . . . Φ̄(z̄) . . . |0i∗ =
2πi w
n∈Z
I
X
1
−m−h̄
dz̄ (z̄ − w̄)m+h̄−1 Φ̄(z̄) . . . |0i∗
·
(z̄ − w̄)
h0| . . .
2πi w̄
m∈Z
X
(z − w)−n−h h0| . . . Φn (w) . . . |0i∗
=
n∈Z
·
X
m∈Z
(z̄ − w̄)−m−h̄ h0| . . . Φ̄m (w̄) . . . |0i∗
being true for an arbitrary insertion of fields away from (z, z̄).
For the hermitian conjugate of a mode operator we get
(4.39)
4.2 Radial Quantization
65
Proposition 4.8:
Φ†n (w) = Φ−n (w)
Proof:
and
Φ̄†n (w̄) := Φ̄−n (w̄) .
(4.40)
We apply definition (4.30) to the mode expansion:
X X
X X
1 1
Φ(z, z̄)† = z −2h z̄ −2h̄ Φr ( , ) =
z n−h z̄ m−h̄ Φn ⊗ Φ̄m =
z −n−h z̄ −m−h̄ Φ−n ⊗ Φ̄−m .
z z̄
n∈Z m∈Z
n∈Z m∈Z
From the definition of the mode operators, the proposition follows immediately.
(4.41)
Remark. We will sometimes use the notation
Φn,m(w, w̄) := Φn (w) ⊗ Φm (w̄)
(4.42)
Φn,m := Φn,m (0, 0) .
(4.43)
and accordingly
Then, the mode expansion can also be written
X X
(z − w)−n−h (z̄ − w̄)−m−h̄ Φn,m(w, w̄) .
Φ(z, z̄) =
Z m∈Z
(4.44)
n∈
Hermitean conjugation is given by
Φ†n,m(w, w̄) = Φ−n,−m .
(4.45)
Next, we would like to introduce commutators of our operators and their modes. Commutators
are most useful if taken at equal times. The commutator of two local fields can be defined to
fulfil
h0|[a(z), b(w)]|0i ≡ h0| [a(z)b(w) − b(w)a(z)] |0i .
(4.46)
However, this will not be of much use since we do not really know how to calculate the correlator
on right hand side. We only know how to calculate the correlator of radially ordered fields, from
ha(z) b(w)i ≡ h0|R {a(z)b(w)} |0i
(4.47)
(remember that the fields on the right hand side are operators, while those on the left–hand side
are path integral variables). Let us see what happens if we contour integrate the right–hand side
along a circle around 0 (anti–clockwise) with radius greater |w|:
I
I
dz h0|a(z)b(w)|0i
(4.48)
dz h0|R {a(z)b(w)} |0i =
|z|>|w|
|z|>|w|
using the definition of radial ordering. Similarly, integrating along a circle with radius smaller
|w| yields
I
I
dz η h0|b(w)a(z)|0i .
(4.49)
dz h0|R {a(z)b(w)} |0i =
|z|<|w|
|z|<|w|
Because ha(z) b(w)i is holomorphic except for poles at z = w, we know from (4.47) that the same
has to be true for h0|R {a(z)b(w)} |0i. Therefore, from Cauchy’s theorem
I
I
dz h0|b(w)a(z)|0i
(4.50)
dz h0|R {a(z)b(w)} |0i =
0=
|z|<|w|
|z|<|w|
and
I
|z|<|w|
dz h0|R {a(z)b(w)} |0i =
I
w
dz R {a(z)b(w)}
(4.51)
66
Chapter 4 – Operator formalism
where the last contour is a small circle around z = w. If we now subtract (4.50) from (4.48) and
use (4.51), we end up with
I
I
I
dz h0|R {a(z)b(w)} |0i . (4.52)
dz h0|b(w)a(z)|0i =
dz h0|a(z)b(w)|0i − η
w
|z|<|w|
|z|>|w|
In the limit |z| → |w| the left hand side gives something very similar to A b(w) − η b(w) A which
is what we would have expected for a naive commutator (at equal times). The right hand side is
the contour integral over ha(z) b(w)i and we know how to calculate it: using Cauchy’s theorem,
we only need to know the (singular part of the) OPE of a(z) with b(w).
This proves and motivates the following lemma and definition.
Lemma
H 4.9: Let Φ(z), Ψ(z) be the holomorphic components of primary conformal fields. Let
A := 0 dz a(z) be an operator obtained from the local operator a(z) by contour integration along a
small circle around z = 0. It holds that
) I
(I
I
[A, Ψ(w)]−η := lim
ǫ→0
|w|+ǫ
dz a(z) Ψ(w) − η
dz Ψ(w) a(z)
|w|−ǫ
=
w
dz R {a(z)Ψ(w)} .
(4.53)
η = +1 for commuting fields and η = −1 for anti–commuting fields. the last integral is performed
along a circle around w with infinitesimal radius. [·, ·] is called the equal–time (anti–)commutator or
simply (anti–)commutator.
Remark. We will use the standard notation:
[A, Ψ(w)] := [A, Ψ(w)]−
{A, Ψ(w)} := [A, Ψ(w)]+
(4.54)
Theorem 4.10: Let Φ(z), Ψ(z) be the holomorphic components of the operators corresponding to
primary conformal fields. Following statements are true
(i) The commutator of the modes (at z = 0) of Φ(z) with Ψ(w) is
I
1
[Φn , Ψ(w)] =
dz z n+hΨ −1 R{Φ(z) Ψ(w)} .
2πi w
(ii) The commutator of their modes (at z = 0) is given by
I
I
1
dz z n+hΦ −1 wm+hΨ −1 R {Φ(z) Ψ(w)} .
[Φn , Ψm ] =
dw
(2πi)2 0
w
(4.55)
(4.56)
Proof: (i) Inserting the definition of the modes (4.34) into (4.53) gives the proposition directly.
(ii) Integrating both sides of (4.55) along a small circle around w = 0 and using the definition of the
modes (4.53) gives the proposition.
By replacing all z, w with z̄, w̄ and holomorphic with anti–holomorphic, we obtain a similar
theorem for the anti–holomorphic components of primary fields:
Theorem 4.11: Let Φ̄(z̄), Ψ̄(z̄) be the anti–holomorphic components of the operators corresponding
to primary conformal fields. Following statements are true
(i) The commutator of the modes (at z̄ = 0) of Φ̄(z̄) with Ψ̄(w̄) is
I
1
dz̄ z̄ m+h̄Ψ̄ −1 R{Φ̄(z̄) Ψ̄(w̄)} .
[Φ̄n , Ψ̄(w̄)] =
2πi w̄
(4.57)
4.3 Central Charge and the Virasoro Algebra
67
(ii) The commutator of their modes (at z̄ = 0) is given by
I
I
1
n+h̄Φ̄ −1 m+h̄Ψ̄ −1
dz̄
z̄
w
R
Φ̄(z̄)
Ψ̄(
w̄)
.
d
w̄
[Φ̄n , Ψ̄m ] =
(2πi)2 0
w̄
(4.58)
Proof: (i) Inserting the definition of the modes (4.34) into the anti–holomorphic version of (4.53) gives
the proposition directly.
(ii) Integrating both sides of (4.57) along a small circle around w̄ = 0 and using the definition of the
modes (4.53) gives the proposition.
Remark. Note that this justifies to keep only the divergent part of the operator product expansions: any term regular at z = w in the product on the right–hand side will drop out of the
integration due to Cauchy’s theorem.
4.3 Central Charge and the Virasoro Algebra
In the last section we introduced the mode expansion. Of special importance are the modes of
the energy–momentum tensor, which correspond to the generators of the Virasoro algebra. We
will see that it is the quantum version of the Witt algebra we encountered in chapter 2.
The components of the quantum energy–momentum tensor have conformal dimensions (2, 0)
and (0, 2) respectively—except for the mentioned anomalies, which we shall ignore at this point.
Hence, the mode expansions of their corresponding operators (for which we use the same symbols
T, T̄ as before) are given by
I
X
1
−n−2
dz z n+1 T(z)
(4.59)
z
Ln ⊗ 1
with Ln =
T(z) =
2πi
Z
n∈
T̄(z̄) =
X
Z
z̄ −n−2
n∈
1 ⊗ L̄n
with L̄n =
1
2πi
I
dz̄ z̄ n+1 T̄(z̄) .
(4.60)
Because of chiral splitting, T and T̄ act independently. Usually, this statement is expressed in
the vanishing of the commutator of their modes. However, this is a trivial statement in our
notation
[(Ln ⊗ 1), (1 ⊗ L̄n )] ≡ [Ln , 1] ⊗ [1, L̄n ] = 0 .
(4.61)
To get an impression of the physical importance of the Ln , L̄n , look at the transformation of
a primary field Φ(w, w̄) of weight (h, h̄). From the conformal Ward identity we have
I
I
1
1
dz ε(z) R {T(z)Φ(w, w̄)} +
dz̄ ε̄(z̄) R T̄(z̄)Φ(w, w̄) .
(4.62)
δε,ε̄ Φ(w, w̄) =
2πi w
2πi w̄
Using lemma 4.9 and the chiral decomposition Φ(w, w̄) = Φ(w) ⊗ Φ̄(w̄), this is equivalent to
δε,ε̄ Φ(w, w̄) = [Qε , Φ(w)] + [Q,ε̄ , Φ̄(w̄)]
where
Qε :=
1
2πi
I
dz ε(z) T(z)
0
and
Qε̄ :=
1
2πi
I
dz̄ ε̄(z̄) T̄(z̄) .
(4.63)
(4.64)
0
So we see that the energy–momentum tensor defines the generator of conformal transformations
in the operator formalism in HC. In accordance with standard quantum field theory, we call
Qε,ε̄ := Qε ⊗ 1 + 1 ⊗ Qε̄
(4.65)
68
Chapter 4 – Operator formalism
the (quantum) conformal charge of conformal transformations. Qε , Qε̄ will be called the chiral
and anti–chiral conformal charges. Expanding ε(z)and ε̄(z̄) into1
ε(z) =
X
Z
z n+1 εn
ε̄(z̄) =
X
Z
z̄ n+1 ε̄n ,
(4.66)
n∈
n∈
gives
Qε =
1
2πi
I
dz
0
X
Z
z n+1 εn T(z) =
n∈
1
2πi
I
dz
0
X
Z
z n+1 εn
n∈
Similarly, it is
Qε̄ =
X
Z
X
Z
z −m−2 Lm =
m∈
X
Z
εn Ln . (4.67)
n∈
(4.68)
ε̄n L̄n .
n∈
Recall that ε ∈ {1, z, z 2 } corresponds to the generators of translations, dilatations and rotations
on the complex plane. Recall that
z := et e−ix
(4.69)
we have for dilatations (α ∈ R+
0)
z = et e−ix 7−→ α z = et+ln α e−ix .
(4.70)
Hence, dilatations on the complex plane correspond to time translations in Minkowski space–
time. From (4.66) we see that for dilatations
(4.71)
ε(z) = const. z = ε0 z .
The corresponding conformal charge generating this transformations is determined from equations (4.67) and (4.68). We obtain
L0 ⊗ 1 + 1 ⊗ L̄0
: generates time translations
(4.72)
In quantum field theory, the generator of time translations is the Hamiltonian. Therefore, we
associate L0 ⊗ 1 + 1 ⊗ L̄0 to the Hamiltonian. However, these association is not unique and in
c
.
fact it turns out that that L0 ⊗ 1 + 1 ⊗ L̄0 and the Hamiltonian differ by 12
Now, let us determine the algebra of modes, i.e. the commutation relations of the Ln , L̄n with
themselves. We have from (4.56)
I
I
1
[Ln , Lm ] =
dz z n+1 wm+1 R {T(z)T(w)} .
(4.73)
dw
(2πi)2 0
w
Inserting the most general OPE (3.213) of a CFT with central charge2 c
T(z) T(w) ∼
c/2
2 T(w)
∂T(w)
+
+
4
2
(z − w)
(z − w)
z−w
(4.74)
yields
1
[Ln , Lm ] =
(2πi)2
1
I
dw
0
I
w
dz z
n+1
w
m+1
c/2
2 T(w)
∂T(w)
+
+
4
2
(z − w)
(z − w)
z−w
.
(4.75)
We expand in accordance with corollary 4.7. Recall that ε(z) (ε̄(z̄)) is the component of a holomorphic (anti–
holomorphic) vector field, a (−1, 0) tensor (a (0, −1) tensor).
2
Recall that we always assume c = c̄
4.3 Central Charge and the Virasoro Algebra
69
To perform the z integral, we need to Laurent expand z n+1 to third order in z − w:
1
(n + 1) n wn−1 (z − w)2
2!
z n+1 = wn+1 + (n + 1) wn (z − w) +
1
+ (n + 1) n (n − 1) wn−2 (z − w)3 + O((z − w)4 ) .
3!
(4.76)
Applying Cauchy’s theorem to (4.75) gives
c
2 (n + 1)
[Ln , Lm ] =
n (n2 − 1) δn,−m +
12
2πi
I
dw w
n+m+1
0
1
T(w) +
2πi
I
dw wn+m+2 ∂T(w) .
0
The second term is by definition of the modes
I
2 (n + 1)
dw wn+m+1 T(w) = (2n + 2) Ln+m .
2πi
0
(4.77)
(4.78)
Differentiating the mode expansion for T(w) (4.59) yields
∂T(w) =
X
∂ X −k−2
w
Lk =
(−k − 2)w−k−3 Lk .
∂w
Z
Z
k∈
(4.79)
k∈
With this, the third term in (4.77) is
I
I
1 X
1
n+m+2
dw w
∂T(w) =
(−k − 2) dw wn+m−k−1 Lk
2πi 0
2πi
0
k∈Z
X
(−k − 2) δn+m,k Lk
=
(4.80)
Z
k∈
= −(m + n + 2) Ln+m .
Finally, we have
c
n (n2 − 1) δn,−m .
(4.81)
12
This algebra is known as the Virasoro algebra [Vir70]. It almost agrees with the Witt algebra
[Ln , Lm ] = (n − m) Lm+n +
[ℓn , ℓm ] = (n − m) ℓm+n
(4.82)
which is fulfilled by the classical generators of conformal symmetry, as we have shown in chapter
2.
The analogous calculation for the anti–holomorphic modes L̄ reveals another copy of the same
algebra.
Before exploring this algebras further, we will determine the commutator of the modes Ln , L̄n
with the holomorphic, anti–holomorphic components of primary fields Φ(z), Φ̄(z̄).
Using theorem 4.11 (i) we have
I
1
dz z n+1 R{T Φ(w)} .
(4.83)
[Ln , Φ(w)] =
2πi w
Inserting the OPE of T with Φ from proposition 3.27 gives
I
1
∂Φ(w)
h Φ(w)
n+1
[Ln , Φ(w)] =
dz z
+
= h (n + 1) wn Φ(w) + wn+1 ∂Φ(w) . (4.84)
2πi w
(z − w)2
z−w
The anti–holomorphic relation reads
[L̄n , Φ̄(w̄)] = h̄ (n + 1) w̄n Φ̄(w̄) + w̄n+1 ∂¯Φ̄(w̄) .
We summarize the calculations above:
(4.85)
70
Chapter 4 – Operator formalism
Theorem 4.12: The the holomorphic and antiholomorphic modes Ln , L̄n of T(z), T̄(z̄) both fulfil
the Virasoro algebra Vir:
[Ln , Lm ] = (n − m) Lm+n +
c
n (n2 − 1) δn,−m
12
(4.86)
[L̄n , L̄m ] = (n − m) L̄m+n +
c
n (n2 − 1) δn,−m .
12
(4.87)
For a primary field Φ(z, z̄) = Φ(z) ⊗ Φ̄(z̄) of weight (h, h̄) it holds that
[Ln , Φ(w)] = h (n + 1) wn Φ(w) + wn+1 ∂Φ(w)
(4.88)
[L̄n , Φ̄(w̄)] = h̄ (n + 1) w̄n Φ̄(w̄) + w̄n+1 ∂¯Φ̄(w̄) .
(4.89)
and
Remarks.
• The appearance of the Virasoro algebra from the classical Witt algebra can be understood
from the abstract theory of algebra:
Our classical symmetry was represented by the Witt algebra and physical objects have to
be representations of W. After quantization, the main objects are vectors in a Hilbert
space H. It is natural that these are still representations of the Witt algebra. However,
since two vectors ψ, φ ∈ H give rise to the identical vacuum expectation value if
φ = eiθ ψ ,
we have to identify all such elements. The resulting representations are called projective
representations of W.
The theory of abstract algebra tells us that every projective representation of an algebra
can be obtained through central extension under certain circumstances. This means, the
algebra is extended by an algebra of elements in the center of the algebra, i.e. elements
which commute with all the others.
It turns out, that in case of the Witt algebra there is only a one–parameter family of central
extensions, corresponding to the addition of a multiple of the identity:
c·1.
The resulting algebra is precisely the Virasoro algebra [PS86].
• We have seen before, that the non–zero central charge leads to an anomalous transformation
behavior of T and T̄. However, there still remained two subalgebras of non–anomalous
sl(2, C) transformations. These are generated by ε ∈ {1, z, z 2 }, ε̄ ∈ {1, z̄, z̄ 2 } and therefore
correspond to
{L−1 , L0 , L1 }
and
{L̄−1 , L̄0 , L̄1 } .
(4.90)
Indeed, the Virasoro algebra and the Witt algebra coincide for n, m ∈ {−1, 0, 1} since the
central charge term vanishes.
Furthermore, it is possible to show that sl(2, C) is the only closed subalgebra of Vir.
4.4 Space of States
71
4.4 Space of States
Next, we will work out the consequences of the algebra above and construct a Fock space of
states. First, we will state the following lemma
b be the operator picture of a CFT. There exists a unique state |ψi = |Ωi⊗|Ωi ∈
Lemma 4.13: Let C
HC with
L0 |Ωi = 0
and
L̄0 |Ωi = 0 .
(4.91)
This state is the vacuum of HC
|ψi = |0i .
(4.92)
Proof: The proposition follows from one of Wightman’s axioms, the cluster decomposition principle.
It states that correlators of far separated fields factorize into lower order correlators:
+
+
+*
*
*
Y
Y
Y
Y
|x−y|→∞
Φlj (yj )
(4.93)
Φlj (yj )
Φki (xi )
−→
Φki (xi )
j
i
j
i
We will not perform the proof here, it can be found in [Gab00].
Note that sl(2, C) symmetry forces
h0|T(z)|0i = 0
(4.94)
since the one–point function vanishes for any field with conformal weight (h, h̄) 6= (0, 0) according
to (3.137). Recall that T(z) transforms anomalous with respect to conformal transformations,
but regularly as a (2, 0) quasi–primary field with respect to the sl(2, C) subalgebra.
Therefore, we have
X
Z
n∈
1
z n+2
h0| (Ln ⊗ 1) |0i =
X
Z
n∈
1
z n+2
hΩ|Ln |Ωi = 0 .
(4.95)
Since the right–hand side is independent of z, we obtain from the limit z → 0
Ln |Ωi = 0
∀ n ≥ −1 .
(4.96)
L̄n |Ωi = 0
∀ n ≥ −1 .
(4.97)
hΩ|L−n = 0
∀ n ≥ −1
(4.98)
hΩ|L̄−n = 0
∀ n ≥ −1 .
(4.99)
Similarly, we have
Because of L†n = L−n , L̄†n = L̄−n we also have
Now, consider an incoming state created by the primary field Φ(z, z̄) = Φ(z) ⊗ Φ̄(z̄) of weight
(h, h̄)
Φ(0, 0)|0i = Φ(0)|Ωi ⊗ Φ̄(0)|Ωi .
(4.100)
From (4.96) we have for all n ≥ −1
Ln Φ(0)|Ωi = Ln Φ(0)|Ωi − Φ(0) Ln |Ωi = [Ln , Φ(0)] |Ωi .
(4.101)
Using (4.88) we obtain
Ln Φ(0)|Ωi = h δn,0 Φ(0)|Ωi + δn,−1 ∂Φ(0)|Ωi .
(4.102)
72
Chapter 4 – Operator formalism
A similar calculation holds for L̄n , Φ̄ and we conclude
L0 Φ(0)|Ωi = h Φ(0)|Ωi
Ln Φ(0)|Ωi = 0
∀n > 0
(4.103)
L̄0 Φ̄(0)|Ωi = h̄ Φ̄(0)|Ωi
L̄n Φ̄(0)|Ωi = 0
∀n > 0 .
(4.104)
Therefore, every primary field gives rise to an energy eigenstate. We will use the notation
|h, h̄i ≡ |hi ⊗ |h̄i ≡ Φ(0, 0)|0i = Φ(0)|Ωi ⊗ Φ̄(0)|Ωi .
(4.105)
What about states of the form
L−m |hi
(4.106)
with some m > 0 ? Application of the Virasoro algebra yields
L0 L−m |hi = L−m L0 |hi + [L0 , L−m ] |hi = (h + m) L−m |hi .
| {z }
(4.107)
=(0+m) L0−m
Again, this holds analogously for |h̄i. Hence, application of L−m ⊗ 1 + 1 ⊗ L̄−n for m, n > 0
gives infinitely many more energy eigenstates. Of course, it is possible to apply the operator
repeatedly. We define states obtained in this way as descendants of |h, h̄i.
Definition 4.14: Let |h, h̄i the in–state of a primary field of weight (h, h̄). We call
L−m1 . . . L−mN |hi ⊗ L̄−m̄1 . . . L̄−m̄N̄ |h̄i
(4.108)
a descendant or excited state of |h, h̄i. We employ the convention 1 ≤ m1 ≤ · · · ≤ mN , 1 ≤ m̄1 ≤
· · · ≤ m̄N̄ (which can be shown to be equivalent to normal ordering).
A descendant is an energy eigenstate with eigenvalue
h + L + h̄ + L̄
(4.109)
P
P
of level (L, L̄) = ( mi , m̄i ). The set of descendants of |h, h̄i together with the state itself is
i
i
called conformal family, denoted by [Φ].
Remark. The question how many distinct states are present at a given level, has an interesting
connection to number theory. The number of possibilities to reach a certain level by application
of L−m ⊗ 1 + 1 ⊗ L̄−n is given by
π(L) π(L̄)
(4.110)
where π(L) is the number of ways to write the integer L as the sum of non–negative integers. π
can be obtained from a generating function:
∞
X
L=0
L
π(L) q =
∞
Y
k=1
1
1
≡
.
1 − qk
φ(q)
(4.111)
φ(q) is known as the Euler function. It also appears in the partition function of the free boson
on the torus and therefore in the one–loop vacuum–to–vacuum amplitude of the bosonic string,
see chapter 5.
We have seen that our space of states not only contains the states created by primary fields, but
also infinitely many descendants. This seems to make the task of solving this theory completely
(to determine all of its correlators) quite difficult. However, correlators of descendants can be
reduced to correlators of primary fields.
4.5 The Operator Algebra
73
Proposition 4.15: Let a descendant be given
Φ{m,m̄} |0i ≡ L−m1 . . . L−mN |hi ⊗ L̄−m̄1 . . . L̄−m̄N̄ |h̄i
≡ Φ{m} |Ωi ⊗ Φ̄{m̄} |Ωi .
(4.112)
It holds that:
(i)
hΩ|Φ{m} X|Ωi = L−m1 . . . L−mN hφ(0) Xi
(ii)
with X({zi }) ⊗ X̄({z̄i }) :=
L−ℓ
Proof:
Q
i
hΩ|Φ̄{m̄} X̄|Ωi = L̄−m̄1 . . . L̄−m̄N̄ φ̄(0) X̄
(4.113)
(4.114)
Φki (zi , z̄i ), zi , z̄i 6= 0 and
X (ℓ − 1) hk
1
i
−
∂z ,
:=
(zi )ℓ
(zi )ℓ−1 i
i
L̄−ℓ
X (ℓ − 1) h̄k
1
i
−
∂z̄ .
:=
(z̄i )ℓ
(z̄i )ℓ−1 i
(4.115)
i
This was first shown by [BPZ84]. A proof can also be found in [FMS97].
From this follows directly that two families [Φi ], [Φj ] built over primary fields of different
conformal weights are orthogonal with respect to the scalar product:
{m,m̄}
h0|Φi
{k,k̄}
Φj
|0i = . . . hΦi Φj i = . . . δhi hj δh̄i h̄j .
(4.116)
From the definition of the descendants we see that the families are closed with respect to the
Virasoro algebra. Therefore, they form representations of Vir and are called Verma modules.
4.5 The Operator Algebra
We have seen that representations of the Virasoro algebra—the Verma modules—are orthogonal
with respect to the scalar product of HC. However, they share an intricate algebraic structure
called the operator algebra.3
To simplify expressions in this section, let us choose an orthonormal base of primary states by
setting the two point functions to be
hΦi (z, z̄)Φj (w, w̄)i =
δhi hj δh̄i h̄j
(z − w)2 hi (z̄ − w̄)2 h̄j
.
(4.117)
Clearly, this is no restriction since it can always be achieved by renormalizing the fields appropriately.
Now consider the three–point function of primary fields
hΦi (z1 , z̄1 )Φj (z2 , z̄2 )Φk (z3 , z̄3 )i = Cijk (z1 − z2 )hk −hi −hj (z23 )hi −hj −hk (z13 )hj −hi −hk
(z̄1 − z̄2 )h̄k −h̄i −h̄j (z̄23 )h̄i −h̄j −h̄k (z̄13 )h̄j −h̄i −h̄k
(4.118)
where we have inserted the general solution (3.138). In the limit z1 → z2 we obtain
Cijk (z12 )hk −hi −hj (z1 − z3 )−2hk (z̄12 )h̄k −h̄i −h̄j (z̄1 − z̄3 )−2h̄k ,
where we have used that z13 = z23 for z1 → z2 .
3
Although related, the operator algebra and the OPE should not be confused.
(4.119)
74
Chapter 4 – Operator formalism
Let us now look at the most general operator product of two primary fields:
X
Φi (z1 , z̄1 )Φj (z2 , z̄2 ) =
Arij (z1 − z2 , z̄1 − z̄2 ) ψr (z2 , z̄2 ) .
(4.120)
r
The sum on the right–hand side is allowed to range over the whole space of states and therefore the
summation is over all primary fields and their descendants. Furthermore, conformal invariance
restricts the coefficients to be of the form (simply comparing the conformal weights of both sides)
r
Arij (z1 − z2 , z̄1 − z̄2 ) = Cij
(z1 − z2 )hr −hi −hj (z̄1 − z̄2 )h̄r −h̄i −h̄j .
(4.121)
So the most general operator product is
X X
p{m,m̄}
Φi (z1 , z̄1 )Φj (z2 , z̄2 ) =
Cij
(z1 − z2 )hr −hi −hj (z̄1 − z̄2 )h̄r −h̄i −h̄j φp{m,m̄} (z2 , z̄2 ) .
p {m,m̄}
(4.122)
This relation among the fields in a CFT is called the operator algebra. Inserting this into a
correlator of three fields, we see that this yields
p{0,0}
Cij
(4.123)
= Cijp
the three–point function coefficients.
Since correlators of descendants can be determined from primary field correlators alone, we
p{m,m̄}
expect the constants Cij
to be somehow related to the Cijp . In fact, it can be shown that
[FMS97]
p{m,m̄}
p{0,0} p{m} p{m̄}
Cij
= Cij
βij
β̄ij
,
(4.124)
where the functions β, β̄ are determined from hi , hj , hp and the central charge c only. For
example, in the case hi = hj it is at level 1:
p{1}
βij
=
1
2
p{1}
β̄ij
=
1
.
2
(4.125)
On level 2 we have:
p{1,1}
βij
=
c − 12 hi − 4 hp + c hp + 8 h2p
4(c − 10 hp + 2c hp + 16 h2p )
p{2}
βij
2 hi − hp + 4 hi hp + h2p
=
c − 10 hp + 2 chp + 16 h2p
p{1,1}
β̄ij
p{2}
β̄ij
=
c − 12 h̄i − 4 h̄p + c h̄p + 8 h̄2p
4(c − 10 h̄p + 2c h̄p + 16 h̄2p )
2 h̄i − h̄p + 4 h̄i h̄p + h̄2p
.
=
c − 10 h̄p + 2 ch̄p + 16 h̄2p
(4.126)
(4.127)
In general, there are 2 π(L) coefficients at given level L.
This illustrates how conformal symmetry fixes the operator algebra of a CFT. By its repeated
application, every correlator of primary fields and their descendants can be calculated. The
only necessary inputs are the conformal weights, the central charge and the three–point function
coefficients. Hence, in principle the theory can be solved completely. Such a theory is usually
called integrable.
4.6 Unitarity, Minimal Models and RCFT
In this section we will investigate the consequences of unitary scattering amplitudes and further
explore the integrability of CFTs.
To allow a probability interpretation of the scattering amplitudes requires
{m,m̄}
hφout
{m,m̄}
|φin
i≥0
(4.128)
4.6 Unitarity, Minimal Models and RCFT
75
not only for all primary fields but also for their descendants. Consider in particular the correlator
{1,0}
{1,0}
0 ≤ hφout |φin
i.
(4.129)
Because of chiral splitting, this implies for Φ(z, z̄) = Φ(z) ⊗ Φ̄(z̄) that
0 ≤ hΩ|Φ† (0)L1 L−1 Φ(0)|Ωi .
(4.130)
Since L1 Φ(0)|Ωi = 0, we have
0 ≤ hΩ|Φ|ΩiΦ† (0)L1 L−1 Φ(0)|Ωi = hΩ|Φ|ΩiΦ† (0)[L1 , L−1 ]Φ(0)|Ωi
= hΩ|Φ|ΩiΦ† (0)2 L0 Φ(0)|Ωi = 2 h hφout |φin i .
(4.131)
Similar arguments apply to Φ0,1 and therefore, we deduce that
h≥0
h̄ ≥ 0
(4.132)
for all conformal weights in any unitary CFT. This is rather interesting from a physical point
of view, since we associated h + h̄ with the energy of a state (modulo some overall constant).
Hence, the energy is automatically bounded from below in all unitary CFTs.
So we have seen that unitarity already restricts possible CFTs to possess only primary fields
of positive conformal weights. In the previous section, we discussed that the knowledge of
all conformal weights, the central charge and three–point function coefficients is, in principle,
sufficient to solve the theory completely. However, this is practically impossible to perform if
there appear infinitely many Verma modules in our theory. Hence, theories where only finitely
many Verma modules appear are of special interest. They are called minimal models.
The requirement for a theory to be unitary and minimal restricts the possible values of the
central charge and possible conformal weights appearing in a CFT. This restrictions have been
worked out [FQS84b], [GKO86]. There is a necessary condition on the central charge
c≥0
(4.133)
in addition to our requirement on the conformal weights. All CFTs with c < 0 are non–unitary.
In the case 0 < c < 1 there is only a discrete series of unitary minimal models:
c=1−
hr,s (m) =
[(m + 1) r − m s]2 − 1
4 m (m + 1)
6
m (m + 1)
h̄r,s (m) =
(4.134)
[(m + 1) r − m s]2 − 1
4 m (m + 1)
(4.135)
with 1 ≤ r ≤ m, 1 ≤ s ≤ r, m ≥ 2. The discrete series of minimal models has a wide variety of
successful applications to two–dimensional critical systems in statistical physics. For example, it
contains the Ising model (m = 2), the tricritical Ising model (m = 3), the 3–state Potts model
(m = 4) and the tricritical 3–state Potts model (m = 5).
For applications in the theory of high energy physics—in particular string theory—another
class of solvable models is of importance: the rational conformal field theories. They possess in
general infinitely many Verma modules, but can be reorganized into a finite set of Verma modules
belonging to an extension of the Virasoro algebra. The central charge and conformal weights
in a rational CFT are necessarily rational numbers [Vaf88]. In mathematics, this class of CFTs
is usually called finite models, because the Zhu algebra of rational CFTs is finite dimensional
[Zhu96].
5 String Theory
“If you don’t know, ask. You will be a fool for the moment, but a wise man for the
rest of your life."
Seneca (∼54 BC – 39 AD)
In this chapter we will describe how the developed formalism applies to string theory. It turns
out that bosonic string theory can be described as the CFT of 26 free bosons on the string’s
world–sheet. Superstring theory can be viewed as a supersymmetric extension of a CFT of 10
free bosons.
Standard textbooks on string theory are [GSW87] and [Pol98].
5.1 Sum over Topologies
The trajectory of a relativistic point particle forms a one–dimensional submanifold of space–time,
called the world–line. The dynamics of the particle can be described by a variational principle,
where the action is the length of the world–line
Z
Spoint =
ds .
(5.1)
world line
This can be used as a starting point to define the action of a string: As it propagates through time
it sweeps out a two–dimensional manifold, called the world–sheet. Its embedding into space–time
is given by
X µ (ξ 0 , ξ 1 )
(µ = 0, . . . , D − 1)
(5.2)
with ξ 0 , ξ 1 describing the string’s temporal and spatial extension.
Generalizing the concept of world–line length to world–sheet volume, we have the action
Z
√
1
d2 ξ γ γ αβ ∂α X µ ∂β Xµ ,
(5.3)
Svolume [γ, X] =
4πα′
Σ
with the metric on the world–sheet γαβ and γ := det(γαβ ). α′ is the so–called Regge slope,
it has units of space–time length squared and determines the characteristic scale of the string.
Additionally, we introduce Einstein–gravity (with boundary term) on the world–sheet
Z
Z
1
λ
2 √
d ξ γ R(ξ) +
ds k ,
(5.4)
Sgrav [γ] =
4π
2π
Σ
∂Σ
with the Ricci scalar R, scalar curvature k and a dimensionless parameter λ.
Let us now construct a quantum theory of strings. The String’s partition function should
be obtained from summing over all possible string histories, weighted with e−S/~ (after Wick
rotation). However, it is not sufficient to perform the path–integral over all possible embeddings
78
Chapter 5 – String Theory
X and metrics γ. Consider for example the propagation of a single string. Classically, the world–
sheet will be diffeomorphic to a cylinder. In the quantum description, the string may split to
form an intermediate two–string state. The resulting world–sheet will look like a torus, i.e. have
a different topology. So propagation of strings in a quantum theory will give rise to all possible
topologies for the world–sheet. This motivates to define the string’s partition function as follows:
Z
Z
Z
X Z
λ
1
αβ
µ
2 √
2 √
d ξ γ γ ∂α X ∂β Xµ exp −
d ξ γ R(ξ) .
Z=
DX Dγ exp −
4πα′
4π
topologies
(5.5)
After Wick rotation the world–sheet has become Euclidean.
Every two–dimensional surface admits a complex structure and can be regarded to be a Riemann surface. Furthermore, the Gauß–Bonnett theorem holds true:
Z
Z
1
1
2 √
d ξ γ R(ξ) +
ds k = χE
(5.6)
4π
2π
∂Σ
with the topological invariant χE the Euler characteristic. For compact orientable Riemann
surfaces it is
χE = 2 − g
(5.7)
where g denotes the number of handles.
Hence, the partition function becomes
Z
Z
Z
X
1
2 √
αβ
µ
−λχE
d ξ γ γ ∂α X ∂β Xµ .
Z=
e
DX Dγ exp −
4πα′
(5.8)
topologies
Because of χE ≥ 2 the sum over topologies corresponds to a perturbative expansion in the
coupling eλ , which is, therefore, called the string coupling constant.
5.2 Gauge Fixing and Ghosts
The path integral in (5.8) over γ is divergent because of the diffeomorphism invariance of the
action. This divergence can be removed by gauge fixing the action with the Fadeev–Popov
method. Usually, one adopts the conformal gauge in which the metric is given by
γαβ =
1 σ(ξ)
e δαβ .
2
The resulting action is that of D free bosons
Z
1
√
d2 ξ γ γ αβ ∂α X µ ∂β Xµ .
′
4πα
(5.9)
(5.10)
However, gauge fixing introduces new conformally invariant degrees of freedom in form of the
ghost action
Z
Sghost = d2 ξ bαβ ∂ α cβ
(5.11)
with world–sheet fermions b and c. The central charge of the ghost CFT is -26. As we have
seen in the last chapter, this shows that the ghost CFT possesses negative norm states and is
not unitary. However, the combined central charge of the D free bosons and the ghosts is
C = D − 26 .
(5.12)
5.3 Superstring theory
79
To have the theory free of gravitational anomalies requires c = 0 and hence,
(5.13)
D = 26 .
With the methods introduced in the previous chapters, one can work out the space of states.
It has the drawback of containing a state of negative mass—a tachyon. Further, the spectrum
contains only space–time bosons. Therefore, the above quantum theory of strings is known as
bosonic string theory. Altogether, this theory is phenomenologically unattractive since it fails to
describe fermions in nature.
5.3 Superstring theory
In the 80s, a generalization of bosonic string theory was found leading to space–time fermions
in its spectrum and absence of tachyons. The basic idea is to introduce superpartners of the
energy–momentum tensor components:
and
TF
T̄F .
(5.14)
In doing so one has the freedom to choose either periodic (Ramond) or anti–periodic (Neuveu–
Schwarz) boundary conditions. The fermionic energy–momentum tensor components have the
following OPEs:
∂TF (w)
3 TF (w)
+
(5.15)
T(z)TF (w) ∼
2
2 (z − w)
z−w
D
2 T(w)
.
+
3
(z − w)
z−w
(5.16)
∂ T̄F (w̄)
3 T̄F (w)
+
2
2 (z̄ − w̄)
z̄ − w̄
(5.17)
D
2 T̄(w̄)
+
.
3
(z̄ − w̄)
z̄ − w̄
(5.18)
TF (z)TF (w) ∼
and
T̄(z)T̄F (w) ∼
T̄F (z)T̄F (w) ∼
They imply that the new components are ( 32 , 0) and (0, 23 ) primary fields. This could have been
expected, since they are superpartners of spin 2 fields.
The theory now possesses an extension of the Virasoro algebra as symmetry, the super Virasoro
algebra. It is called the (1, 1) superconformal field theory, where (1, 1) refer to the number of
holomorphic and anti–holomorphic superpartners.
Consequently, we have to introduce superpartners for the ghost fields as well. Alltogether, the
combined central charge is then
c = D − 10 .
(5.19)
Hence, a consistent formulation of superstring theory requires the space–time to be ten dimensional.
There are five distinct, consistent superstring theories that can be constructed from superconformal field theory. They are:
• Type I: open and closed unoriented superstrings, can have an additional SO(32) gauge
group
• Type IIa: closed oriented superstrings, opposite boundary conditions for holomorphic and
anti–holomorphic degrees of freedom
• Type IIb: closed oriented superstrings, opposite boundary conditions for holomorphic and
anti–holomorphic degrees of freedom
80
Chapter 5 – String Theory
• Heterotic SO(32)/Z2 : closed oriented strings, different symmetry algebras for holomorphic
and anti–holomorphic degrees of freedom. SO(32)/Z2 gauge group.
• Heterotic E8 × E8 : closed oriented strings, different symmetry algebras for holomorphic
and anti–holomorphic degrees of freedom. E8 × E8 gauge group.
The heterotic string theories make use of the chiral splitting into holomorphic and anti–
holomorphic degrees of freedom. They form a hybrid of the holomorphic part of bosonic string
theory and the anti–holomorphic part of superstring theory. For consistency, it is necessary to
compactify 16 out of the 26 bosons on a even self–dual Lorentzian lattice. This fixes it to be
the root lattice of SO(32)/Z2 or E8 × E8 , which form the possible gauge groups of the heterotic
strings.
5.4 Compactifications of the heterotic string
In this last section, we would like to sketch some of the aspects making the heterotic string
attractive for building models which contain the standard model of elementary particles as a low
energy approximation. We focus on the aspects of heterotic oribifold compactifications, although
there are many promising possibilities to construct phenomenologically attractive string models.
A first obstacle to realistic models is the necessity for ten dimensional space–time. A solution
to this is to assume that 6 of the 9 space dimensions form a compact space which is small enough
as to have evaded detection until today.
Not all comapact six dimensional spaces are equally suitable for this. For instance compactification on a six–dimensional torus leads to N = 4 space–time supersymmetry. However, it is
known that it is not possible to describe chiral fermions in theories with more than one supercharge. Requiring N = 1 space–time supersymmetry forces the compactification manifold to
be Ricci flat. Such manifolds are known as Calabi–Yau manifolds [Yau78]. Another possibility
is the compactification on orbifolds [DHW85] [DHW86]. These are manifolds that are smooth
except for isolated points. A standard example is the cone, which is a smooth manifold except
for its tip. A compact orbifold is constructed from a torus by identifying points under the action
of a discrete group:
T 6 /P .
(5.20)
Usually, the point group is taken to be abelian and hence P = ZN or P = ZN × ZM . If the
geometry is chosen appropriately, orbifold compactifications have N = 1 space–time supersymmetry.
Although the set of admissible compactification manifolds can be reduced with consistency
arguments, there still remains are huge number of possible geometries, which is believed to
be about 10500 [Sus03] [Sch06]. The problem of finding the right geometry is often called the
landscape problem—in analogy to the problem of finding a specific valley in a vast mountainous
landscape.
In recent works [L+ 07] [L+ 08] [LNRS+ 08] a few hundred promising orbifold geometries were
identified, which give rise to the complete MSSM spectrum without exotics. This greatly reduced
landscape is usually called the mini–landscape. Interestingly, known problems of supersymmetric
GUTs, like the doublet–triplet splitting and the µ problem, can be solved in the mini landscape
models.
6 Conclusions and Outlook
In the present work, we devoted our study to conformal quantum field theory (CFT) with a
special focus on its geometrical aspects.
In chapter 2 we analyzed the group of conformal transformations and pointed out its unique
structure in two–dimensions. We saw that the conformal algebra in two dimensions is infinite
dimensional and closely related to the complexified algebra of vector fields on the circle—the
Witt algebra. Furthermore, we put effort in a careful implementation of the algebra on the Riemann sphere, where conformal transformations are simply the holomorphic and anti–holomorphic
mappings.
Surprisingly, the dimensionality of the conformal group depends on the metric’s signature.
While in Minkowskian space–time it is infinite dimensional, only a finite dimensional part of the
algebra can be integrated to form a group on the Riemann sphere.
The insights from this analysis guided us on our approach to conformal quantum field theory
in chapter 3. Differing from many common approaches, we chose to work in the path integral
formalism rather than using operator methods. We had to face the problem of a mismatch
between our theory’s symmetry group and symmetry algebra. Since we saw that the origin of
this mismatch is the Wick rotation, we chose to implement the concept of infinitesimal symmetry
right from the start. A natural geometrical tool for doing so was provided by the Lie derivative
of a tensor. This lead us to the concept of primary fields, which infinitesimally look like tensors
of the conformal group. Finally, we introduced CFT as the quantum field theory of primary
fields.
We continued our study by examining the role of the energy–momentum tensor in CFT.
We arrived at the observation that every Poincaré and scale invariant theory is automatically
conformally invariant. This remarkable result remains true even after quantization.
In course of studying the free boson, we saw the importance of operator product expansions
in CFT and encountered the conformal anomaly, related to the central charge. It leads to an
anomalous transformation behavior of the energy–momentum tensor. Of special interest is the
fact that this anomaly leaves a sl(2, C) subalgebra of the conformal group unaffected. This
algebra consists of translations, rotations and dilatations. As a consequence, the CFT remains a
sensible quantum field theory, since all fields still have definite energy, spin and scaling dimension.
The path integral approach allowed us some insights into the geometrical structure of CFT.
We have seen that there is a deep connection between conformal invariance and the complex
structure of the manifold we have been working on. In case of the free boson, this can be
understood from the path integral by using Hodge’s theorem, an important result from algebraic
topology [KP09]. Further studies in this direction are planed.
Finally, we gave a brief introduction to the foundations of string theory. We saw how CFT
appears naturally in the world–sheet dynamics of quantized strings after gauge fixing. CFT is
the method of first choice for calculations of string amplitudes using the methods described in
chapter 4.
In future work, we plan to use the techniques developed here in order to calculate arbitrary
correlators in orbifold compactifications of the heterotic string. This would open up many new
possibilities to explore the mini–landscape and might give rise to new insights to physics beyond
the standard model.
7 Acknowledgements
First of all, I would like to thank my supervisor Prof. Michael Ratz for providing me with the
opportunity to perform my thesis on such an interesting topic. I am grateful for his great support
and encouragement during my work.
I would also like to express my gratitude to Prof. Buchner. In his outstanding lectures I
learned the language and tools of geometry, without which this work never would have been
possible.
Special thanks are devoted to my room mate Christian Paleani. His deep knowledge and mathematical abilities greatly enriched this work. Moreover, he helped me very much by proofreading
this work intensively.
I thank Cristoforo Simonetto for thorough and very useful proofreading of this thesis.
Furthermore, I would like to thank all of my companions of the T30e group who I respect,
personally and professionally.
Last but not least, I want to express my gratitude for the kind and loving support of my mother
Eva Klaput and my sister Paulina Klaput. Ohne Euch wäre alles viel schwieriger gewesen, Vielen
Dank.
A Differential Geometry
“He who understands geometry, understands anything in the world."
Galileo Galilei (1564 – 1642)
In this chapter, we quote some useful facts from differential geometry without proof. More
detailed treatments of the material covered here can be found in [Nak03] [Buc06].
A.1 Manifolds
Definition A.1: A C r differentiable manifold M is a topological Hausdorff space1 with:
(i) M is provided with a family of pairs {(Ui , φi )}, called atlas
S
(ii) {Ui } is a family of open sets with Ui = M.
i
(iii) φi : Ui → Vi ⊂ Rn is a homeomorphism2
(iv) For Ui ∩ Uj 6= 0, the map φi ◦ φ−1
j is r times continuously differentiable.
Remarks.
• The (Ui , φi ) are called charts and φ(p) is called local coordinates of p.
• We will always assume r = ∞.
Definition A.2: Let M, N be manifolds with atlases {(Ui , φi )} and {(Vj , ψj )}. A homeomorphism
f : M → N
(A.1)
is called diffeomorphism if the functions
φi ◦ f ◦ ψj−1
ψj ◦ f ◦ φ−1
i
(A.2)
are both infinitely often continuously differentiable.
We denote the set of all tangential vectors of curves passing through p the tangent space at
point p and write Tp M. Its dual space (i.e. the set of linear maps from Tp M to the real numbers)
is denoted Tp∗ M and called the cotangent space.
A basis of Tp M is written as
∂
(A.3)
{eµ } = { µ }
∂x
1
A topological space is a set with a notion of open sets (where unions and intersections of open sets have to be
open again). It is called Hausdorff, if for any two elements in the set there exist two disjoint open sets which
include each one of the two elements.
2
i.e. φ has an inverse φ−1 , φ and φ−1 map open sets to open sets
86
Chapter A – Differential Geometry
where xµ are local coordinates of p.
A basis of Tp∗ M is written as
{θ µ } = {dxµ } ,
and defined through
dxν (
(A.4)
∂
) = δ νµ .
∂xµ
(A.5)
where xµ are local coordinates of p.
A general (p, q) tensor T is a linear map
T : (Tp M)p × Tp∗ M
q
→R.
(A.6)
′
Consider the change of local coordinates xµ 7→ f µ . The relation of the bases { ∂x∂ ν } and
{ ∂f∂ν ′ } is given by
′
∂f µ ∂
∂
=
.
∂xµ
∂xµ ∂f µ′
′
Accordingly, the bases dxµ and df µ are related via
∂xµ
µ′
.
′ df
µ
∂f
dxµ =
A diffeomorphism has a natural action on (p, q)–tensors. If T maps p vectors and q covectors
into the real numbers:
p
1
q
1
p
1
1
q
T : (w, . . . , w, β, . . . , β) 7→ T (w, . . . , w, β, . . . , β)
(A.7)
then the transformed tensor maps the same vectors and covectors to
1
p
q
1
1
p
q
1
f ∗ T : (w, . . . , w, β, . . . , β) 7→ T (df (w), . . . , df (w), β ◦ df −1 , . . . , β ◦ df −1 ) .
(A.8)
df is the covector defined by
df := ∂µ f dxµ .
(A.9)
Another important concept is the following
Definition A.3: A one–parameter family {ft }t∈R is called a flow of diffeomorphisms if
f0 = 1
fs ◦ fr = fs+r
f−s = fs−1 .
(A.10)
There is the important
Lemma A.4: Let {fs }s∈R be a flow of diffeomorphisms. The differential equation
d
fs (p) = X(fs (p))
ds
(A.11)
has a unique solution fs for every vector field X. We say that X is the infinitesimal generator of the
flow fs .
We will need the Lie–derivative of a (p, q) tensor.
Definition A.5: The Lie–derivative of a (p, q) tensor t in direction of the vector field X is given by
£X T (p) := lim
s→0
1
((fs∗ T )(p) − T (x)) .
s
(A.12)
A.2 Complex Integration
87
Proposition A.6: The Lie–derivative is a derivation, i.e.
(i)
£X (a T + b S) = a £X T + b £X S
(A.13)
£X (T ⊗ S) = (£X T ) ⊗ S + T ⊗ (£X S)
(A.14)
(ii)
for all a, b ∈ R and tensors T , S.
For a (2, 0) tensor field, we have in particular:
(£X g)µν = ∇µ Xν + ∇ν Xµ
(A.15)
There is a very special (2, 0) tensor field:
Definition A.7: A symmetric (2, 0) tensor field g is called metric tensor if it is non–degenerate, i.e.
from
∀ X ∈ Tp M : g(X, Y ) = 0
(A.16)
follows always Y = 0. A Manifold equipped with a metric tensor (M, g) is called pseudo Riemannian manifold.
If the metric is positive definite (∀X, Y 6= 0) then we call (M, g) a Riemannian manifold or Euclidean
manifold.
A two–dimensional manifold with complex structure (i.e. a notion of complex conjugation) is
called a Riemann surface. Every two–dimensional surface admits a complex structure and may,
therefore, be viewed as a Riemann surface.
A.2 Complex Integration
Theorem A.8: Stokes’ theorem On a Riemann surface it holds that
I
I
Z
dz̄ vz̄ ,
i d2 z (∂z vz̄ + ∂z̄ vz ) = − dz vz −
ω
(A.17)
(∂ω)∗
∂ω
where ∂ω is the contour along the border of ω (anti–clockwise) and (∂ω)∗ its complex conjugate. This can be used to derive a useful representation of the delta distribution. Because of
ln |z − w|2 = ln(z − w) + ln(z̄ − w̄)
we have
(∂z ∂z̄ + ∂z̄ ∂z ) ln |z − w|2 = ∂z
(A.18)
1
1
+ ∂z̄
.
z̄ − w̄
z−w
(A.19)
Although this expression seems to vanish everywhere, we have to be a bit careful at (z, z̄) =
1
(w, w̄). Indeed, from Strokes’ theorem, we have with vz = z−w
, vz̄ = z̄−1w̄
i
Z
ω
I
1
−
dz
d z (∂z ∂z̄ + ∂z̄ ∂z ) ln |z − w| = −
z
−
w
w
2
2
I
dz̄
w̄
1
= −4πi .
z̄ − w̄
(A.20)
We conclude that
1
1
− (∂z ∂z̄ + ∂z̄ ∂z ) ln |z − w|2 = −
4π
4π
1
1
+ ∂z̄
∂z
z̄ − w̄
z−w
= δ(2) (z − w, z̄ − w̄) .
(A.21)
B Path integrals and two–point functions
Here, we will summarize the most important path integral formulae which we use in the work.
Recall the formula for a finite dimensional Gaussian integral
Z
1
1
ec
−1
(B.1)
(dx) e− 2 (x,Ax)+(b,x)+c = √
e 2 (b,A b)
det
A
n
R
with x, b ∈ Rn , c ∈ R and a symmetric matrix A ∈ Rn×n . A−1 is the inverse matrix. (·, ·) is the
scalar product in Rn , therefore,
n
X
x i bi
(B.2)
(x, b) ≡
i=1
and accordingly
(x, Ax) ≡
n
X
i=1
xi
n
X
Aij xj =
(dx) =
xi Aij xj .
(B.3)
i,j=1
j=1
The integration measure is
n
X
n
Y
(B.4)
dxi .
i=1
The infinite–dimensional generalization of (B.1) is the backbone of the path integral formalism:
Z
1
1
ec
−1
b
b
ZJ ≡ (Dφ) exp − (φ, O φ) + (J, φ) + c = p
(J, O J) .
(B.5)
exp
2
2
b
det
O
H
We have replaced the finite–dimensional Hilbert space Rn with a (in general infinite–dimensional)
b is a linear
Hilbert space H(M) over a (pseudo) Riemann manifold M. Now, φ, J ∈ H(M) and O
−1
b
b
operator O : H(M) → H(M), O its inverse. (·, ·) is the L2 scalar product in H(M):
Z
dx J(x) φ(x) ,
(B.6)
(J, φ) =
M
b is written as an integral operator as
with dx the integration measure on M. The operator O
follows
Z
b : φ(x) 7−→
dy O(x, y) φ(y)
(B.7)
O
M
and analogously its inverse as
b −1 : φ(x) 7−→
O
Z
dy O−1 (x, y) φ(y) .
(B.8)
M
From (B.5) we identify the partition function to be
Z = Z0 = p
ec
b
det O
.
(B.9)
90
Chapter B – Path integrals and two–point functions
Therefore, the two–point function is
δ2
1
Z exp
hφ(x)φ(y)i ≡
Z δJ(x) δJ(y)
1
−1
b
= O−1 (x, y) .
(J, O J) 2
J=0
bO
b −1 = 1H by definition, we have
Because O
Z
Z
!
dy O−1 (x, y) φ(y) = φ(x) .
dỹ O(x, ỹ)
M
(B.10)
(B.11)
M
b will always be a differential operator, i.e. of the form
In our work, O
Z
b
dỹ δ(x − ỹ) Dỹ φ(ỹ) .
O : φ(x) 7−→ Dx φ(x) =
(B.12)
M
Plugged into (B.11) this gives
Z
Z
Z
!
dy Dx O−1 (x, y) φ(y) = φ(x) .
dy δ(x − ỹ) Dx O−1 (x, y) φ(y) =
dỹ
M
M
(B.13)
M
Hence, we conclude that the two–point function is the Green’s function of the differential operator
Dx , i.e.
Dx hφ(x)φ(y)i = δ(x − y) .
(B.14)
For concrete calculations, we will put this in a more convenient form:
Theorem B.1: Let an Euclidean QFT be given, with partition function
Z
Z
1
Z = Dφ exp −
dx φ(x) Dx φ(x)
2 M
(B.15)
where Dx is a differential operator. Then the two–point function of the Euclidean QFT is the Green’s
function of Dx , i.e.
Dx hφ(x)φ(y)i = δ(x − y) .
(B.16)
Nomenclature
gµν
√
g
Riemannian metric
p
:= det(gµν )
CM (r, s) the conformal group on a manifold M of signature r, s
COM (r, s) the connected component of the conformal group containing the identity element
£X
Lie–derivative along X
−1 0
0 1
ηµν
Minkowski metric (ηµν ) =
CP1
the Riemann sphere
W
Witt algebra
δε,ε̄
first variation of a tensor, see definition 3.1
C ∪ {∞}
(h, h̄) conformal weights of a primary field
Σg
Riemann surface of genus g
L
Lagrangian
Z
partition function
h. . .i
correlator (defined through the path integral)
Tµν
energy–momentum tensor
R(x) Ricci scalar
: . . . : normal ordering
∼
equality modulo regular terms
T(z)
:= −2π Tzz , holomorphic component of the energy–momentum tensor
T̄(z̄)
:= −2π T̄z̄z̄ , anti–holomorphic component of the energy–momentum tensor
c
central charge
HC
The Hilbert space of the CFT C
|0i
vacuum of HC, unique state with h, h̄ = 0
|Φin i incoming asymptotic state created by Φ
hΦout | outgoing asymptotic state created by Φ
92
Chapter B – Path integrals and two–point functions
R {. . .} Radial ordering, see proposition 4.2
Ln
mode operators of T
L̄n
mode operators of T̄
Vir
Virasoro algebra
|Ωi
vacuum of the holomorphic and anti–holomorphic subspaces
Φ{m,m̄} a descendant of Φ
Cijk
three–point function coefficient of the primary fields Φi , Φj and Φk
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