LECTURES
on
CONFORMAL FIELD THEORY
Krzysztof Gaw edzki
C N R S I H E S Bures sur Yvette France
Intro duction
Over the last decade and a half conformal eld theory CFT has b een one of the main domains
of interaction b etween theoretical physics and mathematics The present review is designed as
an intro duction to the sub ject aimed at mathematicians Its scop e is limited to certain simple
asp ects of the theory of conformally invariant quantum elds in two space time dimensions The
two dimensional CFT exp erienced an explosive develop ement following the seminal pap er
of Belavin Polyakov and Zamolo dchikov although many of its concepts were intro duced b efore
that date It still plays a very imp ortant role in numerous recent developments concerning
higher dimensional quantum elds From the mathematical p oint of view CFT may b e de ned
as a study of Virasoro algebra or algebras containing it of its representations and of their
intertwiners The theory de es however such narrowing de nitions which obstruct the much
wider view that it op ens and into which we o er here only some glimpses In four lectures we
discuss
conformal free elds
axiomatic approach to conformal eld theory
p erturbative analysis of two dimensional sigma mo dels
exact solutions of the Wess Zumino Witten and coset theories
To signal the omissions whose full list would b e much longer let us p oint out that almost no
mention is made of lattice mo dels whose critical p oints are describ ed by CFT s of the p erturbative
approach to string theory based on the two dimensional CFT of sup erconformal theories The
mo dest goal of these lectures is to make the physical literature on CFT b oth the original pap ers
and the textb o oks e g Conformal Field Theory by Di Francesco Mathieu S en echal Springer
more accessible to mathematicians
Lecture Simple functional integrals
Contents
What is quantum eld theory for me
Euclidean free eld and Gaussian functional integrals
Feynman Kac formula
Massless free eld with values in S the partition functions
Toroidal compacti cations of dim free elds baby T duality and mirror symmetry
Compacti ed d free elds the correlators
What is quantum eld theory
Field theory deals with maps b etween space the space time and space M the target These
spaces come with additional structure e g they may b e Riemannian or pseudo Riemannian
manifolds The case of Minkowski signature on is the one of eld theory prop er whereas the
Euclidean signature corresp onds to static equilibrium situations In many cases however for
example for at the passage from one signature to the other may b e obtained by analytic
continuation in the time variable Wick rotation t it and b oth situations may b e studied
interchangeably the Euclidean setup b eing sometimes more convenient
An imp ortant datum of the eld theory is the action a lo cal functional of For example
R
one may consider S jd j dv where the metric structures on and M and a volume on
must b e used to give sense to the right hand side
In the classical eld theory one studies the extrema of the action functional i e maps
cl
satisfying
S
cl
The extremality condition is a PDE for e g the wave or Laplace equation or the Maxwell
cl
Yang Mills or Einstein ones to mention only the most famous cases One should b ear in mind
that non linear PDE s is a complicated sub ject where our ignorance exceeds our knowledge
Following an extremely intuitive reformulation of quantum eld theory QFT by Feynman
the latter consists in studying functional integrals
Z
S
h
D F e
M ap M
Q
where F is a functional of an insertion and D stands for a lo cal pro duct d x
x
of measures on M The ab ove expression is formal and one of the aims of these lectures is to
show that it may b e given sense and even calculated in some simple situations More generally
however the functional integral written ab ove should b e considered as an approximate expression
for structures which live their own lives di erent and in some asp ects more interesting then the
lives of the ob jects whose symb ols app ear in the integral Still although formal and approximate
the functional integral language proved extremely useful in studying the QFT structures It also
made the relation of QFT to classical eld theory quite intuitive unlike in the case of the latter
all maps called often but somewhat abusively classical eld con gurations or classical elds
S
h
in Minkowski give contributions to QFT each with the probability amplitude p e
case these are not probabilities since S should b e taken imaginary but never mind The
classical physics corresp onds to the stationary phase or saddle p oint approximation in which all
contributions to the integral but those of the stationary p oints of S are discarded Such an
approximation is justi ed when the Planck constant h may b e treated as very small as in the
usual macro scale physics but not for example in sup er uid helium
It should b e stressed that in general the relation of quantum to classical is not one to one as
the formulation could suggest and not even many to many except for sp ecial situations e g
with lots of symmetries on b oth levels Such situations are of sp ecial interest for mathematicians
b ecause they allow to reduce QFT to the more familiar classical structures The QFT approach
allowed in many such cases new insights recall for example the use of top ological or quasi
top ological eld theories as factories of invariants This motivates the utilitarian interest of
mathematicians in QFT The very di culty in making sense out of the integrals is at the
origin of a deep er source of mathematical interest of QFT namely in the mathematics of the
new structures carried by QFT The mathematical structures in integrable or conformal two
dimensional eld theories or in four dimensional SUSY gauge theories just emerging provide here
the examples not sp eaking ab out mathematics which promises to underlie the panoply of string
theory dualities
It may b e go o d to remind brie y what do physicists use QFT for
i It provides a relativistic theory of interactions of elementary particles And so Quan
tum Electro dynamics QED describ es interactions of electrons their anti particles p ositrons
and photons the Glashow Weinb erg Salam theory of electro weak interactions describ es at the
same time the b eta decays and QED Quantum Chromo dynamics QCD deals with the strong
interactions quarks forming nucleons The latter two build what is called the standard mo del
of particle physics
nd
ii QFT in its Euclidean version describ es critical phenomena at the order phase tran
sition p oints like that in H O at temp erature T C and pressure p pascals
c c
atm or that in Fe at T C or in the Ising mo del at its critical temp erature
c
The criticality is characterized by slow decay with the distance of statistical correlations whose
asymptotics is describ ed by Euclidean QFT
iii In string theory aiming at uni cation of gravity with the other interactions from p oint
i two dimensional conformal QFT provides the classical and p erturbative solutions and the
the quantum string theory prop er still to b e non p erturbatively de ned may b e considered as
a deformation of quantum eld theory where particles are replaced by strings the typical size of
the string b eing the deformation parameter
iv Finally many QFT techniques are used in the theory of non relativistic condensed
matter
Euclidean free eld and Gaussian functional integrals
In the rest of this lecture we shall describ e how one may give sense to functional integral in
the simplest case of free eld This is the case where the space of maps M ap M is a vector
space inheriting the linear structure from that of M or is a union of a ne spaces and where
the action functional S is quadratic The corresp onding functional integral is Gaussian plus in
nd
the case an easy but interesting decoration in theta functions The adjective free refers
to the absence of particle interactions which is related to linearity of the classical equations
Let b e a Riemannian d dimensional oriented compact manifold and let M R
we consider the Euclidean case The action functional is taken as
Z
jd j m dv G S
L
where dv is the Riemannian volume and the op erator G m is often called the
Euclidean propagator of free eld of mass m
The simplest functional integral of the typ e is the one with trivial insertion F giving
what is called statistical sum or partition function and denoted traditionally by Z
Z Z
G S
D Z e D e
M ap R M ap R
we have put h for simplicity The names come from statistical physics the integral sums
the probabilities probability amplitudes p of all microscopic states of the system The
space M ap R may b e considered as the Hilb ert space with the L scalar pro duct using the
metric volume dv on Were this space nite dimensional the integral would give
Z det G
Q
d
i
p
where are any orthonormal co ordinates on the Hilb ert space of if we normalized D
i
i
maps It is sensible to maintain the ab ove formula as a de nition of the formal functional integral
for the partition function even in the in nite dimensional case It is necessary then to give sense to
the determinant of the p ositive op erator G whose discrete eigenvalues n b ehave
n
d
as O n A convenient but nonunique way to do it is by the zeta function regularization
de ning
G
det G e
P
d
s
extends to a meromorphic function analytic in for Re s where s given as
G
n
n
the vicinity of zero We shall stick to this de nition of in nite determinants throughout the
present lectures
The next functional integrals we may like to compute are the ones for the correlations functions
n
dep ending on a sequence x of p oints in
i
i
R
S
x x e D
n
M ap R
R
h x x i
n
S
e D
M ap R
Again mimicking the case of nite dimensional Gaussian integrals we may de ne the formal
functional integral on the right hand side by setting
for n o dd
G x x for n
G x x G x x G x x G x x
h x x i
n
G x x G x x for n
Q P
G x x for n even
i i
pairings
i i
f i i g
where G x y denotes the kernel of op erator G which is smo oth for x y and exhibits a coinciding
d
p oints singularity ln dist x y for d and dist x y for d Such a de nition of
the correlation functions is additionally substantiated by the fact that there exists a probability
measure d on the space of distributions D such that
G
Z
h x x i x x d
n n
G
D
where the equality is understo o d in the sense of distributions x may b e then considered as a
random distribution
Feynman Kac formula
Some p eople in the audience may wonder what it all has to do with Minkowski space eld theory
involving Hilb ert space H quantum Hamiltonian H and quantum eld op erators acting in H
since the Euclidean functional integral scheme has led us to entirely commutative structures
as the Euclidean random distributions x which may at most b e considered as a distribution
with values in commuting multiplication op erators acting in L D d The relation of the
G
two schemes is provided by the so called Feynman Kac formula Let us start from a simple
quantum mechanical example
Example d L with the p erio dic identi cation of the ends and the standard
metric In this case d is supp orted by the space of continuous functions C L and is
per
G
essentially a version of the Wiener measure More exactly it di ers from the latter by the density
R
L
m
x
Supp ose that x x x L Then e
n
Z
x H x x H x L H
n
tr e e e
x x d
n
G
LH
tr e
C L
per
where
r r
r r
m d d m m
d m
m m a a H
d m d m d
is the Hamiltonian of a harmonic oscillator acting in L R d a and its adjoint a satisfy the
canonical commutation relation
a a
m
and corresp onds to the zero eigenvalue is The ground state of H is prop ortional to e
annihilated by a and the higher H eigenstates are obtained by aplying p owers of a to each
a raising energy i e eigenvalue of H by m a is called the annihilation and a the creation
op erator Hence the sp ectrum of H is f m m g mZ With the use of the orthonormal
q
m
n
p
a comp osed up to normalizations from the Hermite p olynomials H basis
n
n
n
times L R d may b e identi ed with the Hilb ert space completion of the symmetric
algebra S C the b osonic Fo ck space over C Note that
r
a a
m
Problem Consider formula for the p oint function
a Use the Fourier transform to write the left hand side Show that its L limit G x x
exists
b Prove that for x x and complex numb ers
n n
n
X
G x x
k l k l
k l
c What is the L limit of the right hand side of Eq for n
d Show that b oth sides of Eq with n coincide at L Prove b using this result
e Prove relation for n and nite L
xH
It may b e more natural to read the Feynman Kac formula from the right to left e is
nothing else but the transition probability to pass from to in time x which may b e used
to de ne the Markov pro cess x with the measure on the space of continuous realizations
coinciding with d
G
d
Example d L with p erio dic identi cations Now d is carried by genuinely
G
n
L The Feynman Kac x x b e s t x x distributional s Let x x
i i
n i i
formula now takes the form
Z
x H x x H x L H
n
tr e x e x x e
n
x x d
n
G
LH
tr e
d
D L
where the quantum Hamiltonian H is a p ositive self adjoint op erator in the Hilb ert space H a
tensor pro duct of an in nite numb er of harmonic oscillators one for each Fourier mo de of the
k
R
ik x
e x dx classical time zero eld
d
k
L
O
L C d H
k
d
Z k
L
The annihilation op erators
r
r
d
d k
L
a
k k
d
k d L
k
p
where k k m and the creation op erators a adjoint to them satisfy the canonical
k
commutation relations
a a
k k k
k
with all the other commutators vanishing The quantum time zero eld is
r
X
i k x
e a a x
k
d
k
k L
k
The Hamiltonian
X
H k a a
k
k
k
P
k
P
j j
k
d
k
L
as the ground state annihilated by all a The sp ectrum of H is k Z has e
k
k
There are three natural ways to lo ok at the Hilb ert space H
i H is an in nite tensor pro duct of oscillator spaces L C d how should it b e de ned
k
d d
ii H is the Hilb ert space completion of the symmetric algebra S l S L L Z
L
this is the Fo ck space picture
iii H is a space of functionals of variables or of the time zero classical eld x
k
on the vacuum functional obtained by acting by creation op erators a
k
One may intro duce the Minkowski time dep endence of the quantum eld by de ning
itH itH
t x e x e
Problem Show that
r
X
i t k i k x i t k i k x
e a e a t x
k
d
k
k L
k
The in nite volume limit L of the formulae for H and t x may b e easily taken if
d
we intro duce op erators a k L a In the limit one obtains the op erator valued distributions
k
d
d
S L R d k d k dk and a k and their adjoints a k acting in the Fo ck space
satisfying the commutation relations
d
a k a k k k
p
d
By identifying L R d k by multiplication by k with the space of functions on the upp er
d k
mass hyp erb oloid f k k g square integrable with the Lorentz invariant measure one obtains
k
the Minkowski scalar free eld of mass m constructed in more abstract and explicitly Poincare
covariant way in David Kazhdan s lectures check it
Massless free eld with values in S
Let us pass to the next case where is again a general compact d dimensional
manifold M R Z S and the action functional is that of the massless free eld
R
S jd j dv note that has a natural interpretation of square of the radius of the
R
jd j dv but use the metric d on the target circle if we rewrite the classical action as
This is the case of a conformal invariant eld theory with the conformal group acting pro
jectively in the corresp onding Hilb ert space of states transforming covariantly eld op erators
We shall get there slowly discussing in more detail the d case where the conformal group
is in nite dimensional essentially D if f S D if f S Let us start with an elementary
treatment of the functional integral
How should we view the space of maps from to S A convenient way is to de ne
M ap R Z M ap R Z
H om Z
where M ap R is a a function on the universal cover of equivariant with resp ect
to the action of the fundamental group
ax x a for a
Note that this de nition makes sense for maps of arbitrary class smo oth continuous or distribu
H Z with given by the p erio ds of H Z Each tional H om Z
M ap may b e uniquely decomp osed according to
Z
x
h h
x
where is the harmonic representative of H corresp onding to x is the base p oint of
h
and is a univalued function on For the free eld action we obtain
k k S
h
L
L
there is no mixed term why This suggests the following de nition of the functional integral
for the partition function of the system
Z Z
X
k k
S
h
L
L
Z e D e D e
H Z
M ap R
M ap S
X
vol
k k
h
L
e
det
H Z
p
where in det the zero mo de should b e omitted it contributes the factor vol where vol
R
dv to the functional integral why
n dx and an easy calculation see Example d L In this case
per h
L
Problem b elow shows that
d
L det
dx
Hence the d partition function
X X
L
Ln LH L n
e tr e Z e
d
Poisson
det
dx
n Z n Z
resummation
d
in
is the op erator acting in L R Z d with the eigenvectors e n where now H
d
Z corresp onding to eigenvalues n
Problem Prove for x x L and q Z the Feynman Kac formula
n i
Z
n R
L
Y
d dx
iq x x H iq x x H iq iq x L H
n n
i i
e e D tr e e e e e e
i i
where the functional integral over M ap L S on the left hand side is computed as the one
for the partition function Z treated ab ove Infer that the left hand side may b e also expressed
iq x iq x
n n
as the exp ectation h e e i w r t the Wiener measure on the p erio dic paths on
xH
S constructed from the transition probabilities e
Let us discuss in greater detail the case d when is a Riemann surface of genus
h with a xed metric inducing the complex structure of Let us chose a marking of
h h
i
a symplectic bases a b of H Z with the corresp onding basis of holomorphic
i j
i j i
R R
j ij j ij ij
forms The imaginary part of the p erio d matrix is
a
b
i
i
p ositive The equation
t
m n c c
h
i
g
for m n Z gives the harmonic forms in H Z with a p erio ds m and b p erio ds
i i j
n
j
t
k k m n m n
h
L
and the sum over in eq may b e done explicitly After Poisson resummation over n one
obtains the following result
X
k k
h
h
L
e det
Q
H Z
where the theta function is de ned as follows Let E b e the s dimensional Euclidean space
s
Q
Let Q b e a lattice in E E E j j considered with the inde nite metric j j
s s s s
E E
s s
Then
X
i q q i q q
e
Q
h
q q Q
where the decomp osition q q q is according to that of E Ab ove
s s
m n m n
p p
j m n Z g R R Q f
Inserting the relation into eq we obtain
vol det
ln ln h
Z Z e
Q
det
From eq it is obvious that the right hand side is marking indep endent Technically this is
due to the fact that in the inde nite scalar pro duct the lattice Q is even scalar pro ducts are
integers scalar squares are even and self dual
h
We may discard from Z any factor of the form const by the addition to the action of a
R
r dv h Doing term prop ortional to the integral of the scalar curvature r of since
that we discover a somewhat miraculous equality
Z Z
a consequence of the obvious identity Q Q More directly identity follows from the
Poisson resummation formula applied to the left hand side of eq and the fact that the lattice
H Z with the L scalar pro duct is isomorphic to its dual the isomorphism is induced by
the intersection form Eq is the simplest manifestation of the so called T duality which
states that the dimensional massless free elds with values in the circles of radius and of
radius are indistinguishable This identi cation of inverse radia of free eld compacti cation
has a deep meaning in the string theory context and we shall return to it in a later discussion
On the genus curve T C Z Z with in the upp er half plane and the standard
metric
j j det
Q
i in
where e e is the Dedekind eta function
n
Problem a relatively complex calculation going back to Kronecker
R
s t s
annd t e dt show that a Using the identity s
P
s
n for Re s analytically continued where is the Riemann zeta function s
n
elsewhere
b For i with real show using the identity from a and the Poisson
i
resummation that for Re s su ciently large
p
Z
X X
s t n t s s in
A
t e dt s j nj e
s
n
n
Note that the right hand side is analytic in s around s Using easy relations s
R p p
x x t t
and x e obtain t e dt s O s
X
d
s
j nj ln j q j
ds
s
n
i
with the standard notation q e
ln c By taking in the last formula show that
d
on L the zeta regularized determinant d Prove that for the p erio dic b c op erator
dx
d
det L
dx
e Show that the sp ectrum of the Laplacian on the torus C Z Z in the metric jdz j
j m nj for n m Z is given by
m n
f Pro ceeding as in b decomp ose
X X
s s
j m nj j m nj s
m n
m n
Z
p
X X
s m t n t s s imn
A
s t e dt m e s
s
m m n
and show that after analytic continuation
Y X
m s
q j s ln s s ln j s
m n
m
m n
Infer that
Y
m
q j ln ln j
m
and that
j j det
Q
n
where the Dedekind eta function q q
n
Hence in the genus case
Z Z j j Z
Q
The marking indep endence together with the indep endence of Z on the normalization of the
at metric on T see b elow implies that Z is a mo dular invariant function
a b
Z Z
c d
a b
SL Z for
c d
Toroidal compacti cations the partition functions
The ab ove discussions may b e easily generalized to the case of toroidal compacti cations i e to
N N
the case of massless free eld on with values in the N dimensional torus T R Z
P P
i j N i j
b d d on T and g d d and a constant form Fix a constant metric g
ij ij
ij ij
N
de ne the classical action of the eld T as
Z
kd k i S
L
Applying the same metho d as b efore do it results in the formula
N
vol det
Z Z Z
d
d
Q
det
d
T duality again where d d g b and the lattice
ij ij ij
t
dm n
d m n
N N N
p p
j m n Z g R R f Q Q
d
d
N N
is an even self dual lattice in R R with the inde nite scalar pro duct j x y j x g x
y g y At genus
a b
N
Z Z Z j j Z
d d
d
Q
c d
d
Example Let T b e the Cartan torus of a simply laced simple simply connected Lie group
the compact form of the A D E groups By spanning the Lie algebra of T by the coro ots
N
we may identify T with T where N is the rank of the group Let g where tr tr
ij
i j i
is the Killing form normalized so that g We may write
ii
g d d
ij ij j i
for some integers d and set
ij
b d d
ij ij j i
so that d g b The corresp onding action of the toroidal compacti cation coincides mo d
ij ij ij
i with the action of the WZW mo del with elds taking values in the corresp onding simple
R
h
group the term is the remnant of the top ological WZ term De ning for p P
where P is the coweight lattice dual to the ro ot lattice
X
t
i tr p q p q
e
Q p
h
q Q
one obtains
X
j j
Q
Q p
b
h
p P Q
In particular at genus
X X
Q p
jch j Z
d
p
N
p P Q p P Q
Q p
runs through the characters of the level representations of the where ch
N
p
corresp onding Kac Mo o dy algebra In particular for the E case P Q and the partition
function is the absolute value squared of ch which is a cubic ro ot of the mo dular invariant
function j In general the right hand side of eq coincides with the genus partition
function of the level WZW mo del This remains true for higher genera and for the complete
CFT s which is another miraculous coincidence of eld theories with elds taking values in quite
di erent target spaces e g the SU WZW mo del at level is equivalent to the free eld with
values in S of radius
The fact that the toroidal partition function is a nite sesqui linear combination of
expressions holomorphic in is a characteristic feature of rational conformal theories
Problem Show that the free eld compacti ed on a circle of rational radius squared
is rational in the ab ove sense
For free elds with values in the Cartan tori of simply laced groups describ ed ab ove the general
partition functions are hermitian squares with resp ect to Quillen like metric of holomorphic
sections of pro jectively at vector bundles over the mo duli spaces of curves We shall return to
these issues during a more detailed discussion of the WZW mo dels
Example Consider the toroidal compacti cation to T equipp ed with the complex structure
induced by the complex variable T T is in the upp er half plane and with a constant
K ahler metric g R T d d with R and a constant form i R T d d Set
R R iR The partition function of the corresp onding free eld is
vol det
Z Z
R T
Q
det
R T
where
R m T R m T n n
R m T R m T n n
i i
p p
Q f j m n Z g C C
R T
R T R T
with the inde nite quadratic form j z z j jz j jz j Note that Q may b e obtained
T R
from Q by complex conjugation on the second C We infer that
R T
Z Z
R T T R
which is the simplest instance of mirror symmetry claiming identity of CFT s with elds in two
di erent Calabi Yau manifolds with the role of mo dular parameters of complex and p olarized
K ahler structures interchanged
Problem Show that b esides the relation the partition function satis es the identities
Z Z Z Z Z
R T R T R T
R T R T
which imply the separate SL Z invariance in R and T
In general the mo duli space of N dimensional toroidal compacti cations is a double coset
I I I
N N N N N N
j Z O R R R R O N O N O R R
I I I
in a hop efully self explanatory notations which coincides with the mo duli of even self dual
lattices in R R with the inde nite scalar pro duct
The action functional of the compacti ed two dimensional massless free eld uses only
the conformal class of the metric on The regularization of the free eld determinants
reintro duces however the dep endence on the conformal factor of the metric an e ect called
conformal anomaly More exactly one has
det
ln r x
x vol
where r is the scalar curvature of or denoting the metric dep endence of the partition function
by the subscript
N
Z r x Z
e
x
R
s t
t tr e and the Problem Prove the relation using the identity s s
short time expansion of the heat kernel of
t
e x x r x O t
t
Problem Prove that the in nitesimal relation is equivalent to the global one
R
N
r dv kd k
L
Z Z e
e
Toroidal compacti cations the correlation functions
Besides the functional integrals for the partition functions we would like to study the ones for
the correlation functions of the massless eld with values in S of the typ e
Z
n
R
Y
jd j dv
i q x
i i
D e e
i
for integer q see Problem for the d example This may b e attempted by the same strategy
i
as b efore by separating the eld into the harmonic and univalued part as in eq and then
summing over the rst and integrating over the second This gives the expression
Z
P
P
X
k k i q x
i q x
i i
h h
i i
L i
L i
e D e
H Z
The sum over H may b e expressed by partial Poisson resummation as an explicit theta function
As for the functional integral it may b e formally p erformed by mimicking the nite dimensional
formulae
Z
P
P
n
vol
q q G x x
i q x
i j i j
i i
i j
L
i
P
e D e
q
det
i
i
where the Kronecker delta is contributed by the integral over the constant mo de of and
the constant G x y G y x is a Green function of satisfying G x y x y
x
vol
P
ambiguity should drop out ab ove due to the vanishing of q The obvious problem with the
i
ab ove formula is that
G x y ln dist x y nite
when y x so that G x x is not de ned This is a standard problem with the short distance
singularities due to distributional character of typical con gurations in the Gaussian free eld
measure A p ossible treatment is to renormalize the ab ove expression by replacing the divergent
contributions by their nite parts
ln dist x y G x x lim G x y
y x
Up on division by the partition function all this leads to a well de ned renormalized expression
for the correlation functions which we shall denote by
i q x i q x
n n
h e e i
where the colons remind the renormalization pro cedure which is closely related to the Wick
ordering discussed in Kazdan s lectures The correlations dep end on the charges q and p osi
i
tions x but also on the metric on which is signaled by the subscript In particular on CP
i
ln jz x z y j in the standard complex variable and with H one may take G x y
for the metric g e dz dz with a conformal factor e we obtain setting z z x
i i
q q
i j
P
q
Y
i
x
i q x i q x
i
n n
i
P
jz z j h e e i e
i j
q
i
i
i j
The dep endence on the conformal factor of the metric is solely due to the renormalization
and p ersists in general
P
x i q x i q x i q x i q x
n n n n
i i
i
h e e i h e e i e
e
q
i q
i
i
where are the conformal dimensions of the Euclidean elds e The generalization
i
i q
to the toroidal compacti cations is straightforward The conformal dimensions of elds e
N
q g q where q Z is now
The op erator picture of the the free eld compacti ed on S is as follows The quantum
Hib ert space is
Z
H L S d F F
Z
Ab ove L S d is the in nite sum of copies of L S each lab eled by an integer winding
numb er w The Fo ck space F is generated by applying op erators n to a
n
vector annihilated by with n
n
n
n m n m n
n
ipx th
p
and F is another copy of F Let jp w i denote the function e in the w copy of
d
Z
H may b e generated L S d Integer p is the eigenvalue of the op erator p
i d
by applying op erators with negative n to vectors jp w i The multivalued free eld
n n
op erator with time dep endence is given by
X
i
n n
i t x n i t x n
p
e e t x p t w x
n n
n
The relation to the m limit of the massive free eld on p erio dic interval of length L
given by eq should b e evident Mo dulo the constant and winding mo des for n
p p p p
n a i n a i n a i n a i
n n n n n n
n n
The relab eling allows to separate the left moving part involving from the right moving one
n
containing The one handed parts of t x are called chiral elds the zero mo des can b e
separated to o For the uncompacti ed massless free eld the k mo de contributes to the
acting on L R d which has continuous sp ectrum The Hamiltonian the term d d
compacti cation of the eld and consequently of its zero mo de restores the discreteness of the
energy sp ectrum
Hilb ert space H carries a representation of two commuting copies of the Virasoro algebra
with generators L and L Explicitly
n n
X
L
n m n m
m Z
p
w creator to the left of annihilators and similarly for L where we have set
n
p
p and w p The quantum Hamiltonian H L L P L L
generates the space translations j i is the ground state of H it is also annihilated by
P
Problem Show that L s indeed satisfy the Virasoro algebra relations with unit central
n
n n L L n m L
n m n m n m
Let us intro duce the vertex op erators
i q t x
V t x e
q
de ned as the formal p ower series in and reordered by putting the creation op erators
n n
with negative n indices to the left of the annihilation op erators corresp onding to p ositive n and
also op erators to the left of p
A B B A A B
Problem Using the op erator relation e e e e e holding if A B commutes with
A and B show that for t t t
n
q q
i j
Y
P
jz z j V it x V it x
i j q q n n
n
q
i
i
i j
t ix
j j
where z e This together with eq provides a spherical version of the Feynman Kac
j
formula
i t x i t x
In variables z e and z e the commutation relations of L s and L s with the
n n
vertex op erators take the form
n n
L V z z n z V z z z V z z
n q q z q
n n
L V z z n z V z z z V z z
n q q z q
q
where ab ove or q g q stands for the conformal dimension of the op erator Later
on we shall see that these relations essentially follow from eq and the general covariance
of the corresp onding correlation functions In the professional jargon the elds satisfying such
commutation relations are called primary Virasoro op erators
On the level of the Hilb ert space H H T duality b ecomes the unitary transformation
U H H such that
T
uw
U ju w i jw ui and U U U U
T T n n T T n n T
Z iu
p
where for u w Z ju w i denotes the function in the w comp onent of L S d e
U intertwines the action of the Virasoro algebras in H and H and maps the vertex op erators
T
in H to new op erators which should b e considered on the equal fo oting with the original ones
Up to now we have considered only S valued free elds with p erio dic b oundary conditions
In string theory aplications one also considers elds on space with b oundaries and with xed
b oundary conditions like the Neumann ones op en strings Quantizing such elds on space
which is an interval d one obtains quantum eld
X
N N
i
i t x n i t x n N
n n
p
e e t x p t
n n
n
which may b e realized in the subspace of the p erio dic b c Hilb ert space H generated by applying
N
p
with negative n to vectors ju i The T duality maps this eld op erators
n n
n
into the one corresp onding to the Dirichlet b oundary conditions
X
D D
i
i t x n i t x n D
n n
p
e e t x w x
n n
n
D
where is xed mo dulo Z t x acts in the subspace of H generated by applying
D
p
to vectors j w i In toroidal compacti cations with more dimensions of the
n n
n
target one may have mixed D brane typ e b oundary conditions with some co ordinates of the
eld xed to prescrib ed values at the ends of the space interval
Problem Massless fermions on Riemann surface
Let b e a Riemann surface Spin structure on may b e identi ed with the square ro ot
A Dirac spinor is an element of L L L of the canonical bundle K T
where L is the bundle complex conjugate to L The conjugate spinor is L L
and in the euclidean Dirac theory it should b e treated as an indep endent eld
for Ma jorana fermions Denote by the op erator of L and by its complex conjugate
L
L
which may b e naturally identi ed with The action is a function on the o dd vector space
L
L L L L
Z
S
L
L
note that the integrand is naturally a form Partition functions of the Dirac fermions are
given by the formal Berezin integral
Z
S
det det Z e D D det
L L L
L
L
The last determinant may b e zeta regularized giving a precise sense to the partition function Z
L
of the Dirac eld on
On the elliptic curve C Z Z with in the upp er half plane the canonical bundle K
may b e trivialized by the section dz and spin structures corresp ond to the choice of p erio dic or
anti p erio dic b oundary conditions under z z and z z
L pp pa ap aa
a Show that the eigenvalues of are
L
L
j m nj
m n
with
m Z n Z for L pp
m Z n Z for L pa
n Z for L ap m Z
m Z n Z for L aa
b Infer that
Z
pp
c Show that
s
s s s
pa
pa
Infer from Eq that
Y
n
q j Z jq
pa
n
In the Hilb ert space picture
L L
Z tr q q
pa
H H
R R
The Ramond sector Hilb ert space is H H with
R R
H C C
R
n
and H is another copy of H L acts in the rst copy It has eigenvalue on C the
R R
th
Ramond ground states and the o ccupied n mo de in the fermionic Fo ck space adds n to it
The p erio dic partition function is
F F L L L L
Z tr q q str q q
pp
H H H H
R R R R
F
where C corresp ond to the eigenvalues of and each o ccupied
fermionic Fo ck space mo de adds to F Z vanishes since mo des with o dd and even Fermi
pp
numb ers are paired
d Show that
Y
n s s
q j s s s and Z jq
ap
ap
ap
n
The Hilb ert space interpretation is
L L
Z str q q
ap
H H
NS NS
where the Neveu Schwarz sector Hilb ert space is
H C
NS
n
th
The Neveu Schwarz ground state has eigenvalue zero of L and the n o ccupied zero mo de
to it The fermion numb er of the NS ground state vanishes and each o ccupied contributes n
fermionic mo de adds to it
e Show that
Y
n s
q j and Z jq s s s
aa
pa aa pa
pa aa pa
n
and that
L L
q q Z tr
aa
H H
NS NS
f Prove the mo dular prop erties
Z Z Z Z Z Z
pa pa ap aa aa ap
Z Z Z Z Z Z
pa ap ap pa aa aa
Problem Bosonization
The spin structure is called even o dd if the dimension of the kernel of is even o dd Denote
L
by L the parity of L The b osonization formula asserts that
X
L h
Z C Z
L
L
where on the right hand side we have the partition function of the b osonic free eld with values
in the circle of radius squared C is a constant and h the genus of the Riemann surface
These equalities extend to correlations For example the fermionic elds x corresp ond
i x i x
to b osonic elds e and x to e What are their conformal weights Prove
identity for C Z Z using the expression for Z and the classical pro duct
expressions for the theta functions
Y X
n iz n iz n i n inz
q e q e q z j e
n
n Z
What is the Hilb ert space interpretation of the left hand side of Eq on the elliptic curve
In summary by calculating the functional integrals for compacti ed massless free elds we
have constructed mo dels of two dimensional CFT sp eci ed by giving the partition functions and
correlation functions on general Riemann surfaces In the next lecture s we shall examine the
emerging CFT structure on a more abstract level
References
A set of Gaussian integration formulae may b e found e g in Quantum Field Theory and
Critical Phenomena by J Zinn Justin Sects and See also Sects and
for the discussion of of the path integral and the Feynman Kac formula Of course for the
latter topic Quantum Mechanics and Path Integral by Feynman Hibbs is the physics classic
th
Rigorous theory of in nite dimensional Gaussian integrals may b e found e g in the volume
of Gelfand Vilenkin See also Simon s The Euclidean P Quantum Field Theory
The free elds with values in S are discussed brie y e g in the Ginspargs contributions to Les
Houches Scho ol Session XLIV Fields Strings and Critical Phenomena eds Brezin Zinn
Justin or in Drou e Itzykson Th eorie Statistique des Champs InterEditions Sects
and For the case of elds with values in a torus see the pap er by Narain Sarmadi Wittem
in Nucl Phys B p and for the case of complex torus read Vafa s contribution
to Essays on Mirror Symmetry ed S T Yau International Press Hong Kong
Free fermions and b osonization on a Riemann surface are discussed in the pap er by Alvarez
Gaum e Bost Mo ore Nelson Vafa in Commun Math Phys p
Lecture Axiomatic approaches to conformal eld theory
Contents
Conformal eld theory data
Conformal Ward identities
Physical p ositivity and Hilb ert space picture
Virasoro algebra and its primary elds
Highest weight representations of V ir
Segal s axioms and vertex op erator algebras
Conformal eld theory data
In the rst lecture we have discussed a functional integral construction of the simplest mo dels
of CFT the toroidal compacti cations of massless free elds In this lecture we shall present
a more general approach to CFT which although not overly formalized will b e axiomatic in
spirit using only the most general prop erties of the free eld mo dels We shall assume that the
basic data of a CFT mo del sp ecify for each compact Riemann surface its partition function
x Z and a set of its correlation functions h x i of the primary elds from
l l n
n
a xed set f g The correlation functions are symmetric in the pairs of arguments x l are
l i i
de ned for non coincident insertion p oints x and are assumed smo oth We shall also need
i
later some knowledge of their short distance singularities The dep endence of b oth the partition
and the correlation functions on the Riemannian metric will b e assumed regular enough to
assure existence of distributional functional derivatives of arbitrary order The basic hyp othesis
are the following symmetry prop erties
i di eomorphism covariance
Z Z
D
D x h x D x i h x i
l l l n l n D
n n
ii lo cal scale covariance
R
c
r dv kd k
L
Z e Z
e
n
Q
x
i
l
i
x h x h x i x i e
l l l l n n
n n
e
i
where c is the central charge of the theory In i we limit ourselves to orientation preserving
di eomorphism assuming that under the change of orientation of the surface
Z Z
x i x h h x x i
n l l n
l l
n
n
where is an involution of the set of primary elds preserving their conformal weights
l
l
iq iq
e e for the toroidal compacti cations In what follows we shall rst explore the
implications of the ab ove identities which we shall jointly call conformal symmetries Other
imp ortant prop erties of the correlation functions for example those resp onsible for the Hilb ert
space interpretation of the theory will b e intro duced and analyzed later
Let us de ne new correlation functions with insertions of energy momentum tensor by
setting
h T y T y x x i
m l l n
m m n
m m
Z Z h x x i
l l n
n m m
y y
m
where is the inverse metric In complex co ordinates energy momentum tensor
has the comp onents
T and T T T T
z z z z z z z z z z
y T By de nition the correlation functions h T x y x i are distri
l m l n
m m n
butions in their dep endence on y y As we shall see b elow they are given by smo oth
m
functions for non coincident arguments and away from x s but we shall also have to study their
i
distributional b ehavior at coinciding p oints
Conformal Ward identities
Symmetries in QFT are expressed as Ward identities b etween correlation functions Eqs
and are examples of such relations for group like conformal symmetries It is often useful to
work out also Ward identities corresp onding to in nitesimal Lie algebra version of symmetries
We shall do this here for the in nitesimal conformal symmetries The resulting formalism was
the starting p oint of the Belavin Polyakov Zamolo dchikov s pap er The approach presented
here is close in spirit to the article by Eguchi Ooguri to some extend also to Friedan s
Les Houches lecture notes The general strategy is to expand the global symmetry identities to
the second order in in nitesimal symmetries This will b e a little bit technical so you might wish
to see rst the results listed at the end of this section
Let us start by exploring the in nitesimal version of the lo cal scale covariance Using the
de nition we obtain the relation
c
z z z z z z
Z h T i h T i h T i r Z
e z z z z z z
z z z z z z
Note that if jdz j then and Besides the
z z z z z z
scalar curvature of vanishes In such a metric eq reduces to the equality
h T i
z z
which states that energy momentum tensor in a CFT is traceless in the at metric tr T
T It is the rst example of Ward identities expressing the in nitesimal conformal invariance
z z
on the quantum level We shall see further identities of this typ e b elow
called also stress tensor in a static view of the Euclidean eld theory
Notice that if e with around the insertion p oints then the correlation functions
do not change Let us x the complex structure of and holomorphic complex co ordinates
around the insertion p oints of a correlation function Call a metric lo cally at if it is compatible
with the complex structure of and of the form jdz j around the insertions For such a choice
of we shall drop the subscript in the notation for the correlation functions like in eq
We may restore the full dep endence on the conformal factor by using the covariance relations
and For example for h T i we obtain
z z
Z
c
r dv k k h T i h T i
z z z z
z z
L
e
In order to compute the functional derivative on the right hand side we shall need the following
z z z z z z
Lemma Let To the rst order in
z z z z
r
z z
z z z z
Pro of Consider the inverse metric To compute the curvature to
z z
z z
z z
the rst order in we shall change the variables to z z z z so that in the new co ordinate
Since the metric is
z z
z z z z z z z z z z
z z
then retaining only the terms of the rst order in and their complex conjugates we
obtain
z z z z z z z z
x z z z
z
z z z z
z z z z z z
z z z z z z z z
z z z z
z
we have kept more terms then needed for the Lemma for a future use The requirement that
z z
means in the leading order that Hence to the rst order in
z z z
z z
r v i log i dz dz
z z z z z z
z z z z
v
z z
i
th
where v is the new volume form equal to dz dz in the order
R
Using the Lemma and the relation kd k dv we obtain the relation
L
Z
r dv kd k
z
z z
z L
which substituted into eq gives the dep endence on the conformal factors of the exp ectation
value of T
z z
c
h T i h T i
z z z z z
z
e dz dz
What are the transformation prop erties of h T i under holomorphic changes z z f z
z z
of the lo cal co ordinate Under such replacements the notion of a lo cally at metric changes
accordingly By the di eomorphism covariance and eq we have
dz
i h T i h T
z z z z
jdz dz j dz dz
dz
c
h T i log dz dz log dz dz
z z z
z
d z dz d z dz c c
h T i fz z g h T i
z z z z
dz dz dz dz
The function fz z g is the Schwarzian derivative of the change of variables As we see in
the correlation functions with lo cally at metric T do es not transform as a pure quadratic
z z
di erential under general holomorphic changes of variables The transformation law de nes
what is called a pro jective connection on
a b
az b
Problem Show that the Schwarzian derivative fz z g vanishes i z with
c d
cz d
SL C i e for the M obius transformations
The further information ab out the correlation functions with energy momentum tensor inser
tions will b e obtained by studying deviations of the metric from the lo cally at one Applying
to eq the op erator Z at lo cally at and using eq we obtain
w w
Z
c
z w z w h T i h T T i
z z w w z z
z
where stands for the two dimensional function Let us explore now the implications of the
di eomorphism covariance and Under an in nitesimal transformation
D z z z z z
the change in the inverse metric where D may b e read from eq
z z z z z z z z
z z z z
z z z z z z
z z z z
z z z z z z z z
z z z z
The di eomorphism covariance implies that
Z
z z z z z z
h T i h T i h T i dv
z z z z z z
Inserting the expressions for stripping the resulting equation from the arbitrary function
z z
and retaining only the rst order terms in around a lo cally at metric we obtain
z z z z
h T i h T i h T i
z z z z z z z z z
w and z refer to the complex co ordinates of two nearby insertions taken in the same holomorphic chart
z z z z
h T i h T i h T i
z z z z z z z z z
z z
Sp ecializing to we infer that
h T i h T i
z z z z z z
More generally the comp onent T T of energy momentum tensor is analytic anti analytic
z z z z
in correlation functions in a lo cally at metric and away from other insertions Eq is another
conformal Ward identity
At coinciding p oints the correlation functions of energy momentum tensor give rise to sin
Z at lo cally at to eq gularities which we shall study now Application of
w w
Z
gives
z w h T i z w h T i h T T i h T T i
z z z z w w z z z w w z z z w w
Using eq di erentiated with resp ect z in order to replace h T T i in the last relation
z z z w w
we obtain
c
h T T i z w h T i z w h T i z w
z z z w w z z z z w w
z
c
z w z w h T i z w h T i
z w w w w w
z
This is a distributional equation Since z w in the sense of distributions it
z
z w
follows that
c
h T i h T i h T T i
w w w w w z z z w w z
z w z w z w
which is still another conformal Ward identity Since the only solutions of the distributional
equation f are analytic functions one may rewrite the identity as a short distance
z
expansion enco ding the ultraviolet prop erties of the CFT
c
h T T i h T i h T i
z z w w w w w w w
z w z w z w
where stands for terms analytic in z around z w The complex conjugation of eq
gives the singular terms of h T T i this time up to anti analytic terms Expansions of
z z w w
the typ e are usually called the op erator pro duct expansion OPE in accordance with
the op erator interpretation of correlation functions to b e discussed in the next section We shall
follow this terminology
w w
What ab out the mixed insertions Di erentiating Z eq with resp ect to at
lo cally at we obtain
h T T i h T T i z w h T i
z z z w w z z z w w z z z
With the use of the complex conjugate version of eq to eliminate h T T i this reduces
z z w w
to
c
z w h T T i
z z z z w w
z
which stripp ed of gives
z
c
z w h T T i
z z z z w w
i e a contact term with supp ort at z w plus a function analytic in z and anti analytic in w
The other source of singular contributions to the correlation functions of T or T are
z z z z
insertions of the primary elds x Let us compute these singularities Pro ceeding similarly
l
as b efore we apply to Z h x i at and lo cally at obtaining with the help
e l
e
Z
of eqs and the relation
h T w w i z w h w w i
z z l l l
we have replaced the p oint x in the argument of by its lo cal co ordinate w and its complex
l
conjugate to stress the non holomorphic dep endence on x of the x insertion Next we
l
exploit the di eomorphism covariance For D z z z z z and D
h w w i Z Z h w w i
l l
Since for jdz j
z z z z z z
z z
to the rst order in see eq we infer that
z w h w w i h T w w i h T w w i
w l z z z l z z z l
Using the last equation to eliminate h T w w i from eq acted up on by we obtain
z z z l z
the relation
h T w w i z w h w w i z w h w w i
z z z l l l w l
which may b e conveniently rewritten as an OPE of the pro duct of the T comp onent of energy
z z
momentum tensor with a primary eld
l
h T w w i h w w i
z z l l w
z w z w
Finally note that under the holomorphic change of the lo cal co ordinate z z f z
dz
l
j h z z i j h z z i h z z i
l l l
jdz dz j dz dz
dz
or
l l l l
h z z i dz dz h z z i dz dz
l l
so that b ehaves like a form in the correlation functions with lo cally at metric
l l l
One often needs to consider also primary elds with weights and integer or
l l l l
half integer d is the scaling dimension of such a eld and s its spin
l l l l l l
th
Geometrically the correlation functions of such elds are sections of the s p ower of the sphere
subbundle in the cotangent bundle T
Let us collect the relations obtained in this section for low p oint insertions in the correlation
functions Since all the considerations were lo cal the same equalities hold in correlation functions
with other insertions as long as their p oints stay away from the insertions taken together Adding
also the relations involving the complex conjugate comp onents of energy momentum tensor and
intro ducing simpli ed notation T T T T we obtain
z z z z
i identities
T T
z z z z
T T
z z
ii op erator pro duct expansions
c
T w T w T z T w
w
z w z w z w
c
T w T w T z T w
w
z w z w z w
c
T z T w z w
z z
l
T z w w w w
l l w
z w z w
l
T z w w w w
l l w
z w z w
iii transformation laws
c
fz z g dz T z dz T z dz
c
T z dz T z dz fz z g dz
l l l l
z z dz dz z z dz dz
l l
Physical p ositivity and Hilb ert space picture
Up to now we have analyzed abstract conformal elds in the Euclidean formalism probabilistic
in its nature and distinct from the traditional op erator approach The op erator formalism of
QFT ts into the general quantum mechanical scheme with the
i Hilb ert space of states
ii representation of the symmetry group or algebra
iii distinguished family of op erators
as its basic triad This is a fundamental fact of QFT that the passage b etween the Euclidean and
the op erator formalisms which we have discussed already for free elds may b e done in quite
general circumstances This fact is resp onsible for the deep relation b etween critical phenomena
and quantum elds and it has strongly marked the developments of QFT CFT which is not an
exception in this resp ect has largely pro ted from the unity of two approaches In the present
section we shall discuss how the op erator picture may b e recovered from the Euclidean formulation
of CFT presented ab ove assuming the physical or Osterwalder Schrader p ositivity formulated
as a condition on correlation functions on the Riemann sphere CP Analysis of the genus zero
situation will allow to recover the Hilb ert space of states and to translate the op erator pro duct
expansions of the last section into an action of the Lie algebra of conformal symmetries and of
the primary eld op erators in the space of states Later we shall describ e the op erator formalism
on higher genus Riemann surfaces which p ermits to relate naturally CFT in di erent space time
top ologies
Let us consider the map CP CP z z interchanges the disc D f jz j
g with D f jz j g and leaves invariant their common b oundary f jz j g Supp ose that
we are given a Riemannian metric on D compatible with the complex structure which is of
the form jz j jdz j around D we shall call such a metric at at b oundary is a metric
on D and it glues smo othly with on D to the metric on CP Consider formal
expressions
Y
z z X
i i l
i
i
for distinct p oints z in the interior of D with the empty pro duct case included Denote
i
l l
z z z z
l
l
z z
where l l is the same involution that app eared in eq For X as ab ove we set
Y
z z X
l i i
i
i
The physical p ositivity requires that for each family of complex numb ers each family X
of expressions and each family of metrics on D at at b oundary
X
Z h X X i
These prop erties hold for the free eld compacti cations The condition may b e rewritten
using correlation functions with energy momentum insertions and a xed metric Set
T z z T
z
T T z z
z
and extend the de nition to expressions
Y Y Y
Y T z z z T z
m l i i n
i
m n
i
with all p oints in the interior of D and distinct for which
Y Y Y
Y T z z z T z
m l i i n
i
m n
i
One may infer from the prop erty that
X
h Y Y i
where h i denotes the correlation functions in a lo cally at metric
Problem Show that implies
The construction of the Hilb ert space H of states is now simple The expression
X
Y i h Y
de nes a hermitian form on the space V of formal linear combinations of pro ducts Due
D
to this form is p ositive and b ecomes p ositive de nite on the quotient by its null subspace
null
V One sets
D
null
V V H
D
D
We shall denote by the canonical map from V to H by H its image H and by Y the
D
image Y of Y The empty pro duct in gives rise to the vacuum vector The scalar
pro duct is given by
Y Y h Y Y i
H carries an anti unitary involution I mapping vector Y to Y where Y corresp onds to
Y Y Y
Y T z z z T z
m i i n
l
i
m n
l
Virasoro algebra and its primary elds
De ne the action of dilations by q C jq j on the elds by setting
l l
q z q z S T z q T q z S T z q T q z S z z q q
l q q q l
For Y given by we put
Y Y Y
S Y S T z S T z S z z
q q m q n q l l l
i
m n
l
Problem Using the conformal symmetries of the correlation functions verify that
h Y S Y i h S Y Y i
q q
In the Hilb ert space we may de ne the dilation op erator S by the equality
q
S Y S Y
q q
Note that eq implies that S is well de ned on the dense invariant domain H In fact the
q
family of op erators S forms a semigroup S S S Applying many times the Schwartz
q q q q q
inequality identity and the semigroup prop erty of S one obtains following Osterwalder
q
Schrader
j Y S Y j kY k kS Y k kY k Y S Y
q q q q
n
n
n
kY k kY k Y Y S
q q
Assume now that for each there exists a constant C s t
j h Y S Y i j C t
t
when t What it means is that when the distances of a group of insertions are uniformly
shrunk to zero the singularity of the correlation functions is not stronger then the p ower law
given by the overall scaling dimension of the group Using b ound on the right hand side of
and taking n to in nity we infer that
j Y S Y j kY k kY k
q
i e that the dilation semigroup S is comp osed of contractions of H Eq implies now that
q
S The weak continuity of the semigroup S on H follows from that on H which is S
q q
q
evident By the abstract semigroup theory
L L
S q q
q
for strongly commuting self adjoint op erators L and L s t L L Clearly H is
inside the domain of L and of L and
L Y j S Y L Y j S Y
q q q q
q q
It also follows that S H is dense in H for all q
q
L L are only the tip of an op erator iceb erg To see more of it de ne op erators T z
T z and z z with S H as the dense domain jz j by setting
l z
T z Y T z Y T z Y T z Y z z Y z z Y
l l
It is easy to see that the op erators T z T z and z z are well de ned Note that for Y
l
given by eq and with the absolute values of all insertion p oints di erent
Y Y Y
Y R T z z z T z
m l i i n
i
m n
i
where R reorders the op erators so that they act in the order of increasing jz j This is the
reason why the op erator scheme describ ed here is often called radial quantization Under the
conjugation by the anti unitary involution I of H
I z z I z z IT z I T z I T z I T z
l
l
It will b e useful to intro duce Fourier comp onents of the op erators T z and T z
I
n
z T z dz L
n
i
jz j r
I
n
z T z dz L
n
i
jz j r
Since the insertion of T z in the correlation functions h i is analytic in z as long as the
other insertions are not met the matrix elements Y L Y and hence the vector L Y itself
n n
do es not dep end on r as long as r and the contour jz j r surrounds the insertions of Y
similarly for L Notice that
n
I
n
Y L Y z h Y T z Y i dz
n
i
jz j
I
n
Y i dz z h Y T
i z
jz j
where we have moved the integration contour slightly representing T z with jz j as
z T The right hand side is equal to
z
I
n
dz Y Y T z
z z
i z
jz j
I
n
w T w dw Y Y L Y Y
n
i
jw j
It follows that op erators L and L are closable and their adjoints satisfy
n n
L L L L
n n
n n
L s and L s commute with the anti involution I of H It will b e convenient to somewhat
n n
extend the domain of de nition of the op erators intro duced ab ove Let us admit in expressions
H
n
Y of integrated insertions z T z dz and similarly for T z Denote by H the
jz j r
resulting subspace of H Of course H contains H and is invariant under L s and L s
n n
Op erators T z T z and z z may b e clearly extended to S H and we shall assume
l z
b elow that this has b een done
The calculation which we shall do now is an example of an argument which translates certain
OPE s into commutation relations and is used in CFT again and again A devoted student should
we keep the same symb ols for their closures
let us remark that these op erators are not closable so their domains should b e handled with sp ecial care
memorize its idea once for all We start with the OPE for T z which will give commutation
relations b etween L s Let us consider the matrix element
n
Y L T w Y
n
I I
n
dz z h Y T z T w Y i dz
i
jz j jw j jz j jw j
I
c
n
z dz h Y T w T w Y i
w
i z w z w z w
jz w j
where we have used the fact that the order of op erators is determined by the radial order of
insertions in the correlation function In the last line we have collapsed the contour of integration
n
to a small circle around w and inserted the OPE Expanding z around z w
n n n n
n
n n n
z z w w z w w z w w
n n
n z w w w
and retaining only the terms which contribute to the residue at z w in the last integral of eq
we obtain
c
n n
Y L T w Y h Y f n n w n w T w
n
n
w T w g Y i
w
which is the weak form of relations
c
n n n
L T w n n w n w T w w T w
n w
Similarly the OPE implies that
c
n n n
n n w n w T w w T w L T w
w n
By virtue of eq the mixed commutators L T w and L T w vanish
n n
m
Performing a contour integral over w on b oth sides of eq multiplied by z we obtain
the commutation relations
c
L L n m L n n
n m n m n m
The in nite dimensional Lie algebra with generators L and a central element C called the
n
central charge and with relations where c is replaced replaced by C is known as the
Virasoro algebra We shall denote it by V ir It is closely related to the Witt Lie algebra
n
of p olynomial vector elds V ect S on the circle f jz j g with generators l z and
n z
relations
l l n m l
n m n m
More exactly V ir is a central extension of V ect S i e we have an exact sequence of Lie
algebras
C V ir V ect S
where the second arrow sends to C and the third one maps L to l
n n
Eq gives rise to another set of Virasoro commutation relations
c
n n L L n m L
n m n m n m
L s and L s commute Both and hold on the invariant dense domain H H As we
n m
see the Hilb ert space of states H of a CFT carries a densely de ned unitary i e with prop erty
representation of the algebra V ir V ir with central charges acting as the multiplication
by c
The representation theory of the Virasoro algebra has played an imp ortant role in the con
struction of mo dels of CFT We shall include for completeness a brief sketch of its elements in
the next section But why did the Virasoro algebra app ear in CFT in the rst place As we
have mentioned V ir is the central extension of an algebra of vector elds on the circle But
V ect S V ect S may b e identi ed with the Lie algebra of p olynomial conformal vector elds
on the two dimensional cylinder f t x j x mo d g with the Minkowski metric dt dx
M
By de nition the conformal vector elds X satisfy L f for some function f where
X M X M X
L denotes the Lie derivative w r t X The identi cation assigns to generators l and l the
X n n
n n i t x i t x
conformal vector elds z and z resp ectively with z e and z e
z z
In particular i l l is the in nitesimal shift of the Minkowski time t and i l l the
in nitesimal shift of x Hence V ect S V ect S is the Lie algebra of Minkowskian confor
mal symmetries and representations of V ir V ir describ e its pro jective actions realizing such
conformal symmetries on the quantum mechanical level pro jective representations corresp ond
to genuine actions of symmetries on the rays in the Hilb erts space representing pure quantum
states H L L is the quantum Hamiltonian and P L L is the quantum momentum
op erator The unitarity conditions corresp ond to the natural real form of the algebra com
p osed of real vector elds such vector elds are represented by skew adjoint op erators so that
the corresp onding global conformal transformations act by unitary op erators V ect S may b e
viewed as the Lie algebra of the group D if f S of orientation preserving di eomorphisms of
g
the circle Let D if f S denote the group of di eomorphisms of the line commuting with the
shifts by
g
Z D if f S D if f S
g g
The group D D if f S D if f S Z which acts on the light cone variables x
diag
t x is the group of conformal orientation and time arrow preserving di eomorphism of the
Minkowski cylinder V ir V ir action in H integrates to the pro jective unitary representation
of D
We shall need more information ab out the representations of V ir V ir which app ear in
CFT This may b e obtained by studying the b ehavior of the primary eld op erators with resp ect
to the Virasoro algebra action
Problem a Show by employing the contour integral technique that
n n
L w w n w w w w w w
n l l l w l
n n
L w w n w w w w w w
n l l l w l
later we shall see that it is more natural to shift H by a constant
b Using the ab ove relations and eqs and prove that the op erators L L given by
and coincide with the generators of the semigroup S intro duced earlier
q
Eqs and express on the op erator level the prop erties of the Virasoro primary elds
of conformal weights Comparing them to the last two equations of Lecture we infer
l l
t ix t ix
that op erators w w for w w e e should b e interpreted as the imaginary time
l
versions of Minkowski elds Note that the comp onents T z and T z fail to b e Virasoro
primary elds of weights and resp ectively due to the anomalous term prop ortional
to c in the relations and
Recall that as a generator of a self adjoint semigroup of contractions the self adjoint op era
tor H L L has to b e p ositive In Minkowskian QFT with Poincare invariance the p ositivity
of the Hamiltonian implies the sp ectral condition H P where P is the momentum op
erator The same is true in CFT with its Hilb ert space corresp onding to cylindrical Minkowski
space The Virasoro commutation relations imply
L L
Indeed Let E b e a non vanishing joint sp ectral pro jector of L and L corresp onding to
B
eigenvalues in a small ball B with the L eigenvalues negative and such that E
B
Then for any normalized vector with E we have
B
L L E E L
B B
On the other hand
L L L L L L L
which shows that L cannot have negative sp ectrum Similarly for L Hence only p ositive
energy representations of the Virasoro algebra with L L app ear in CFT with
the Hilb ert space interpretation The techniques of CFT apply however also to certain scaling
limits of statistical mechanical mo dels without physical p ositivity where a wider class of Virasoro
representations intervenes
Relations and provide further sp ectral information ab out L and L They imply
the equalities
L n L n
n n
L n L n
n l n l
L L
l l l l l l
where by de nition lim z z In particular it follows that the vacuum vector
l l
z
is an eigenvector of L and L with the lowest p ossible eigenvalues We shall assume that it
is a unique vector up to normalization with this prop erty although there are CFTs without this
prop erty In fact is annihilated by the sl sl Lie subalgebra generated by L L L and
L but not by the entire symmetry algebra V ir V ir of the theory the conformal symmetry
is sp ontaneously broken
are also eigenvectors of L and L with eigenvalues and it follows that
l l l
In fact vectors are annihilated Also I and I
l l l l
l
by all Virasoro generators with p ositive indices The eigenvectors of L L and C with such
prop erty are called Virasoro highest weight HW vectors
Summarizing we have shown that the Hilb ert space of states in a CFT carries a densely
de ned p ositive energy representation of two commuting copies of the Virasoro algebra with the
same central charge c The primary eld op erators z z applied to the vacuum b ecome in
l
the limit z HW vectors of the Virasoro representations
Highest weight representations of V ir
For completeness we include a brief sketch of representation theory of the Virasoro algebra
An imp ortant class of representations of the Virasoro algebra is constituted by the so called
L L
highest weight HW representations Let CL CC N CL N CL
n n
n n
V ir N N is the triangular decomp osition of the Virasoro algebra Let the
dual space to C c L A V ir mo dule representation M is called a HW
c
mo dule of HW if there exists a vector v V such that
N v
U N v M
c
x v x v for x
where U denotes the enveloping algebra v is called the HW vector c the central charge and
the conformal weight of the HW representation It follows that M is the linear span of the
c
vectors L L L v with n n n but these vectors are not necessarily
n n n r
r
r
r
P
n is called the level of the vector L L L v A level linearly indep endent N
i n n n
r
r
i
N vector is an eigenvector of L with eigenvalue N We shall denote the subspace of the
N
Clearly vectors of di erent levels are linearly indep endent thus we level N vectors by M
c
have
M
N
M M
c
c
N
N N
with the dim M p N the partition numb er of N A HW mo dule with dim M p N
c c
for all N i e where all the vectors of the form L L L v with ordered n s are
n n n i
r
r
linearly indep endent is called the Verma mo dule V It exists for all complex c and is
c
unique up to isomorphisms In the analysis of the HW mo dules of the Virasoro algebra an
N
imp ortant role is played by singular vectors A non zero vector v of level N is called
s
N
singular if L v for all n Any vector v with L v for all n is a a sum of
n s n s
s
N
singular vectors its non zero homogeneous comp onents Any singular vector v generates a
s
submo dule in V isomorphic to V We have
c c N
i Any submo dule of V is generated by its singular vectors
c
ii Any HW mo dule M is isomorphic to a quotient of the Verma mo dule V
c c
iii The factor mo dule of the Verma mo dule by the maximal prop er submo dule is the unique
up to isomorphisms irreducible HW mo dule H
c
A V ir mo dule M is called unitary if there exists a p ositive hermitian scalar pro duct
on M s t
v L w L v w for all n Z and for all v w V
n n
It follows then that v C w C v w for all v w V and that the eigenvalues of C and
L are real On each Verma mo dule V with c and real there exists a unique hermitian
c
Shap ovalov form h i s t hv L w i hL v w i for all v w V and that hv v i
n n c
De ne Nul l fv V j hv w i for all w V g
c c c
N N N N
i hv v i if v v is a level N N vector and N N
ii any singular vector of p ositive level b elongs to Nul l
c
iii Nul l is the maximal prop er submo dule of V i e H V Nul l
c c c c c
Let us investigate the conditions under which H is a unitary V ir mo dule or equivalently
c
under which the hermitian form on H induced by the Shap ovalov form is p ositive Since
c
c
n n necessarily c and Now consider the two n hL v L v i n
n n
v and w L v and supp ose that c The Gram determinant level vectors v L
n
n
hv v i hv w i
det n n
hw v i h w w i
Thus for c a necessary and su cient condition is c corresp ond to the trivial
one dimensional representations So there exists no non trivial unitary HW representation of the
Lie algebra V ect S of the p olynomial vector elds on the circle It is enough to study the
N N
of the given level Let m p ositivity of the Shap ovalov form restricted the the subspaces H
c c
denote the p N p N matrix with the entries h L L L v L L L v i
n n n n
n n
r
r
r
r
P P
where n and n are ordered and n n N Unitarity of H is equivalent to the
i i c
i i
N N
conditions m for all N In particular det m has to b e non negative for all N when
c c
H is unitary Direct calculation for level and gives
c
det m
c
c c det m
c
N
A general formula for the determinant of m was given by Kac and was proven by Feigin and
c
Fuchs
Y
N
p N r s
det m m
N r s
c
r s N
r s N
m r ms
where m m is a ro ot of the equation c and
r s N
m m m m
Q
n r s
s
r s with n r s denoting the numb er of partitions of N in which r app ears s
r s N
r s N
times
N
For m go es to a diagonal matrix with p ositive entries From the Kac formula
c
N N
it follows that det m for c Therefore m is non degenerate and p ositive
c c
for c and non negative for c Thus V is irreducible for c
c
and H is unitary for c Since for c
c
p N r s
Y
r s
N
det m
N
r s N
n
it follows that V is irreducible if and only if n Z
For c the situation is more interesting providing the rst example of the selective
p ower of conformal invariance
Theorem For c with c and unitarity of the irreducible HW representation
H requires that c b elong to the following discrete series of p oints
c
c
m
m m
r m
m r ms
s r
m
r s
m m
The theorem was proven by Friedan Qiu Shenker by a careful analysis of the geometry of lines
c m m in the c plane This in conjunction with the Kac determinant formula
r s
allowed subsequent elimination of p ortions of the c plane were negative norm vectors app ear
by an induction on N At the end only the p oints listed ab ove were left Go ddard Kent Olive
constructed explicitly unitary irreducible HW representations of the Virasoro algebra for the
ab ove series of c employing the so called coset construction
All unitary HW representations integrate to pro jective unitary representations of D if f S
in the Hilb ert space completion of H
c
All unitary representations M of the Virasoro algebra s t the closure of L is a p ositive
self adjoint op erator with a discrete sp ectrum of nite multiplicity in the Hilb ert space completion
M of M is essentially isomorphic to a direct sum of unitary representations the isomorphism
M We shall may require a di erent choice of the common invariant dense domain for L s in
n
L L
see in the next section that in CFT the op erators S q q should b e trace class for jq j
q
from which it follows that L and L have a discrete sp ectrum of nite multiplicity Hence
the Virasoro algebra representations which app ear in CFT are direct sums of the unitary HW
representations
The algebra V ect S V ect S and hence also V ir V ir acts naturally on the space
of forms f dz dz on D n f g by Lie derivatives
n n n n
l f n z f z f l f n z f z f
n z n z
The commutation relations and express the fact that the op erators intertwine the
l
l l
action of V ir V ir in H H and H This gives a representation theory interpretation
of the primary elds of CFT
Segal s axioms and vertex op erator algebras
Up to now we have studied QFT on closed Riemann surfaces Let us consider now a compact
Riemann surface connected or disconnected with the b oundary comp osed of the connected
comp onents C i I We shall parametrize each C in a real analytic way by the standard
i i
unit circle S C Comp onents C may b e divided into in and out ones dep ending on
i
whether the parametrization disagrees or agrees with the orientation of C inherited from
i
This induces the splitting I I I of the set of indices i We may invert the orientation
in out
of C by comp osing its parametrization with the map z z of S To with parametrized
i
b oundary we may uniquely assign a compact surface without b oundary by gluing a copy
of disc D to each b oundary comp onent C with i I and a copy of disc D to each C
i in i
with i I Conversely given a closed Riemann surface with holomorphically emb edded
out
disjoint discs D and D lo cal parameters by removing their interiors we obtain the surface
with b oundary parametrized by the standard circles f jz j g We shall call a metric h
on compatible with its complex structure at at b oundary if for each i it is of the form
jz j jdz j in the lo cal holomorphic co ordinate around C extending its parametrization We shall
i
consider only such metrics on surfaces with b oundary
First let us note that a metric on and metrics on D all at at b oundary give rise
i
by gluing to the metric
i i I i I i
out
in
on It follows from the lo cal scale covariance formula that the combination of partition
functions
Y
Z Z Z
i i
i I
is indep endent of the conformal factors of Besides the transformation of Z under e
i
with vanishing around the b oundary is still given by eq It is sensible to call Z the
partition function of the Riemann surface with b oundary Let Y H Consider the matrix
i
elements de ned by
Y Y
Y Z h Y A Y Y i
i I i i i I i i
out
in
i I i I
out
in
We shall p ostulate that the ab ove formula de nes op erator amplitudes A mapping the tensor
pro ducts of the CFT Hilb ert spaces asso ciated to the b oundary comp onents of In particular
for without b oundary eq should b e read as the identity A Z Amplitudes A
or rather their holomorphic counterparts were considered as the de ning data of CFT in the
work of Segal Let us state the Segal axioms adapted to the real setup
i We are given the Hilb ert space of states H with an anti unitary involution I and for each
compact Riemann surface with parametrized b oundary or without b oundary connected
or disconnected the op erator
O O
A H H
i I i I
out
in
which we assume trace class
ii If is a disjoint union of and then
A A A
iii If D is a di eomorphism reducing to identity around the parametrized b ound
ary then
A A
D
iv If denotes the Riemann surface with conjugate complex structure and opp osite orien
tation then
y
A A
v The inversion of b oundary parametrization acts on the amplitude A by the isomorphism
H induced by I in the corresp onding Hilb ert space factor H
b oundary comp onents then and C vi If is obtained from by gluing of the C
i i
A A tr
i i
where tr denotes the partial trace in the H factors corresp onding to C and C
i i i i
vii For any function on vanishing around the b oundary
R
c
r dv kd k
L
A e A
e
In the approach in which we start from the data sp ecifying partition functions and correlation
functions on surfaces without b oundaries prop erty i in conjunction with eq imp oses
certain new regularity conditions on the correlation functions Prop erty ii may b e viewed as
a de nition of the amplitudes for disconnected surfaces All the remaining prop erties except
for vi follow easily The gluing axiom brings an essential novel element allowing to compare
the correlation functions of di erent surfaces with di erent complex structures and di erent
top ologies In the heuristic approach employing functional integrals it expresses lo cality of the
latter Let us explain this p oint in more detail
the empty tensor pro duct should b e interpreted as C
We shall think ab out the partition function of a QFT on a closed Riemann surface as b eing
given by a functional integral of the typ e
Z
S
Z e D
Q
d x is the formal volume element on where S is the action functional and D
x
the space of elds On a Riemann surface with b oundary we could consider an analogous
functional integral with xed b oundary values j of the elds
C i
i
Z
S
A e D
i i I
j
i
C
i
It is a function of the eld b oundary values The space of functionals F of the elds on the
circle with the formal L scalar pro duct may b e thought of as the Hilb ert space H of states of
the system we have seen a concrete realization of this idea for the free compacti ed eld We
may then interpret the functional A as the kernel of an op erator mapping the tensor
i i I
pro duct of state spaces one for each in comp onent of the b oundary to the similar pro duct for
the out comp onents
F The space H may b e equipp ed with a formally anti unitary involution I IF
where z z which allows to turn the op erators in H into bilinear forms and vice
versa and to identify the op erator amplitudes when we invert the orientation of some b oundary
comp onents
The basic formal prop erty of the functional integral is that one should b e able to compute
it iteratively Supp ose that the surface is obtained by identifying a parametrized out
of a connected or disconnected surface The with an in comp onent C comp onent C
i i
functional integral on may b e done by keeping rst the value of the eld on the gluing circle
xed and integrating over it only subsequently In other words the equality
Z Z Z
S S
e D D D e
j i i i j
i i
C C
i i
i i i
j j
C C
i i
should hold and this is exactly a formal version of the gluing prop erty vi
What is the functional integral interpretation of eq One should interpret vectors Y H
as corresp onding to functions
Z
S
D
F Z Y e D
Y
j
D
of elds on the circle The complex conjugate functions are then given by the D functional
integrals
Z
S
D
Z Y e D
j
D
so that the formal L scalar pro duct for the functions of elds on the b oundary circle gives
Z Z
S
D D
D Y Y e F F D Z
Y Y
h Y Y i Y Y
has b een assumed lo cally at ab ove Formula follows now by iterative calculation of the
functional integral
Z
Y Y Y Y
S
D Y Y e Y Y i Z h
i i i i
i I i I i I i I
out out
in in
by rst xing the values of on and then integrating over them
Any two dimensional QFT should give rise to op erator amplitudes with prop erties i to vi
R
In particular the so called P theories corresp onding to the actions S k k P dv
L
where P is a b ounded b elow p olynomial give undoubtly rise to such structures The sp ecial
prop erty which distinguishes the CFT mo dels from other two dimensional QFT s is of course
the lo cal scale covariance vii conjecturally sp ecial limits of the P theories should p ossess
this prop erty
As already stressed if we consider eq as the de nition of the amplitudes A then the
prop erties i to vii ab ove b ecome further requirements on the correlation functions The p oint
of Segal was however that the amplitudes A satisfying i to vii may b e taken as the de ning
ob jects of CFT from which the rest of the structure like the Virasoro algebra action the primary
elds and their correlation functions etc may b e extracted In this sense the requirements i
to vii may b e viewed as the basic axioms of CFT encapsulating its mathematical structure
Let us brie y sketch Segal s arguments They require lo oking at the amplitudes A for the
simplest geometries
Discs
For D A H and Z A is a metric indep endent vector which according
D D
to eq we should interpret as the vacuum vector
Z A
D
Similarly A is the linear functional on H
D
Z A
D
It follows from the prop erties iv and v that I
Annuli
Let us pass to surfaces with annular top ology Under gluing of an out and an in
b oundary comp onents of two such surfaces their amplitudes A form a semigroup due to the
prop erty vi This is the semigroup which enco des the Virasoro action on H In particular
for the annuli f jq j jz j g C with the out b oundary parametrized by z z and
q
the in one by z q z using the metric jz j jdz j on C we obtain a semigroup almost
q
identical to the semigroup S considered b efore
q
Y A Y Z h Y S Y i Z Y S Y
q q
C
q
where Z is given by eq
c
Problem Show that Z q q
The immediate consequence of this relation and of eqs and is the identity
L c L c
A q q
C
q
iz
Note that the mapping z e establishes a holomorphic di eomorphism b etween the nite
i
cylinder C f z j Im z g Z and C where q e The pullback of the metric
q
iz H i P
by z e is jdz j The amplitude A is equal to e e where i
C jdz j
and H is the Hamiltonian of the theory quantized in p erio dic volume R Z and P is the
momentum op erator Comparison with eq yields
c
P L L H L L
The requirement that the amplitudes b e trace class op erators implies that L and L have dis
crete sp ectrum with nite multiplicity so that their eigenvalue zero corresp onding to eigenvector
is isolated with p ossible multiplicity Note that energy of the vacuum state b ecomes equal
c
to now If we work on the space R LZ instead the energy sp ectrum is multiplied by
L
c
This p ermits to calculate following Cardy the central so that the lowest energy b ecomes
L
charge of the conformal mo dels from the nite size dep endence of the ground state energy in the
p erio dic interval
Gluing together the b oundaries of C one obtains the elliptic curve T C Z Z
with metric jdz j The prop erties iii and vi of the op erator amplitudes imply then the
following expression for the toroidal partition function
L c L c
Z A tr q q
T jdz j
which is necessarily a mo dular invariant function recall why The mo dular invariance of the
partition functions of CFT has played an imp ortant role in the search and the classi cation of
mo dels
The amplitudes for other annular surfaces allow to obtain the action of other generators of the
double Virasoro algebra in the Hilb ert space of states H If D z f z D is a holomorphic
emb edding of D into its interior preserving the origin then D n f int D is in a natural
f
way an annulus with parametrized b oundary one in f D and one out D comp onent
n
z z
z z
Such annuli form a semigroup In particular for f z q e z for n jq j
q n
and j j su ciently small
L L L L
n n
q q e Z e A
f
q n
enco ding the action of the op erators L L for n the complex conjugate annuli
n n f
q n
give amplitudes enco ding L L for n The Virasoro HW vectors X H of conformal
n n l
weights may b e characterized by the equalities
l l
df df
l l
X Z A X
l l
dz dz
f
In particular L X X L X for n and similarly for L s
l l l n l n
Each Virasoro HW vector gives rise to a primary eld and the correlation functions of the
primary elds may b e recovered from the op erator amplitudes by the following construction Let
x x b e a sequence of non coincident p oints in the surface without b oundary Sp ecify
n
lo cal parameters at p oints x by emb edding discs D into so that their images centered at
j
p oints x do not intersect may b e then viewed as a surface with b oundary capp ed with
j
corresp onding to the HW vectors discs D The correlation functions of the primary elds
l
i
X may b e de ned in a lo cally at metric by the formula
l
i
x h x i A X
l l n i l
n
i
Z
in accordance with relation The characteristic prop erty of the HW vectors assures
form in variable x that the left hand side is unambiguously de ned as a
j l l
j j
Problem Using relation and the gluing axiom show that the right hand side of eq
df df
th l l
j j
under the change z f z of the j lo cal parameter of picks only a factor
dz dz
surface
As we see the relation b etween the primary elds and the HW vectors discovered
l l
b efore is one to one if we include among the primary elds the trivial identity eld whose
insertions have no e ect in correlation functions and which corresp onds to the HW vector
Pants
More generally it is p ossible to asso ciate to each vector X in the Hilb ert space of states H
L L
in the domain of q q for some q with jq j an op erator valued eld X w w
de ned for jw j in the following way For jq j jw j jq j jq j consider the
Riemann surface
P f jq j jz j jz w j jq j g
q q w
with the out b oundary comp onent f jz j g parametrized by z z and the in ones by
z q z and by z w q z Such a surface is often called pants for obvious reasons The
op erator X w w will b e essentially de ned as the amplitude of the pants More exactly we
shall put
L L L L
X w w Y A q q Y q q X
P
q q w
Z
Note that X w w Y is indep endent of q and of q as long as X is in the domain of
L L L L
q q and Y in the domain of q q
Problem a Show that X lim X w w X
w
b Show that for a HW vector X X w w coincides with the op erator w w assigned
l l l
to the primary eld corresp onding to X
l l
c Show that L w w T w and that L w w T w
The op erators X w w satisfy an imp ortant identity
Y z z X w w Y z w z w X w w
holding for jw j jz j jz w j Eq follows from the two ways that one may
obtain the disc with three holes by gluing two discs with two holes
Problem Prove eq using the gluing prop erty vi of the op erator amplitudes and treating
with care the normalizing factors Z
The relation may b e viewed as a global form of the OPE s the lo cal forms following from it
by expanding the vector Y z w z w X into terms homogeneous in z w and z w
Since any Riemann surface can b e built from discs annuli and pants the general amplitudes
A may b e expressed by the Virasoro generators and op erators X w w This p ermits
to formulate the basic mathematical structure of CFT in an even more economic and more
algebraic way than through the amplitudes A with the prop erties i to vii getting rid
h
of the Riemann surface burden Instead one obtains a set of axioms for the action of the
V ir V ir algebra and of the vertex op erators X w w in the Hilb ert space H with the
relation the main consistency condition playing a prominent role That was essentially
the idea which in the holomorphic version of CFT has led to the concept of a vertex op erator
algebra develop ed by Frenkel Lep owsky Meurman and by Borcherds The latter by studying
the algebras arising in the context of toroidal compacti cations on Minkowki targets was led
to the concept of generalized Kac Mo o dy or Borcherds algebras which promise to play an
imp ortant role in physics and mathematics
References
The op erator pro duct expansion in the two dimensional CFT is a starting p oint of the seminal
Belavin Polyakov Zamolo dchikov pap er in Nucl Phys B p see also Eguchi
Ooguri Nucl Phys B p
The treatment of the op erator formalism based of the notion of physical p ositivity was adapted
from Osterwalder Schrader Commun Math Phys p and p
For the theorem ab out the unitary representations of the Virasoro algebras see Friedan Qiu
Shenker Phys Rev Lett p and Commun Math Phys p
and also Go ddard Kent Olive Commun Math Phys p
For the Segal axiomatics of CFT see G Segal s contribution to IXth International Congress
in Mathematical Physics Simon Truman Davies eds Adam Hilger and G Segal The
de nition of conformal eld theory preprint of variable geometry A general discussion of CFT
designed for the mathematical audience may b e found in Gaw edzki Asterisque
p
Vertex op erator algebras are the sub ject of Frenkel Lep owsky Meurman s b o ok Vertex Op
erator Algebras and the Monster Academic Press see also a review by Geb ert in Int J
Mo d Phys A p
Lecture Sigma mo dels
Contents
PI e ective action and large deviations
Geometric sigma mo dels
Regularization and renormalization
Renormalization group e ective actions
Background eld e ective action
Dimensional regularization
Renormalization of sigma mo dels to lo op
Renormalization group analysis of sigma mo dels
PI e ective action and large deviations
It will b e useful to describ e another relation b etween a eld theoretic and a probabilistic concept
Consider a p ositive measure
S
d e D
on a nite dimensional euclidean space E We shall assume that S grows faster than linearly
at in nity and for convenience that the measure is normalized The characteristic functional of
d
R
W J h J i
e e d
P
N
is then an analytic function of J in the complexi ed dual of E Let N b e the
j
sum of N indep endent random variables equally distributed with measure d The probability
distribution of N is
N N
R
Q P
d P N
j j N
j j
P
N
R R R
Q
N h J i NW J h N J i
j
d e D J DJ e
j
j
with DJ denoting the prop erly where the J integration is over an imaginary section of E
C
normalized Leb esque measure on it We shall b e interested in the b ehavior of P for large N
N
It is not di cult to see that
h J i W J o N N inf
N o N
J E
e P e
N
th
another reading of eq says that the N fold convolution b ecomes the N p ower in the Fourier language
sup h J i W J h J i W J
W J J E
W denotes the derivative of W is called the large deviation rate function It is the Legendre
transform of W J which is a strictly convex function on E It controls the regime where
P P
O N as opp osed to the central limit theorem which prob es h i O N where
j j
h i is the mean value of Since J the minimum of o ccurs at W h i
j
The central limit theorem sees only the second derivative of at h i
h i
lim P h i N e
N
N
W may b e recovered from by the inverse Legendre transform
W J sup h J i h J i
J E
We may compute W J and as formal p ower series intro ducing a co e cient h the Planck
constant Taylor expanding S and separating the quadratic contribution to it
Z
h J i S
W J
h
D e e
h
Z Z
S h J S i S S
h h h h
e d e D e
h S
h
where the Gaussian measure
S
h
D e
d
R
h S
S
h
e D
Expanding the exp onential under the d integral into the p ower series and p erforming the
Gaussian integration we obtain an expansion in p owers of h which as discussed in Kazhdan s
lectures gives up on exp onentiation the relation
P
flo ops of Gg
S h I J S
G
Z
h
vacuum
S
W J
graphs G
h
D e e e
h
where by de nition graphs G are connected graphs made of leg vertices J or S leg
vertices S leg vertices S etc with propagators S on the internal lines
and no propagators on the external lines The vacuum graphs are the ones without external
lines The amplitudes I J S are asso ciated to the graphs in a natural way with the symmetry
G
factors of the graphs included If S is a p olynomial and S then there is only a nite
numb er of graphs with a given numb er of J vertices and a given numb er of lo ops and we may
recall that the exp onential of the sum over connected graphs is the sum over connected and disconnected
graphs
we count lines ending at the leg vertices as internal
view W J as a formal series in J and in the numb er of lo ops More exactly comparing the left
and the right hand sides of eq we infer that
R
P
S
D W J S I J S ln e
G
vacuum
graphs G
As discussed by Kazdan by cutting all the lines of the graphs whose removal makes the graph
disconnected we obtain the second representation for W J
X
W J I J
T
vacuum
trees T
where the PI e ective action is de ned by its formal Taylor series
X
R
P
S
D I S S I S ln e
G G
PI vacuum
PI graphs G
graphs G
with external line
X
P
I S S I S
G G
PI graphs G
PI graphs G
with external lines
with external lines
where PI stands for amputated particle irreducible graphs without J vertices Rewriting
R
h J i
h
eq with S replaced by i e as an expansion for e D but keeping only the
leading terms at h small we obtain nally the equality
X
sup h J i I J W J
T
vacuum
E
trees T
Comparing this to eq we see that eqs provide a p erturbative interpretation of the large
deviation function
Geometric sigma mo dels
We have already discussed a simple way to write down a conformal invariant action for maps
M where is a Riemann surface and M g a Riemannian manifold The
functional
Z
i
i j
g kd k S
ij g
L
summation convention dep ends only on the conformal class of We could add to S also
g
a top ological term
Z Z
i i
i j
b S
ij top
i j
involving a form b d d on M which do es not dep end on but only on the
ij
orientation of Renormalizability forces addition of two more terms to the action which break
classical conformal invariance
Z Z
u dv and S w r dv S
dil tach
where u w are functions on M called resp ectively the tachyon and the dilaton p otentials dv
stands for the volume measure and r for the scalar curvature of
One may also consider a sup ersymmetric version of the mo del see Problem set with the
action
Z
SUSY
g D D dz dz d d S
ij
g
i
where the sup er eld
F
and D D In comp onents after elimination of the auxiliary eld F
z z
through its equation of motion one obtains
Z
i
SUSY i j i j i j
S g g r r
ij z z ij z z
g
l k i j
dz dz R
ij k l
j j
j
j i k l
where r with f g standing for the Levi Civita connection
z z z
k l k l
k l
i
and where R denotes the curvature tensor of M Addition symb ols and similarly for r
ij k l z
i j
of the form b d d term corresp onds to the change g g b in eq In
ij ij ij ij
the comp onent formula it results in the same replacement of g and additionally in the
ij
j j
by symb ols of a metric connection with replacement of the Levi Civita symb ols in r r
z z
j
j m
torsion f g g H resp ectively where H is the antisymmetric tensor representing d
k lm k lm
k l
The curvature R in eq b ecomes that of the connection with the plus sign
ij k l
The two dimensional eld theory with action is usually called a sigma mo del The
stationary p oints of S are harmonic maps from to M and corresp ond to the classical
solutions Can one quantize sigma mo dels by giving sense to functional integrals
Z
S
F e D
M ap M
Q
with S S S S S where for F one may for example take u x for
g top tach dil j j
j
some functions u on M We have already seen that this was easily doable for M a torus with
j
a constant metric and a constant form with vanishing or constant tachyon and dilaton
p otentials The corresp onding functional integral was essentially Gaussian and the resulting
theory was a little decoration of the free massless eld Here we would like to examine the case
with an arbitrary top ology and geometry of the target by treating the functional integrals of
the typ e in p erturbation theory and also p ossibly going b eyond the purely p erturbative
considerations employing p owerful metho ds of the renormalization group approach to quantum
eld theories
the names come from the string theory context
it is equal to R for the minus sign connection
k lij
Regularization and renormalization
We may anticipate problems with the de nition of functional integrals even in a p erturbative
approach We shall attempt to remove these problems by using freedom to change the parameters
of the theory namely the metric on M and the tachyon and dilaton p otentials for the sake of
simplicity we shall discard the top ological term in the action The strategy to make sense of
functional integrals of typ e will then b e as follows
regularization we mo dify the theory intro ducing a short distance cuto into it to
make functional integral exist
renormalization we try to cho ose the metric g and the tachyon and dilaton p otentials
entering the action in a dep endent way so that the cuto versions of integrals p ossibly after
further multiplication by a dep endent factor converge to a non trivial limit
There are many ways to intro duce a short distance ultra violet cuto into the theory To
simplify the problem further let us assume that is the p erio dic b ox L that will do away
with the contribution of the dilaton p otential One p ossibility to intro duce the UV cuto is to
consider the lattice version of the sigma mo del Let L b e comp osed of p oints with
Z where L is a p ower of The lattice version of L M is the map co ordinates in
M and for the cuto action we may put
X X
x y d u x S
g g u
x y
x
jx y j
where d stands for the metric distance on M If is the restriction of a xed smo oth p erio dic
g
M valued map on L and u is continuous then in the limit we recover the value
of the original action S S S The cuto version of the normalized integral with
g tach g u
Q
F u x b ecomes now
j j
n
R
Q
S
g u
u x e D
j j g
j
R
S
g u
e D
g
Q
dv x with dv denoting the metric volume measure on M The integral where D
g g g
x
N
is nite e g for compact M and u say continuous The lattice sigma mo dels for M S
j
N
with a metric prop ortional to that of the unit sphere in R are essentially well known spin
mo dels in classical as opp osed to quantum statistical mechanics N corresp onds to the
Ising mo del N to a version of the XY mo del N to a slightly mo di ed classical
N
Heisenb erg mo del u prop ortional to a co ordinate in R describ es the coupling to the magnetic
eld
We would like to study if after renormalization the cuto may b e removed in the correlation
functions More precisely we would like to show that the limits
n
R
Q
S
g u
D Z u x e
j j
g
j
lim
R
S
g u
e D
g
from the p oint of view of statistical mechanics which reformulates the problem in terms of the system with a
xed lattice spacing this is a question ab out the large distance b ehavior of correlation functions
exist for a cuto dep endent linear map Z on the space of functions on M and for cuto
dep endent choices of the metric g and of the tachyon p otential u on M We would
also like to parametrize p ossible limits de ning the correlation functions of the quantum
two dimensional sigma mo dels
Renormalization group e ective actions
One could study the questions raised ab ove rst by p erturbative metho ds applied directly to
the lattice correlation functions It is imp ortant however to set the p erturbative scheme in
the way that do es not destroy the geometric features of the mo del i e in a way covariant under
di eomorphisms of M One way to assure this is to study instead of correlation functions
ob jects known under the name of renormalization group e ective actions
Fix such that is a p ower of and for y denote by B y the set of x in the
L Call a p oint M a barycenter of a set of p oints M j N if square y
j
P P
extremizes d Clearly if M is a euclidean space then Supp ose that
j j
N
j j
g
we x a map M and compute the integral
Z
X Y
S
S
g u
ef f
r y x e D d e D
g
y
g
y
x B y
It essentially computes the probability The right hand side is naturally a measure on M
the distribution of the barycenters y of spins x with x in blo cks B y S
ef f
logarithm of the the density of the right hand side w r t to some reference measure D on
M is called the blo ck spin renormalization group RG e ective action on scale
The renormalizability problem may now b e reformulated as the question ab out existence of
more exactly of the normalized measure the limit of S
ef f
S
ef f
e D
d
R
ef f
S
ef f
e D
on M if we cho ose bare g and u in the dep endent way and keep xed With the
describ ed in the fo otnote one may show that the two mo di cation of the de nition of S
ef f
formulations of the renormalizability problem are essentially equivalent
may b e viewed as describing the theory averaged over The limiting measures d
ef f
variations of the elds on distance scales Pictorially they describ e the system viewed
from far away when we do not distinguish details of length An imp ortant observation at
the core of the RG analysis is that this averaging may b e done inductively by rst eliminating the
variations on the smallest scales then on the larger ones and so on until scale is reached In
recall that even the free eld case required a multiplicative renormalization of the correlation functions of
exp onents of eld
R
P
this would hold if the barycenters were unique and with the normalizing factor
1
2
y x D d r
g
(y )
g 2
x2B (y )
under the integral on the right hand side of eq
this is not exactly the case for our de nition of S but ignore this for a moment
ef f
the in nite volume L the pro cess may b e viewed as a rep eated application of a map on
a space of unit lattice actions If under the iterations the e ective actions are driven to a simple
attractor like an unstable manifold of a xed p oint then the renormalization consists of cho osing
the initial bare actions so that the e ective actions S end up on the attractor In the
ef f
vicinity of a xed p oint this would b e p ossible if the family of the bare actions parametrized by
bare couplings crosses transversally the stable manifold The renormalized couplings parametrize
then the unstable manifold a drawing would b e helpful here This dynamical system view
of renormalization develop ed by K G Wilson is extremely imp ortant and will hop efully b e
explained in much more details in future lectures
We have suppressed in the notation the dep endence on the size L of the b ox If the choice
of g and u involved in the limit can b e done in an L indep endent way we
automatically obtain a family of measures parametrized by and L The measures with the
pro duct L xed to a p ower of are related by the rescaling of space time distances If the
in nite volume limit L of the theory exists b ecomes a continuous parameter Supp ose
of p ossible continuum limits i e the attractor of the RG map that the e ective actions S
ef f
in the dynamical system view may b e parametrized by dimensionless renormalized metrics
is equal to S plus less imp ortant g and tachyon p otentials u You should think that S
ef f
g u
higher dimension terms separated by a precise rule We would say then that the theory is
renormalizable by a metric and a tachyon p otential renormalization This is the scenario realized
in p erturbation theory see b elow In such a situations the L theories are characterized
by the running metric g and tachyon p otential u describing S on di erent
ef f
scales in the passive view of the scale dep endence of the renormalized theory In the active
view the dep endence of g and u is generated by the action of rescalings of distances on the
limiting theory The in nitesimal scale transformations generate a vector eld in the
g u
space of g u de ned by
g g u u g u
g u and g u are called in the physicists jargon the RG b eta and gamma functions
In the dynamical system language is a vector eld on the attractor of the RG map and
g u
it extends the map to a ow The imp ortance of the RG functions lies in the fact that computed
in p erturbation expansion they allow to go b eyond it providing for example a consistency check
on the latter by solving the RG eqs with and given by few p erturbative terms we
may check whether the tra jectories g u stay or are driven out for large that is
at short distances from the region of the g u space where the p erturbative calculation may b e
trusted It is clear that the zeros of the vector eld should play an imp ortant role They
corresp ond to scale invariant and hence conformal invariant eld theories and in the dynamical
system picture to xed p oints of the RG map since they lie already on the attractor
S
ef f
How to generate the p erturbation expansion for the RG e ective actions e A helpful
observation is that the delta function in the de nition can b e rewritten in simple terms if
we use the exp onential map e T M M
i e of the metric on
Problem geometric Show that for vectors in a small ball in T M
j
N N
X X
j
d e r
j
g
j j
P
j
It follows that is a barycenter of the set of p oints fe g i
j
x
Substituting in eq x e y for x T M if x B y or in a shorthand
y
notation e where x y for x B y we obtain
Z
Y X
S S e
g u
ef f
h
e x e D e
g
h
y
x B y
P
x g Note that the lattice eld takes values in a vector space f j x T M
y
x B y
may just b e generated in the standard way by expanding in p owers of The lo op expansion for S
ef f
on the right hand side of eq all terms except for the quadratic contribution to S which
g u
is used to pro duce a Gaussian measure At each lo op order the result will b e invariant under the
simultaneous action of di eomorphisms of M on g and u When divergences will
app ear in the p erturbative expressions The p erturbative renormalizability of the theory may b e
studied by replacing the bare g and u on the right hand side of eq by
X X
n n
h u g u h g g u u u g g
n n
n n
We may attempt to x the ab ove series by cho osing some way to extract the renormalized metric
g and the renormalized p otential u from e ective actions S We would then like to show that
ef f
the ab ove substitution cancels the divergences in each lo op order of S resulting in a
ef f
family of p erturbative RG e ective actions parametrized by running metric g and tachyon
p otential u Di erentiation of the series over ln with g u xed would then
pro duce in the limit the lo op expansion for the b eta and gamma RG functions
Background eld e ective action
In practice the lattice p erturbative calculations are prohibitively complicated It would b e
easier to work with continuum regularization and renormalization which allow to calculate the
diagram amplitudes by momentum space integrals and to make use of rotational invariance We
have seen in Witten s lectures on p erturbative renormalization of the scalar eld theories with
the or interactions that it was convenient to analyze directly the PI e ective action
given by the Legendre transform of the free energy functional W The latter was de ned as
h J i
the logarithm of the integral of typ e of with F e The de nition of b oth W
and however as well as their p erturbative analysis used heavily the linear structure in the
space of maps from the space time to the target M inherited from the linear structure of M
Such structure is missing if M is a general manifold It is p ossible nevertheless to intro duce
the fact that the exp onential parametrization may work only lo cally do es not imp ede the p erturbative analysis
for sigma mo dels an e ective action mimicking the construction of the large deviations function
see eq and somewhat similar in spirit to the RG e ective actions for the lattice version
of sigma mo dels discussed in the previous section Instead of xing the blo ck barycenters in a
single lattice theory we shall take N indep endent copies of continuum theories with elds
j
and shall x for each x the barycenters x of x de ning the functional
j
Z
N
Y X Y
S
g u
j
e D d x x P r
g j j N
x
g
x
j j
by a formal functional integral Note that the right hand side reduces to a well de ned integral
for a lattice version of the theory For a map M and for a section of the pullback
TM of the bundle tangent to M denote by e the map from to M whose value at p oint
x is obtained by applying the exp onential map to x T M Reparametrizing in eq
x
j
e we obtain
j
Z
N
Y X
j
S e
j
D e e P
j N
j j
We may try to extract the background eld e ective action from the leading con
b
tribution to P at large N
N
N o N
b
P e
N
It should b e clear that coincides then with the e ective action of the eld
b
theory dep ending on as a parameter corresp onding to the functional integral
Z
S e
e D e
Fields take values in a vector space of sections of TM so that the p erturbative treatment
of the theory is more standard Note that only the geometric structure on M was used in the
formal de nition of
b
We shall reformulate the renormalizability problem for the second time as the question
ab out existence of the limit of the regularized version of the background eld e ective
action for a cuto dep endent theory with the action S We could regularize
g u
b
the functional integral by putting elds on a lattice with spacing while keeping
j
as a continuum eld This would not pro duce a big computational gain in comparison to the
p erturbative calculation of the RG e ective actions It is p ossible however to regularize the
lo op expansion of the background eld e ective action just by intro ducing an ultraviolet cuto
in the momentum space integrals for the PI vacuum amplitudes in the eld theory whose elds
form a vector space of sections of TM obtained this way will b e covariant under the
b
di eomorphisms of M in each order of the lo op expansion The p erturbative renormalization
will consist of cho osing the bare parameters of the theory in a cutto dep endent way as in eqs
and such that the limit of exists order by order in the lo op expansion The
b
p erturbative limits will b e parametrized by the running metric g and p otential u on
M with the change of induced by rescaling of distances on
Dimensional regularization
We shall prove the p erturbative renormalizability of the background eld e ective action in the
D sigma mo del only in the leading order of the lo op expansion concentrating instead on the
discussion of the renormalization group asp ects of the lo op result To avoid calculational but
not conceptual di culties we shall work in the at euclidean space time E We shall also
use a sp eci c scheme for regularization of divergent diagrams the dimensional regularization
and a particular way to renormalize the theory i e to chose g and u the minimal
subtraction Brie y the idea is to
regularize the momentum space integrals by rewriting them as integrals into which the
space time dimension D enters as an analytic complex parameter then
to calculate the integral for the values of D where it converges and nally
to analytically continue to the physical values of D extracting the p ole parts of the result
at the physical dimension as the divergence to b e removed by the renormalization
In order to gain some practice let us compare how the simplest divergent diagram of the
dimensional theory O is regularized and renormalized rst in the more conventional
momentum space regularization used in Witten s lecture and then in the dimensional regulariza
tion minimal subtraction scheme The momentum space amplitude I k corresp onding to the
amputated graph was given by the integral
Z Z Z
d q d q
g g
d I k
D
q m q k m q k m
D
d q
In where k is the external momentum q is that of the lo op b oth euclidean and d q
D
space time dimension D the q integral diverges logarithmically It may b e regularized by
restricting the integration to jq j
Z Z
d q
m
g g k
I k d K ln
q k m
jq j
where
Z
m
k g m k
K lim d ln
The renormalization idea is then to substitute
P
n
h r r r g m
n
n
for the coupling constant and the mass squared in the initial action recall that the theory do es
not need renormalization of g and only lo op counterterm would do The p owers of make
and r dimensionless With a choice
ln r
the contribution to the lo op amplitude I k diverging when is canceled resulting in
the renormalized value of the amplitude
Z
k
d ln r I k
r en
d d
r r describing the scale dep endence of the The RG functions r
d d
renormalized couplings are obtained by di erentiating eq over ln with g and m
held xed
d
d
d r d
g
ln O h r h O h h r
d d
so that
r r r h O h
Let us see how the same problem is treated in the dimensional regularization minimal
R
a
subtraction scheme Using the relation e d a we may rewrite the integral for
I k in the form
D
Z Z Z
g
q k m
I k d e d q d
D
Performing the q integral rst we obtain
Z Z
g
D k m
d e d I k
D
D D
The latter integral converges for any complex D with Re D It gives explicitly
Z
D
D
g
I k d k m
D
D
D
The divergence in four dimensions manifests itself as a p ole in the expression at D
Z
g g
d ln k m ln C O D I k
D
D
where C is the Euler constant In the minimal subtraction renormalization scheme
the amplitude is renormalized by substituting for the original coupling and mass squared
P
n
D
h r r r g m
n
n
for those who do not rememb er Feynman s famous formula I don t we could have used twice the identity
R
a
i i
e d a in the original expression for I k changing then the variables to where
i D
i
with the pure p ole dep endence of r on the dimension
n
k
n
X
r r r
n j n
j
D
j
chosen to cancel exactly the p ole part of the dimensionally regularized amplitudes In reality
only the lo op amplitude has a p ole which is simple With
r
D
we obtain the renormalized amplitude
Z
k
I k d ln r ln C
r en
note how has entered under the logarithm The di erence b etween the two renormalizations
may b e absorb ed into a nite rede nitions of the renormalized parameters r see Problem
b elow
Now the RG functions r r are obtained by di erentiating eqs with resp ect
to ln while keeping g and m xed
d D
D D
d
D
d d
g
r h r h
d D d D
d r
r h
d
so that
D
r r h r
The gain is that we have obtained formulae for and in general dimension They may
serve as an indication of how the eld theory b ehave in smaller or larger dimension then the
one considered For example the vanishing of the linear contribution to in dimensions
signals that the theory b ecomes only renormalizable there Note that eqs reduce to
at D This did not have to b e the case since we have changed the parametrization of the
limiting theories and what is geometrically de ned is the vector eld
r
Problem Find to the lo op order the transformation b etween the co ordinates r
of the renormalized dimensional theory corresp onding to the passage b etween the two
renormalization schemes discussed ab ove Show that it preserves the form of the vector eld
r
Problem Find the running couplings r using the lo op approximations to the
functions What can one tell ab out the e ect of higher lo op corrections to the large
UV b ehavior of the running couplings
The four dimensional theory in spite of its sup er renormalizability only a nite numb er of
divergent PI graphs and self consistency of its p erturbative calculations has a non p erturbative
stability problem related to the lack of a lower b ound for the cubic p olynomial This should serve
as a warning that even the RG improved p erturbative analysis is not enough to assure existence
of a renormalized QFT
Renormalization of the sigma mo dels to lo op
As mentioned ab ove the background eld e ective action of the sigma mo del is equal to
b
the value of the e ective action of the theory with the action S given by the relation
S e S
g u
e D e e D
g
st
The of eqs implies then that in the p erturbation expansion
Z
X
S
D I S ln e S
G b
PI vacuum
graphs G
One of the simplifying features of the dimensional regularization is that we may disregard the
D e
g
terms in S coming from the logarithm of the Radon Niko dym derivative Formally
D
R
they are prop ortional to d q which vanishes in the dimensional regularization
R
q
dq in Problem a for Pasha Show by calculating the D dimensional integral e
D
D
two ways that the volume of the unit sphere in D dimensions is equal to
R
dq converges for Re D b Show that in the radial variables the D dimensional integral
jq j
Z
D
D
D
d q
D D
jq j
De ning the value of the integral by analytic continuation for integer D and taking to
R
zero we infer that d q vanishes in p ositive dimensions in the dimensional regularization
Expanding in lo cal co ordinates
i j k i i i
O e
j k
i
i
where f g is the Levi Civita symb ol we obtain after a little calculation you may do it
j k
j k
Z
i j i j i
S e g u g r u
g u ij ij i
i j i j i k j l
dx O r u g r r R
i j ij ij k l
th
Ab ove r denotes the covariant derivative over the i co ordinate and
i
i i i j k
r
j k
The vector bundle TM has a natural metric given by g It will b e convenient to cho ose a
d
of TM d denotes the dimension on M In co ordinates global orthonormal frame e
a
a
i b
e e i Di erent choices of e are related by gauge transformations e e with an
a a b
a a a
i
SO d valued function on the plane E Of course e dep end also on eld It will b e more
a
convenient to rewrite
a i a i
e or e
a
a
a
with a sequence of functions on E The expansion b ecomes then
Z
i j a i a i a
g u u e r e S e
ij i g u
i a
j
a i k a b a b i
r R dx O r u e e
iak b i j
b a
l j i a
R and e R e is the matrix inverse to e where e
ij k l iak b
b a a i
a b a a a a j k i i
e with A r A e e
i b b b j k b
a
dx transforms as an SO d connection form under the gauge transformations Clearly A
b
b
e e The p erturbation expansion for b ecomes now to the lo op order
a b b
a
Z
i j
g u dx
ij b
h
Z
R
j
i a b a i k a b
e dx e r u r R
i j
iak b
a
b
D O h ln e
the terms linear in on the right hand side of do not contribute to The functional
b
integral gives a determinant and we could use the zeta function prescription to make sense out of
it in an SO d gauge invariant way We shall b e however more interested in the divergent part
of the determinant than in its renormalized value The dimensional regularization will allow to
extract the divergence in a convenient SO d gauge invariant manner and b esides it works to
all orders
We shall obtain the expression for the lo op contribution to expanded around a
b
constant value of in the form in co ordinates around
Z
n
X Y
lo op
x dx K x x
j j b n D n
n
j
with the translationally invariant kernels K regularized dimensionally i e meromorphically
n D
dep endent on D with p ossible p oles at D and with vanishing fast at in nity In
order to generate the expansion for the lo op contribution to we shall separate the
b
term
Z
j
a a b i
dx r u j e e
i j
b a
in the action as pro ducing the Gaussian measure from
Z
j j
a b i i a a b a a b c
e e r u e e r u j A A A
i j i j
a a b b c
b b
i k a b
dx R
iak b
treated as an interaction Now it is easy to enumerate the divergent graphs First there are
logarithmically divergent contributions coming from the graphs
O A A O
A A
They cancel each other there are no divergences in D gauge theory The only divergent terms
we are left with are
j
i
O O r u j and e ln det e
ab i j
b a
R
e e r u e e r u j
All other contributions are easily checked to have nite limits at D Since in the dimensional
regularization
R R R R
d q
D m q m D D
d e d e d q
q m
D D D
m D part regular at D
D
the p ole part of the lo ops in is equal to
Z
i k
u u dx R
g g iak a
D
ij j i
r u g r u u where denotes the Laplacian on M Similarly e since e
i j i j g g
a a
R R
j
i
dx tr ln q d q ln det e e r u j e e r u j
ab i j
a b
R R
const dx dt tr d q e e r u j
t
e e r u j q
R
const u dx
g
D
We infer that the p ole part of to the lo op order is
g
Z Z
lo op
i j
R dx u dx
b ij g
div
D D
lo op
is an integral of dimension and where R R is the Ricci tensor on M
ij iaj a b
div
dimension op erators and this result in accord with a simple p ower counting remains true at
higher orders
j
i
is p ositive which is the case for example in the vicinity of a we assume that the matrix e e r u j
i j
0 a
b
minimum of u
The minimal subtraction renormalization scheme adds counterterms to bare metric g and
bare p otential u which cancel the ab ove p oles More exactly one substitutes in the initial action
S of the mo del with the bare metric g and bare tachyon p otential u
g u
h
D
g O h g g
D
h
D
u u u O h
D
The added lo op counterterms change the e ective action by
Z
D
i j
g u dx O h
b
ij
D
and we put
u R u g
g ij
ij
canceling the p oles at D This proves the renormalizability of the sigma mo del background
eld e ective action to lo op
Renormalization group analysis of sigma mo dels
Let us compute the vector eld on the space of metrics and p otentials as given by the
g u
minimal subtraction version of eqs
g u g u g u
d d
g const g const
u const u const
d
to eq we obtain Applying the derivative
d ln
h h d d
D D
g g R O h R O h
ij ij ij
ij
d D d D
D
g D g h R O h
ij ij ij
from which we infer that
g D g h R O h
ij ij ij
Similarly
d h d h
D D D
u O h u O h u u
g g
d D d D
h
D
u O h u D u
g
so that
h u O h u D u
g
The vector eld in the space of metrics may b e used to nd out in which situations we
may exp ect the p erturbative calculation to b e self consistent The condition is that the running
metric g satisfying the RG equation
d
g g
d
stays on all scales in the p erturbative regime Let us illustrate this on the example where
N N
times the induced metric g of the unit sphere in R Due to M S with the metric
the rotational symmetry the renormalized metric is g and the eq for its dep endence
reduces in D to
d
h N O h
d
Clearly for N is driven to zero for large approximately as O The p erturbative
ln
regime corresp onds to small so that the p erturbative expansion is self consistent for the sigma
N
mo del with M S for N The phenomenon is called the asymptotic freedom of the
spherical sigma mo del since corresp onds to a free theory It p ermits to exp ect that the
theory may b e constructed non p erturbatively at least in nite volume Such a non p erturbative
theory would break the conformal invariance of the classical sigma mo del In fact there are
strong reasons to b elieve that its in nite volume version is massive an exact expression for its
S matrix is conjecturally known
The prop erty of asymptotic freedom is shared by all the sigma mo dels which have compact
symmetric spaces as their targets and also more imp ortantly by the non ab elian dimensional
gauge theories with not to o many fermion sp ecies like Quantum Chromo dynamics QCD de
scribing the strong interactions of quarks mediated by SU gauge elds
Problem Consider the ow
d d
u u u
d d
in R Show that there exist only one p erturbative solution
X
n
a u
n
n
for the invariant manifold of the ow Study the non p erturbative invariant manifolds
Sp ecially interesting cases corresp ond to manifolds with vanishing Ricci curvature The N
spherical sigma mo del is the simplest example coinciding with free eld with values in S
As we know it corresp onds to a CFT One may then read from the function the dimensions
iq
equal to d q of the comp osite op erators given by the exp onential functions e on S d
q g
are equal to the eigenvalues of the tree contribution to the comes from the fact that we
g
have considered integrated insertions of the comp osite op erator into the action For Ricci at
targets there is no renormalization of the metric in the lo op order and in the sup ersymmetric
versions up to lo ops lo ops excluded No renormalization of the metric means that the
b eta function vanishes and the scale invariance is preserved to lo ops One may then argue
in p erturbation theory studying the Hessian of the lo op that in the K ahlerian case the
Ricci at metric may b e p erturb ed as to give rise to a scale invariant quantum sigma mo del with
N sup erconformal symmetry as discussed by Ed Witten during the lecture Thus Calabi
K ahler Ricci at manifolds should corresp ond to sup erconformal N eld theories Yau
This observation resulted in a conjectured mirror symmetry b etween Calabi Yau manifolds now
established in many instances
In the case of SUSY sigma mo dels with hyp er K ahler targets i e with the N sup er
symmetry vanishes to all orders of the lo op expansion
The inclusion of the form term into the action of the sigma mo dels mo di es the ab ove
d
g b is given by the Ricci curvature of results In the lo op order the b eta function
ij ij
d
i
il i
g H where the antisymmetric tensor H f g the metric connection with torsion
j k l j k l
j k
j k
corresp onds to d In mo dels in which the connections with torsion are globally at the b eta
function vanishes to all orders even without sup ersymmetry The WZW mo del of CFT which
we shall discuss in the next lecture corresp onds to such a situation Addition of the form
which is closed do es not mo dify the function but may change the long distance b ehavior of the
mo del that seems to happ en for the sigma mo del with S target where the addition of the term
with equal to times the volume form of the unit sphere should render the mo del massless
As for the renormalization of the p otentials whose scale dep endence is governed by the RG
equation
d
u u h u O h
g
d
note that on a symmetric space u approximately repro duces itself up to a normalization if it
b elongs to an eigen subspace of the Laplacian The RG analysis allows then to predict the short
distance b ehavior of the correlation functions involving insertions of the corresp onding comp osite
op erators somewhat similarly as for M S
References
For the rudiments of the p erturbative approach to functional integrals Feynman graphs etc
see again the b o ok by Zinn Justin Sects and Kazhdan s Witten s and Gross lectures
in the present series
The original reference to the renormalization of geometric sigma mo dels is Fiedan s thesis
published with few years delay in Ann Phys p The SUSY case is discussed
in Alvarez Gaume Freedman Mukhi Ann Phys p with further developments
in Alvarez Gaum e Ginsparg Commun Math Phys p Alvarez Gaum e S
Coleman Ginsparg Commun Math Phys p and Grisaru Van De Ven Zanon
th
Nucl Phys B p and p the last pap ers discovered order contributions
to the sup esymmetric b eta function For the case of sigma mo dels with a form in the action see
Braaten Curtright Zachos Nucl Phys B p and also Callan Friedan Martinec
Perry Nucl Phys B p
Lecture Constructive conformal eld theory
Contents
WZW mo del
Gauge symmetry Ward identities
Scalar pro duct of non ab elian theta functions
KZB connection
Coset theories
WZW factory
Let us recall the logical structure of this course In the rst lecture we studied the free
eld examples of CFT s In the second one we analyzed the scheme of two dimensional CFT
from a more abstract axiomatic p oint of view In the third one we searched p erturbatively
among geometric sigma mo dels for non free examples of CFT s Finally in the present lecture
compressed due to lack of time we shall analyze a sp ecially imp ortant sigma mo del the Wess
Zumino Witten WZW one whose correlation functions may b e constructed non p erturbatively
with a degree of explicitness comparable to that attained for toroidal compacti cations of free
elds constituting the simplest examples of WZW theories The WZW mo del app ears to b e
a generating theory of a vast family of CFT s whose correlations can b e expressed in terms
of the WZW ones The comparison of the non p erturbative mo dels obtained this way with
the p erturbative constructions of sigma mo dels allows for highly non trivial tests of di erences
b etween the geometry of Ricci at Einstein spaces and that of CFT s replacing the Einstein
geometry in the stringy approach to gravity
WZW mo del
The target space of the WZW sigma mo del is a compact Lie group manifold G and the two
dimensional theory may b e considered as a generalization of quantum mechanics of a particle
moving on G In the latter case the Euclidean action functional is
Z
S g tr g g dx
x
where tr denotes the Killing form Let R denote an irreducible unitary representation
g g of G in a nite dimensional Hilb ert space V The path integral for the quantum
R R
mechanical particle on G corresp onding to the Wiener measure on G may b e solved with the
use of the Feynman Kac formula taking the form
Z Z
Y Y
n
k S g k S g
g x e dg x e dg x
i
R
i
i
x x
M ap L G M ap L G
per per
normalized so that the long ro ots have length squared
LH L x H x H x x H
n
Tr e e Tr e g e g g
R R R
n
where x x x L k H is the Laplacian on G and g is viewed as a matrix of
n
R
multiplication op erators b oth acting in L G dg dg is the Haar measure Compare Problem
in Lecture dealing with the case G S The theory p ossesses the G G symmetry which
may b e used to solve it the right hand side of is calculable in terms of the harmonic analysis
on G
Problem Compute explicitly the and point functions in
The Euclidean action of the WZW mo del is a functional on M ap G where is a compact
Riemann surface If for simplicity we assume G to b e connected and simply connected then
Z Z
i i
S g g tr g g g g
where the form is de ned on op en subsets O G with H O and satis es there
tr g dg The dep endence of the second term of S g on the choice of makes d
k S g
S g de ned mo dulo iZ so that e is well de ned for integer k To have the energy
b ounded b elow we shall take k called the level of the WZW mo del p ositive
C
Problem Assuming more general ly that g takes values in the complexi ed group G nd
the equations for stationary points of S g
The correlation functions of the WZW mo del are formally given by the functional integrals
Z Z
n n n
k S g k S g
h g x i g x e D g e D g End V
i i
R R R
i i i
i i i
M ap G M ap G
where D g stands for the formal pro duct of the Haar measures dg x over x
As we have mentioned at the end of Lecture the renormalization group b eta function com
puted for the WZW sigma mo del vanishes to all orders due to the atness of the connections
with torsion generated from the metric and the form on G Thus the mo del is conformally
invariant and do es not need renormalization of the action in p erturbation theory The con
formal invariance holds in fact also non p erturbatively due to the LG LG symmetry of the
theory where LG denotes the lo op group M ap S G of G The WZW mo del may b e solved
exactly by
harmonic analysis on LG
or by
exact functional integration
As we shall see the matrix valued comp osite op erators g x need multiplicative renormaliza
R
c
R
tion and acquire scaling dimensions where c denotes the quadratic Casimir of R and h
R
k h
stands for the dual Coxeter numb er of G the quadratic Casimir of the adjoint representation
Gauge symmetry Ward identities
We shall sketch here the functional integral approach to the WZW theory It will b e convenient
to extend a little the mo del by coupling it to an external gauge eld A A A a form
C
with values in the complexi ed Lie algebra g of G De ne
Z
ik
tr A g g g g A g A g A S g A S g
C
Under the chiral gauge transformations corresp onding to maps h G the action
S g A transforms according to the Polyakov Wiegmann formula
h h
S h g h A A S g A A S h A S h A
h h
where A h A h h h and A h A h h h
Problem Prove the Polyakov Wiegmann formula
In the presence of the external gauge eld A the partition function of the WZW theory will b e
formally de ned as
Z
k S g A
Z e D g
A
M ap G
x i by eq with S g replaced by S g A no func and the correlation functions h g
i
A R
i
tional integration over A Using the formal extension to functional integrals of the simple
invariance prop erty
Z Z
f h g h dg f g dg
G G
C C
holding for h G if f is an analytic function on G we obtain
Z
h h
n
A A k S h g h
x i h g Z D g x e h g h
i i
R R
h h h h
i i
A A A A
i
k S h A k S h A
e e h x Z h g x i h x
i i i
R A R A
R
i i
i
i i
This is the chiral gauge symmetry Ward identity for the correlation functions recall the di eo
morphism group and the lo cal rescaling Ward identities discussed in Lecture
The identity factorizes into a holomorphic A dep endent and an anti holomorphic A
dep endent ones Hence in order to study the A dep endence of the correlation functions it is
enough to lo ok for holomorphic maps on a Sob olev space of forms A with values in g
n
V A V
R R
i
i
in general we shall not assume the unitarity A A of the gauge eld
a more standard de nition subtracts also A A inside
what follows do es not dep end on the assumed degree of smo othness of forms provided it is high enough
satisfying the factorized Ward identity
n
h k S h A
A e h x A
i
R
i
i
C
The relation describ es the b ehavior of along the orbits of the group G of complex
C
Sob olev class gauge transformations in A The orbit space A G is the mo duli space
C
of holomorphic G bundles which up on restriction to semi stable bundles and appropriate
treatment of semi stable but not stable ones b ecomes a compact variety N of complex di
mension rank G and dim G h for genus h equal to and resp ectively
The space W x R k of s coincides with the space H V of holomorphic sections of
a vector bundle V over N with typical b er V V A V essentially In another
R C R
G
description W x R k H L where L is a line bundle over the mo duli space of holo
C
morphic G bundles with parab olic structures at p oints x and s may b e interpreted as a
i
non ab elian generalization of theta functions The essential implication of these identi cations is
that W x R k is a nite dimensional space Its dimension dep ends in fact only on h k
and R and is given by the celebrated Verlinde formula W x R k may b e also identi ed
with the space of quantum states of the Chern Simons theory
Out of the global Ward identities one may extract the in nitesimal ones by taking h
i
i
and Taylor expanding in similarly as we analyzed the in nitesimal consequences of the e
i
di eomorphism and rescaling Ward identities in Lecture De ne the insertions of currents into
the correlations by
a
Z h i h J i
A A z A
a
Z A
z
A
a
h J i Z h i
z
A A A
a
Z A
A z
ab a a b
Denote by the subscript a corresp onds to a basis t of the Lie algebra g s t tr t t
J z J z the insertions of J J into correlations with A vanishing around the insertion
z z
p oint and the metric lo cally at J z J z dep ends holomorphically anti holomorphically
on z away from other insertions Expanding to the second order one obtains from the Ward
identities the op erator pro duct expansions
ab abc
k i f
a b c
J z J w J w
z w z w
ab abc
k i f
a b c
J z J w J w
z w z w
a b
J z J w
P
n
J z which imply for the mo des of the corresp onding Hilb ert space op erators J z
n
n
P
n
J z the Kac Mo o dy algebra relations J z
n
n
ab a b abc c
k n J J i f J
n m
n m n m
and similarly for J with the commutators b etween J and J vanishing
n n m
Problem Prove the operator product expansions
Subtraction of the singular part from the expression tr J z J w gives the Sugawara con
struction of the energy momentum tensor of the WZW theory
dim G k
lim tr J z J w T w
k h z w
z w
and similarly for T w In mo des this b ecomes
X
L tr J J
n n m m
k h
m
where the normal ordering puts J with p ositive p to the right of the ones with negative p
p
Scalar pro duct of non ab elian theta functions
x i coincides Since the A dep endence of the unnormalized correlation functions Z h g
i
A A R
i
with that of V we must have A recall that the complex conjugate space V
R
R
W x R k x i W x R k Z h g
i
A A R
i
as a function of A or more explicitly
A x i H A Z h g
i
A A R
i
where is a basis of W x R k H is an x dep endent matrix and the summa
tion convention is assumed From formal reality prop erties of the functional integral de ning
x i one may see that H should b e a hermitian matrix In fact one may argue Z h g
i
A A R
i
that the inverse matrix H corresp onds to the scalar pro duct on the space W x R k of
non ab elian theta functions
H
where is formally given by
Z
k
kAk
L
A A e DA
V
R
with the integration over the unitary gauge elds A with A A The scalar pro duct
is exactly the one which gives the probability amplitudes b etween the states of the Chern
Simons theory Expressions and for the correlation functions may b e expressed in a
denote the evaluation map basis indep endent way as follows Let e
A
e
A
W x R k A V
R
e may b e considered as an element of V W x R k and using the scalar pro duct dual
A
R
to on the second factor we obtain the equality
e e Z h g x i
i
A A
A R A
i
V valued functionals of A viewed as a relation b etween the V
R R
Let us present a functional integral pro of of the relation Denote Z h g x i A
i
A R A
i
Consider the integral over the unitary gauge elds B
Z
k
kB k
L
DB B A A B e
Z
R
n
ik
k S g k S g tr A g g g g A
g g x e e
i
R
i
i
R
ik
g g B g A g g B g A g B B B tr B g
D g D g DB e
Z
R
n
ik
k S g k S g A g g g tr A g
x e g g
i
R
i
i
R
ik
tr g g g A g g g g A g
e D g D g
Z
n
k S g g A
g g x e D g D g A
i
R
i
i
nd
where the equality is obtained by a straightforward Gaussian integration over B Up on the
substitution of relations and the last identity b ecomes
A H A A H H A
or H H H from which the relation follows if we also assume that H
is an invertible matrix
The ab ove expressions reduce the calculation of the correlation functions of the WZW mo del
to that of the functional integral The latter app ears easier to calculate then the original
functional integral In the rst step the integral may b e rewritten by a trick resembling
the Faddeev Pop ov treatment of gauge theory functional integrals The reparametrization of the
gauge elds
h
A n A
C
by chiral gauge transforms of a lo cal slice n A n in A cutting each G orbit once
gives
Z
k h S hh A n
k k A n hh A n e D hh d n
V Q
R
i R
The term h S hh in the action comes from the Jacobian of the change of variables
contributing also to the measure d n The latter is de ned as follows Denote by S the
Q
comp osition of the derivative of the map n A n with the canonical pro jection of A onto
the cokernel of A n Then the volume form n on the slice is the comp osition of the
Q
C
in genus and h G should b e additionally restricted
determinant the maximal exterior p ower of S with the Quillen metric on the determinant
bundle of the family A n of op erators
Unlike in the standard Faddeev Pop ov setup the integral over the group of gauge transfor
mations did not drop out since the integrand in is invariant only under the G valued gauge
transformations Instead we are left with a functional integral similar to the one for the
original correlation functions except that it is over elds hh which may b e considered as taking
C C
values in the hyp erb olic space G G D hh is the formal pro duct of G invariant measures
C
on G G The gain is that the functional integral may b e reduced to an explicitly doable
iterative Gaussian integral For example for G SU and at genus where we may take
A n
Z Z
i i
v v S hh
e v
in the Iwasawa parametrization h u of the dimensional hyp erb oloid
e
SL C S U by R and v C u SU Field v enters quadratically into the action
and p olynomially into insertions Hence the v integral is Gaussian and its explicit calculation
e requires the knowledge of the determinant of the op erator e e
and of the propagator
Z
y
e d y
z z
z z e
z y y z
The dep endence of ln det is given by the chiral anomaly or lo cal index
theorem and is the sum of a lo cal quadratic and a linear term The resulting eld integral
app ears to b e also Gaussian of the typ e encountered in functional integral representations of
a dimensional Coulomb gas correlation functions in statistical mechanics Similar iterative
C
pro cedure based on the Iwasawa parametrization of G G works for arbitrary G and also at
higher genera A result b ecomes a nite dimensional integral over parameters y in the
a
expressions of the typ e for the v eld propagators p ositions of the screening charges in
the Coulomb gas interpretation and at genus h over a part of the mo duli parameters
n
C
At genus the G orbit of A is dense in A As a result W CP x R k is
G
fully determined by V the G invariant subspace of V Hence
R R
G
W CP x R k V
R
canonically For G SU the representations R are lab eled by integer or half integer spins j
i i
l
where v is the highest and the representation spaces V are spanned by vectors f v
j j j
i i i
l j
i
weight HW vector annihilated by e with e f h the usual basis of sl One has using the
standard complex variable z on CP to lab el the insertion p oints
Y
n
z e SU
i
i i
e e v if N J k g W CP z j k f v V j v
j
i
i
j
i
again the cases of genus or require minor mo di cations
P P
e
where e N n and J j In particular for or
i i i
i i
i
p oints
SU
V if J k
j
W CP z j k
f g if J k
and do es not dep end on z The scalar pro duct is given by
Z
J
Y
U z y
k
d y j j v z y e kv k f z j k
a
a
J
C
where
P
j j
i i
z kd k
det
i
k k
L
i
f z j k e
area
CP
carries the dep endence on the metric e jdz j on CP y y y z y is a meromorphic
J
V valued function
j
n J
X Y
n
v f z y
j i
i
y z
a
i
i
i a
and U z y is a multivalued function
X X X
ln y y j ln z y ln z z j j U z y
a a i i a i i i i
i a
a a i i
Integral is over a p ositive density with singularities at coinciding y and the question arises
a
as to whether it do es converge A natural conjecture is that the integral is convergent if and
G
only if v W CP x R k V the only if part is easy For or p oint functions the
R
integrals can indeed b e computed explicitly con rming the conjecture Numerous other sp ecial
cases have b een checked However the general case of the conjecture remains to b e veri ed Note
that the dep endence of the scalar pro duct on the conformal factor agrees with the value
j j
k
of the central charge of the SU WZW theory and with the values of
j j
k k
the conformal dimensions of eld g x it is the inverse of f x j k which enters the WZW
j
correlation functions
Explicit nite dimensional integral formulae for the scalar pro duct have b een also ob
tained for general groups and at genus and for G SU for higher genera The pro of of the
convergence of the corresp onding integrals is the only missing element in the explicit construction
of all correlation functions of the WZW theory although several sp ecial cases have b een settled
completely
it is clear that the case of general group and genus could b e treated along the same lines
KZB connection
The spaces W x R k of non ab elian theta functions dep end on the complex structures
of the surface and on the insertion p oints The complex structures J End T J
form a complex in nite dimensional Fr echet manifold on which the group of Di eomorphisms
of acts naturally The holomorphic tangent vectors to the quotient mo duli space J
C
corresp ond to sections of End T satisfying J J i Lo cally may b e rep
z
resented as dz in J complex co ordinates The family of spaces W x R k forms a
z
z
complex nite dimensional bundle W R k over the space of complex structures and n tuples
of noncoincident p oints x in
The bundle W R k may b e supplied with a natural w r t the action of di eomorphisms
of connection r provided that we cho ose smo othly for each J a compatible metric on
The connections for di erent choices of the metric are related by the conformal anomaly If
J x A J x A dep ending holomorphically on A A iJ A A is assumed
represents a lo cal section of W R k then
Z
k
d tr A A r
r A
z
i z z
i
i i
Z
z
r d T z d z
z
i
a a
t t
a
i i
J z lim r
z z
i i
k h z z
i z z
i
Ab ove z denotes a J complex co ordinate on and d or d stands for the directional
derivatives of when the p oints x and A are kept constant The rst two equations equip
W R k with a structure of a holomorphic vector bundle In the last equations the metric on
z z z z z
and A is taken vanishing around is assumed for simplicity to satisfy
i
z
the supp ort of or around the insertion p oint x and O z z
i i
G
In the genus case W R k is a subbundle of the trivial bundle with the b er V and
R
the connection r extends to the bigger bundle and is given by
a a
X
t t
i
i
H z r r
i
z z z z z
z z
k h k h i i i i i
i
i
i i
for the metric at around the insertions The commuting op erators H z End V are known
i
R
as the Gaudin Hamiltonians The corresp onding at connection app eared implicitly in the
work of Knizhnik Zamolo dchikov on the WZW theory The higher genus generalizations of the
KZ connection were rst studied by Bernard We shall call the connection de ned by eqs
the KZB connection In general it is only pro jectively at
One of the basic op en questions concerning the KZB connection is whether it is unitarizable
In other words whether there exists a hermitian structure on the bundle W R k preserved
by r It was conjectured that the answer to this question is p ositive and that it is exactly the
scalar pro duct on spaces W x R k discussed ab ove that provides the required hermitian
structure Note that a unitary connection on a holomorphic hermitian vector bundle is
uniquely determined Recall that the scalar pro duct given formally by the gauge eld functional
integral may b e reduced to a nite dimensional integral which if convergent de nes a
p ositive hermitian form on W x R k and determines the unitary connection and the energy
momenstum tensor of the WZW theory For genus and G SU where the scalar pro duct
is given by integral the unitarity of the KZ connection requires that
H v v v v
i z z
i i
k
SU
for a holomorphic family z v z W CP z j k V Assuming the convergence of the
j
integrals p ermitting to di erentiate under the integral and to integrate by parts the ab ove is a
consequence of the relation
U z y U z y
k k
e z y e H z
i a y i z
a
i
k
where
n
Y X
n
z y f f v
i a i i j
i
z y y z
a
i
a i
i
a a i
Identity is equivalent to two relations
z y z y
y z i a
a
i
U z y z y U z y z y H z z y
y z i a i
a
i
The rst one is immediate whereas the second more involved one implies that
U x y z y if H z z y U x y
y i z
a
i
i e the Bethe Ansatz diagonalization of the Gaudin Hamiltonians H z vectors z y are
i
U z y provided that y satis es common eigenvectors of H z i n with eigenvalues
i z
i
the Bethe Ansatz equations U x y The relations b etween the Bethe Ansatz and the
y
a
limit of the KZB connection when k h app ear in the context of Langlands geometric
corresp ondence These relations seem also to b e at the heart of the question ab out the unitarity
of the KZB connection at p ositive integer k
Coset theories
There is a rich family of CFT s which may b e obtained from the WZW mo dels by a simple
pro cedure known under the name of a coset construction On the functional integral level
the pro cedure consists of coupling the G group WZW theory to a subgroup H G unitary
gauge eld B which is also integrated over with gauge invariant insertions Let us assume for
H
b e simplicity that H is connected and simply connected as G Let t Hom V V
i
r
R
i
i
intertwiners of the action of H in the irreducible G and H representation spaces resp ectively
The simplest correlation functions of the G H coset theory take the form
Z Z
n n
Y Y
k S g B k S g B
e D g D B tr t g x t e D g D B tr t g x t i h
i i i i
V R i R i
i i
r
i
i i
Note that the g eld integrals are the ones of the WZW theory and are given by eq Denoting
R
k S g B
Z e D g D B we obtain
G H
Z
n
Y
k
kB k
L
t B t B e D B tr t g x t i H Z h
i i i i
i
V R
G H
i
r
i
For W x R k the map B t B V is a group H non ab elian theta
i
r
function b elonging to W x r k the normalization of the Killing forms of G and H may
di er hence the replacement k k Denote by T the corresp onding map from W x R k
to W x r k Eq may b e rewritten as
n
Y
tr t g x t i H T T Tr T T Z h
i i
i
R
G H
i
i
or cho osing a basis of W x r k
n
Y
x t i H tr t g Z h T h T
i i
i R
G H
i
i
where T is the branching matrix of the linear map T in bases and h
Since the ab ove formula holds also for the partition function itself it follows that the
calculation of the coset theory correlation functions reduces to that of the scalar pro ducts
of group G and group H non ab elian theta functions b oth given by explicit nite dimensional
integrals
Among the simplest examples of the coset theories is the case with G SU SU with
level k for pro duct groups the levels may b e taken indep endently for each group and
with H b eing the diagonal SU subgroup The resulting theories coincide with the unitary
rst considered by minimal series of CFT s with Virasoro central charges c
k k
Belavin Polyakov Zamolo dchikov The Hilb ert spaces of these theories are built from the unitary
heighest weight representations of the Virasoro algebras with c discussed in Lecture
is b elieved to describ e the continuum The simpliest one of them with k and c
limit of the Ising mo del at critical temp erature or the scaling limit of the massless theory In
particular in the continuum limit the spins in the critical Ising mo del are represented by elds
tr g x where g takes values in the rst SU The corresp onding correlation functions may
b e computed as ab ove One obtains this way for the p oint function an explicit expression in
terms of hyp ergeometric functions
Similar coset theories but at level k give rise to the sup ersymmetric N minimal
unitary series of CFT s the simplest one with k app earing also at k in the previous
series corresp onds to the so called critical Ising mo del
The G H coset theory with H G is a prototyp e of a two dimensional top ological eld
theory As follows from eq the correlation functions of elds tr g x are equal to the
R
dimension of spaces W x R k normalized by the dimension of W k and are given
by the Verlinde formula In particular they do not dep end on the p osition of the insertion
p oints
WZW factory
As we have seen ab ove the coset construction allows to obtain new soluble CFT s from the
WZW mo dels Let us brie y discuss further re nements which p ermit a chain pro duction of
conformal mo dels whose partition functions and correlation functions may b e computed exactly
at least in principle The most interesting cases of such mo dels corresp ond to situations when
two di erent constructions give rise to the same CFT as in T duality mirror symmetry and
other numerous instances
If the group G is not simply connected the original de nition of the action of the
WZW mo del requires a mo di cation The result is p ossible further restrictions on the levels and
the app earance in some cases of di erent quantizations of the same classical theory vacua
or discrete torsion The mo dels are still exactly soluble although only the partition functions
and the correlations of untwisted elds have b een worked out in detail for general G
Let H G and Z H b e a subgroup of the center Z of G Let P b e a principal
G
H
H bundle where H H Z and Q P G b e the G bundle asso ciated to P via the
G
Ad
H H
H
adjoint action of H on G For appropriate k and for a section g of Q and a connection
G
k S g B
B on P one may de ne the amplitude e The unnormalized correlation functions of
H
the coset G H mo del may then b e obtained by integrating gauge invariant insertions weighted
k S g B
with e over g and B and summing the result over inequivalent H bundles P Hence
H
for given H G there are as many coset theories as subgroups of H Z some of them might
G
have a non unique vacuum
If H is a discrete subgroup of G then P carries a unique canonical at connection and is
H
given by a conjugation class of homomorphisms of the fundamental group of is into H The
construction from the preceding p oint gives rise to the orbifolds of the WZW mo dels
Sup ersymmetric WZW mo dels One adds to the G valued eld g the anticommuting
Ma jorana Fermi elds in the adjoint representation i e sections of L g and L g
resp ectively where L is a square ro ot of the canonical bundle of and one considers the action
Z
A tr A S g A k S g A
L
L
with the external group G gauge eld A The fermionic part of the theory is free and the
complete theory may b e easily solved
Sup ersymmetric coset mo dels The action is as in eq except that A is replaced by
a group H gauge eld B and the Ma jorana elds are taken with values in g h rather than
in g Both the sup ersymmetric WZW mo dels and the sup ersymmetric coset mo dels p ossess the
N sup erconformal symmetry
N coset mo dels If G H is a K ahler symmetric space then the sup ersymmetric
G H coset mo del p ossesses the N sup erconformal symmetry including the U lo op group
symmetry The simplest examples are provided by the SU U mo dels which at level k
k
give rise to the minimal N sup erconformal theory with central charge c
k
Orbifo ds of tensor pro ducts of conformal eld theories may give rise to essentially new
mo dels The famous example are the Z Z orbifolds of pro duct of ve k minimal N
sup erconformal mo dels Two di erent orbifolds may give equivalent conformal sigma mo dels
corresp onding to a mirror pair of Calabi Yau quintic targets
The theories with U lo op group symmetries like the N sup ersymmetric coset mo dels
may b e twisted by considering their elds as taking values in bundles asso ciated with the sphere
subbundle of the spin bundle L and coupled to the spin connection Such twisting of N
sup erconformal mo dels may b e done in two essentially di erent ways A and B twist and it
pro duces top ological eld theories The genus correlations of the A twisted N sigma
mo dels compute the quantum cohomology of the target
For non compact groups G the WZW action S g is not b ounded b elow but one may
try to stabilize the Euclidean functional integral by analytic continuation or and coset typ e
gauging of subgroups of G Such stabilization pro cedures may however destroy the physical
p ositivity Hilb ert space picture of the theory The b est studied mo dels of the non compact
typ e corresp ond to nite coverings of SL R with the U subgroup twisted and the nilp otent
subgroup gauged away the construction of minimal mo dels a la Drinfeld Sokolov the SL R
N
generalizations thereof the Liouville and To da theories and the SL R U black hole mo del
Our knowledge of non compact WZW mo dels is certainly much less complete than that of the
compact case note that this is true also on the level of quantum mechanics where we know
everything ab out harmonic analysis of compact Lie groups but the harmonic analysis of non
compact ones has still op en problems
References
The basic pap ers on the Wess Zumino Witten mo del are Witten Commun Math Phys
p Polyakov Wiegmann Phys Lett B p Knizhnik
Zamolo dchikov Nucl Phys B p Gepner Witten Nucl Phys B p
E Verlinde Nucl Phys B p Tsuchiya Kanie Adv Stud Pure Math
p Tsuchiya Ueno Yamada Adv Stud Pure Math p
The present lecture follows a geometric approach develop ed in author s pap er in Nucl Phys
B p where the relations b etween the scalar pro duct of non ab elian theta
functions and WZW correlations as well as the conjecture ab out the unitarity of the KZ con
nection were rst formulated See also Functional Integration Geometry and Strings Haba
Sob czyk eds Birkh auser p The scalar pro duct formulae for arbitrary Lie group
were obtained at genus in Falceto Gaw edzki Kupiainen Phys Lett B p and
at genus in Falceto Gaw edzki hep th The case G SU genus was studied in
Gaw edzki Commun Math Phys p and Lett Math Phys p
The functional integral pro of of eq was b orrowed from Witten Commun Math Phys
p
The KZ connection at higher genera app eared in Bernard Nucl Phys B p
and B p and was further discussed in Hitchin Commun Math Phys
p in Axelro d Della Pietra Witten J Di Geom p and by other
authors For the relations b etween the KZ connection Bethe Ansatz and geometric Langlands
corresp ondence see Feigin E Frenkel Reshetikhin Commun Math Phys p and
th
E Frenkel s contribution to XI International Congress of Mathematical Physics Iagolnitzer
ed International Press
The original pap ers on the coset mo dels are Go ddard Kent Olive in Phys Lett B
p short version and Commun Math Phys p The functional integral
approach discussed ab ove follows Gaw edzki Kupiainen Nucl Phys B p see
also Bardakci Rabinovici S aring Nucl Phys B p and Karabali Schnitzer
Nucl Phys B p
General WZW mo dels with non simply connected groups were discussed in Felder Gaw edzki
Kupiainen Commun Math Phys p see also Gaw edzki Nucl Phys B
Pro c Suppl B p
The extension of the coset construction mentioned in Sect is due to Hori hep th
see also Fuchs Schellekens Schweigert hep th for the representation theory counterpart
For orbifolds of WZW mo dels see Kac To dorov hep th The relation of tensor
pro ducts of N CFT s to Calabi Yau sigma mo dels was discovered in Gepner Phys Lett B
p The mirror pair of quintics app eared in this context in Greene Plesser Nucl
Phys B p
The N sup ersymmetric coset mo dels are discussed in Schnitzer Nucl Phys B
p the N ones in Kazama Suzuki Nucl Phys B p
Twisting of N sup erconformal eld theories is discussed in Witten s contribution to
Essays on Mirror Symmetry Yau ed International Press
The SL R WZW mo dels are considered e g in Bershadsky Ooguri Commun Math Phys
n
p See Frenkel Kac Wakimoto Commun Math Phys p for
the representation theory asp ects For To da theories see e g Bilal Gervais Nucl Phys B
p The SL R U black hole is the sub ject of Witten Phys Rev D
p