A Review of Conformal Field Theory in 2 D

In this review we study the elementary structure of Conformal Field Theory in 2d which is a recipe for further studies of critical behavior of various systems in statistical mechanics and quantum field theory. We briefly review CFT in d dimensions which plays a prominent role for example in the well-known duality AdS/CFT in string theory where the CFT lives on the AdS boundary. We also describe the mappingof the theory from the cylinder to a complex plane which will help us gain an insight into the process of radial quantization and radial ordering. Finally we will develop the representation of the Virasoro algebra which is the well-known "Verma module". Indexing terms/


INTRODUCTION
Symmetry principles play an important role with respect to the laws of nature.They summarize the regularities of the laws that are independent of the specific dynamics.Thus invariance principles provide a structure and coherence to the laws of nature just as the laws of nature provide a structure and coherence to the set of events.With the development of quantum mechanics in the 1920s, symmetry principles came to play a fundamental role.In the latter half of the 20th century symmetry has been the most dominant concept in the exploration and formulation of the fundamental laws of physics.Today it serves as a guiding principle in the search for further unification and progress.
Compared to ordinary quantum field theories in four dimensions, conformal field theories in two dimensions can be defined in a rather abstract way via operator algebras and their representation theory.In fact, there are many examples of CFTs where the usual description in terms of a Lagrangian action with resulting perturbative expansion is not even known.Instead, following a so-called boot-strap approach, one can define these theories without making reference to an action and sometimes one can even solve them exactly.Such a procedure is possible because the algebra of infinitesimal conformal transformations in two dimensions is very special in contrast to its higher dimensional counterparts -it is infinite dimensional and therefore highly constraining.
The main feature of a conformal field theory is the invariance under conformal transformations.Roughly speaking, these are transformations leaving angles invariant and a particular example is the scaling x ax  of a distance x by some constant a .A field theory exhibiting such symmetry has no preferred scale and one can only expect a physical system to have this property, if there are no dimensionful scales involved.Polyakov [1] conjectured that systems (with assumptions such as isotropy and possibly locality of interactions) exhibiting scale invariance in 2D possess a symmetry larger than simple scaling.This symmetry group is called the conformal group.As an example, the field theory of a free boson encounters a conformal symmetry for the case of vanishing mass.And even for interacting theories it is known that at the fixed point of a renormalization group flow, there are only long-range correlations.Therefore, the natural mass scale at this point, that is, the inverse of the correlation length, vanishes and a conformal field theory description might be available.Physical systems with a conformal symmetry are thus more common than one would have naively expected.
Another important instance featuring conformal symmetry is string theory, which is a candidate theory for the unification of all interactions including gravity.Here, the CFT arises as a two-dimensional field theory living on the world-volume of a string which moves in some background space-time.The dynamics of this string is governed by a non-linear sigma model whose condition for conformal invariance, that is, the vanishing of the  function, gives the string equations of motion.The sigma model perturbation theory is governed by an expansion in / sR , where s isthe natural string length and R a typical length scale of the background geometry.With the help of CFT techniques, one can sometimes solve such theories exactly to all orders in perturbation theory and one can sum all contributions of so-called worldsheet instantons.Therefore, conformal field theory is a very powerful tool for string theory, not only in the perturbative regime but also at small length scales where genuine string effects become important and geometric intuition often fails.

CONFORMAL SYMMETRY IN D-DIMENSIONS
In this section we discuss the conformal symmetry in arbitrary dimension.We investigate the generators and their commutation relations and further, the conformal group is identified with the non-compact group ( 1,1) SO d  .We study the action of a conformal transformation on fields at the classical level and relate the scale invariance to tracelessness of the energy momentum tensor.We then look at the consequences of conformal invariance at the quantum level on the structure of correlation functions and derive the form of two-and three-point functions and the Ward identities.
Given a metric tensor g  , a conformal transformation of the coordinates is an invertible mapping ' xx  , such that the metric tensor is invariant up to a scale: The set of conformal transformations manifestly form a group.It has the Poincare group as a subgroup which corresponds to ( ) 1 x  .Conformal field theories care only about angles and not about distances and the physics of the theory looks the same at all length scales.
We begin with the consequence of the definition 1. .So to first order in  , the metric will transform in following manner: For this transformation to be conformal, it requires that         is proportional to the metric.Hence N o v e m b e r 0 7 , 2 0 1 4 3) The proportionality factor can be derived by contracting both the sides with the metric: For the sake of simplicity, we assume that the metric is the Cartesian metric , where (1,1,...,1) diag

  
. By applying an extra derivative   on 1.3, permuting the indices and taking a linear combination, one gets And upon contracting with Using this and 1.3, we finally get We can get a good idea of the form of the function f using the above equation for different dimensions.When 1 d  , no constraint is imposed on f and therefore any smooth transformation is a conformal transformation in one dimension.It is quite obvious as the notion of angle doesn't exist in one dimension.For the case A and B  are constants.Now we insert this expression into 1.5 to get the general form of  .As f is linear in x , () x  can be at most quadratic in x .We can write the general expression Here c  is symmetric in  and  .Now we can consider each of the terms in the expansion separately.
which is called the special conformal transformation (SCT).This may also be viewed  , a combination of inversion and translation.Note that the points on the surface  have their distances from the origin preserved while the points on the exterior of this surface are sent to the interior and vice versa.Now the generators of the conformal group (discussed in Appendix A), take the following form N o v e m b e r 0 7 , 2 0 1 4 This admits a total of  generators which as we will shortly demonstrate, is isomorphic to ( 1,1) SO d  .These generators obey the following commutation rules: To make the above commutation rules more familiar, we define the following generators:

Conformal Invariance in Classical Field Theory
We now define the effect of conformal transformations on classical fields.Given an infinitesimal conformal transformation paramerized by g w , a multicomponent field () x  transforms as '( ') ( 1) ( ) The generators g T are added to the space-time part given in 1.13 to obtain the full generator of the symmetry.The method to find the allowed form of these generators is to start by studying the subgroup of the Poincare group that leaves the origin 0 x  invariant, which is the Lorentz group.The action of infinitesimal Lorentz transformation on (0)  is given as: N o v e m b e r 0 7 , 2 0 1 4 S  is the spin associated with the field  .Next, by commutation relations of Poincare group, we translate L  to a nonzero value of x : where we used the Hausdorff formula to evaluate the lhs.We proceed in similar fashion to evaluate the whole group.We obtain the following form for all the generators: 2 ) ( )  is the scaling dimension of the field  .Under a conformal transformation ' xx  , a spinless field  ( 0 S   ) of scaling dimension  , transforms as We can write this expression in terms of the scale factor  from 1.
Now since the infinitesimal volume element should be invariant under the conformal transformation, we have

The Energy-Momentum Tensor
Under an arbitrary transformation ' ( ) x x x   , the action changes as: where T  is the energy momentum tensor.We assume here that T  is symmetric as we know it can always be made symmetric by transforming it to Belinfante form.Now using equation 1.3 and 1.4 for an infinitesimal conformal transformation, we reduce the above equation to the following form: The tracelessness of the energy momentum tensor implies the invariance of the action under conformal transformation.
The converse is however not true as     is not an arbitrary function as   can only be of form shown in 1.9, i.e. at most quadratic in x .Current for a general infinitesimal transformation is given as:  being the canonical energy-momentum tensor.Now, one can show that by using the symmetries of a theory, we can write down a symmetric and traceless energy momentum tensor which is related to the dilation current as We notice that 0 j    since the energy-momentum tensor is traceless.It can be shown that in 2 dimensions no modification is required to the canonical energy-momentum tensor and it is already traceless and similarly related to the current as in above equation.( ' ) ( ' )... ( ' ) ( ( )) ( ( ))... ( ( ))

Conformal Invariance in Quantum Field Theory
where the mapping F describes the functional change of the field under the transformation.

2, 3 and 4-Point Functions
Conformal invariance puts restriction on the form the two and three-point correlation functions of quasi-primary fields.We know how a quasi primary field should transform under conformal transformation and this will put a set of constraints on the structure of the correlation functions.Let us consider a two-point function where  and 2  are quasi-primary fields and [] S e  is the action which is conformally invariant.We also assume that the functional integration measure is conformally invariant.The correlation functions will transform in the following manner Here i  is the scaling dimension of field i  .Now translation and rotation invariance require that In other words we can write '' '' (1 2 . ) (1 2 .) Covariance of 1.39 under special conformal transformation implies This is true only when 12    .This means that two quasi-primary fields are correlated only if they have same scaling dimensions.Similarly we can argue that a three-point correlation function should have the following form: where ij i j x x x  and a,b,c are such that N o v e m b e r 0 7 , 2 0 1 4 Demanding invariance under special conformal transformation, we have This is true when we have the following set of constraints Therefore the correlator of three quasi-primary fields can be written as The conformal invariance however does not provide enough constraints to fix the structure of four-point correlation function or beyond.This is because the global conformal transformations only allow us to fix threecoordinates, so the best we can do is to take say , , , ,1, ,0 x x x x x  .Indeed, with four points, it is possible to construct conformal invariants, the anharmonic ratios and the residual x will be a function of these ratios.The best we can do is write the four-point correlation function in terms of these ratios where     and the function f is undetermined.

Ward Identities
The Ward identities are used to exhibit the consequence of the symmetry of action and measure on the correlation functions.As shown in the appendix [C], the ward identity for a current a j  is given as .Hence if we define X as a product of n local fields, we have () Now we find the Ward identity associated with Lorentz invariance.The current is written as where T  is the symmetrized tensor.The generator of Lorentz transformation is given as N o v e m b e r 0 7 , 2 0 1 4 S  is the spin generator.Consequently the Ward identity is On the lhs, we use 1.50 to reduce the above identity to which states that the energy-momentum tensor is symmetric within correlation functions except at the positions of the fields.
Finally we derive the Ward identity for the scale invariance.As shown in 1.19 and 1.32, the dilation current is      respectively.We insert these in 1.49 ( )( ) Again using the 1.47, this reduces to () The trace of the energy-momentum tensor vanishes except at the location of the fields.The 3 equations 1.50, 1.54 and 1.56 are the Ward identities associated with the conformal symmetry.

CONFORMAL TRANSFORMATIONS IN 2 DIMENSIONS
Unlike other dimensions, where the conformal group is finite dimensional and global, in 2D , it is local as well as infinite dimensional.The condition for a transformation to be conformal is the same as the Cauchy-Riemann condition for an analytic function.Since there are an infinite number of analytic functions on a plane, this implies that the conformal group is infinite dimensional.We develop the language of holomorphic and antiholomorphic coordinates on a plane which is a prominent tool in 2 D  CFT.We exhibit the distinction between local and global transformations and introduce generators for local conformal transformations.

Conformal Group in 2D
In 2 dimensions, it is convenient to work with complex coordinates.We introduce complex coordinates In this coordinate system, the metric tensor and its inverse look like where  takes the values z and z .Hence, the line element can be written as 2 ds dzd z  .Also the antisymmetric tensor   and   are Now using the form of metric tensor as in eq.2.2, we find Solving these equations, we find they are equivalent either to We notice these are the Cauchy-Riemann equations for holomorphic and antiholomorphic equations.In terms of the complex coordinates these equations are written as The solutions to these two equations are any anti-holomorphic or holomorphic mapping respectively. () Two dimensional conformal transformations thus coincide with the analytic coordinate transformations, the local algebra of which is infinite dimensional.

Conformal Generators
To find the commutation relations of the generators of the conformal algebra, i.e. the mapping of the form 2.8, we take the infinitesimal transformation ' ( ) ' ( ) where The corresponding infinitesimal generators are These generators obey the following commutation relations: [ , ] ( ) In quantum case the algebra will be corrected by adding an extra term related to the central charge.This is called the Witt's algebra.Since n l commute with m l , the conformal algebra is direct sum of two isomorphic algebras.
We notice that the above algebra is a local algebra as the generators are not all well-defined globally on the Riemann sphere.Holomorphic conformal transformations are generated by

Primary fields
The global conformal algebra generated by A field whose variation under any local transformation is given by above equations is called a primary field.All primary fields are quasi-primary but reverse is not true.

Correlation Functions
Expressed in terms of holomorphic and anti-holomorphic coordinates, the conformal transformation of correlation function of n primary fields i  with conformal dimensions i h and i h is written as: This relation fixes the form of two-and three-point functions.We can use the conformal constraints similar to how we did in earlier sections and find the two-and three-point functions.However here we follow another method just to get a better insight.Now a two-point function will satisfy the following equation under infinitesimal transformation: (2.21) Now using 2.19, this gives We know  and  are at most quadratic in the coordinates.Let us first take G depends only on  For a spinless field, from 2.17, we get /2 hh    .Then the above equation is equivalent to We can similarly find the three-point function

{} {} ii h h h h h h h h h h h h h h h
Note that any three points ,,, z z z z also known as anharmonic ratios .The general expression for four-point function can be written as

Ward Identities
In previous chapter we derived the Ward identities corresponding to translation, rotation and scale invariance in 1.50, 1.54 and 1.56 respectively.Let us assemble the three equations here: ()

29)
Here i s stands for the spin of the field i  .Now we want to rewrite these equations in terms of complex coordinates and complex components.We use the expressions 2.2 and 2.3 for the metric and antisymmetric tensor.For the delta function One may in principle use either one of the above representations but we will use the first one if the integrand is holomorphic and vice-versa.The Ward identities can now be written as N o v e m b e r 0 7 , 2 0 1 4 We add and subtract the last two equations to get where now the holomorphic and anti-holomorphic separate and we use the appropriate definition of delta function from 2.30 and conformal dimensions from 2.17.Inserting these relations into the first two equations from 2.32, we get where we have introduced the normalized energy-momentum tensor ( , ) 2 ( , ) 2 Now by definition, the expressions inside braces in the above equations are holomorphic and anti-holomorphic respectively.Hence we can write where reg.refers to holomorphic functions of z, regular at i zw  .And we will have a similar expression for anti- holomorphic term too.

Conformal Ward Identity
We can bring all the derived Ward identities 2.27, 2.28 and 2.29 into a single equation as follows.Given a conformal transformation where we have used N o v e m b e r 0 7 , 2 0 1 4 We note here, the first term is the translation transformation, (1/ 2) Here integral is taken over a domain M which contains the position of all the fields in the string X .Since the integrand is a divergence, we use the Gauss's theorem to reduce this to the form where we have used 2.2 and 2.3.Further using the 2.34, we write down the conformal Ward identity: The contour C needs to include all the positions ( , ) ii wwof the fields contained in X .

Operator Product Expansion
The equation 2.35 delivers important information about the product of energy momentum tensor with primary fields.It yields a singular behavior of the correlator of the field () Tzwith a primary field ( , ) iii ww , as i zw  .For a single primary field of conformal dimensions h and h , this is written as Here "~" means equality up to terms which are regular as zw  .The operator product expansion (OPE) implies that two local operators inserted at nearby points can be closely approximated by a strings of operators at one of these points [5].
In general language, if we denote all the operators of CFT by i O , then the OPE is C are a set of functions which on ground of translation invariance depend only on the separation between two operators.It is important to note that the above equations are always to be understood as operator insertions between correlation functions.Also the correlation functions are always assumed to be time-ordered.The singular behavior of OPE as zw  will really be the only thing we care about!It will turn out to contain the same information as commutation relations, as well as telling us how operators transform under symmetries.We now proceed with specific example to familiarize ourselves with simple but important systems and basic techniques.

The Free Boson
A simplest example of CFT is a free massless bosonic theory.The action for such a bosonic field is The two-point function or the propagator for this action is given by This OPE reflects the bosonic character of the field as interchanging the two fields doesn't change the correlator.The energy momentum tensor for this theory is given by In terms of complex coordinates, we write the energy-momentum defined in 2.34 as where we have normal ordered the fields.More explicitly, the above expression means We now find the OPE of ()  .This is written as   r 0 7 , 2 0 1 4 We get the first term by two double contractions and second term by four single contractions.Again by expanding the second term around w , we find the above equation is equivalent to We note that the energy-momentum tensor is not a primary field as its OPE has an anomalous term which doesn't appear in 2.41.

The Central Charge
In CFT, the most prominent example of an operator which is not primary is as we saw earlier, the energy-momentum tensor.We worked it out for the free scalar model, but it remains true for all CFT.It is in fact a quasi-primary operator of weight ( , ) (2, 0) hh  . Generally in any CFT, the TT OPE takes the following form We can already get some intuition for the information contained in these two numbers.Looking back at the free scalar field, we see thatit has From the definition it seems natural that the energy-momentum tensor should transform as a rank-2 covariant tensor, T is called the Schwinger term.Notice that the Schwarzian derivative vanishes for a global conformal map which is true for a quasi-primary field.Now the Schwinger term is independent of T and only effects the constant term or the zero mode in the energy.In other words, it is the Casimir energy of the system.We look at an example to understand the physical significance of the central charge.Consider a Euclidean cylinder, parameterized by We make a conformal transformation from cylinder to a complex plane by  The fact that the cylinder and the plane are related by a conformal map means that if we understand a given CFT on the cylinder, then we immediately understand it on the plane.Notice that constant time slices on the cylinder are mapped to circles of constant radius.The origin, 0 z  is the distant past,   .Now the Schwarzian for this mapping can be calculated and it gives ( , ) 1/ 2 S z w  .This gives us the way Now suppose the ground state energy vanishes when the theory is defined on the plane.We then calculate the energy on the cylinder () The conformal transformation then tells us that the ground state energy on the cylinder is This is indeed the (negative) Casimir energy on a cylinder.For a free scalar field, we have

THE OPERATOR FORMALISM
Throughout the previous sections, all our manipulations were assumed to hold inside correlation functions.The consequences of conformal symmetry on two-dimensional field theories were embodied in constraints imposed on these correlation functions via the Ward identities.These Ward identities were most easily expressed in the form of an OPE of the energy-momentum tensor with local fields.Up to now, we only used the path-integral representation of the theory in which all correlation functions could in principle be obtained.We would now like to give an operator interpretation in terms of states in a Hilbert space.

Radial Quantization
The operator formalism distinguishes a time direction from a space direction.This in natural in Minkowski space-time, but somewhat arbitrary in euclidian space.This allows choosing the radial direction from the origin as time direction, and the space direction being orthogonal to it.This choice leads to the so-called radial quantization of two-dimensional conformal field theories.
We may start from a two-dimensional Minkowski space with coordinates t and .One usually takes the space direction  to be periodic,

2[0, ] L  
, defining this way the theory on a cylinder.We continue to Euclidian space, ti   and then perform the conformal transformation which maps the cylinder onto the complex plane

{} C  
, topologically the Riemann sphere.Surfaces of equal Euclidian time  on the cylinder will become circles of equal radii on the complex plane.This means that the infinite past    gets mapped onto the origin of the plane ( 0) z  and the infinite future becomes z .

Mapping sends constant time to constant radii
We assume the existence of a vacuum state 0 upon which a Hilbert space is constructed by application of creation operators.In free field theories, the vacuum may be defined as the state annihilated by the positive frequency part of the field.For an interacting field  , we assume the same Hilbert space except that the actual energy eigenstates are different.
We suppose then that the interaction is attenuated as t   and the asymptotic field lim ( , ) is free.Within radial quantization, this asymptotic field reduces to a single operator, which upon acting on 0 creates a single asymptotic "in" state: We define a bilinear product in this Hilbert space.This can be done by defining an asymptotic "out" state together with action of Hermitian conjugation on the conformal fields.In Minkowski space, Hermitian conjugation does not effect the space-time coordinates.However in Euclidian space since the time it   , it must be reversed upon Hermitian conjugation if t is to be left unchanged.In radial quantization this can be done by mapping * 1/ zz  .This will justify the following definition of Hermitian conjugation on the real surface where  is a quasi-primary field of dimensions h and h .The above equation can be justified by demanding that the asymptotic "out" state †

Mode Expansion
A conformal field ( , ) zz  of dimensions ( , ) hh can be expanded in modes in following manner: We use the fact that holomorphic and anti-holomorphic degrees of freedom decouple and we will drop the dependence of fields upon the anti-holomorphic coordinates for sake of ease.We rewrite the mode expansion in simplified form

Radial Ordering
In a quantum field theory, we are interested in time-ordered correlation functions.Time ordering on the cylinder becomes radial ordering on the plane.Operators in correlation functions are ordered so that those inserted at larger radial distance aremoved to the left.

Subtraction of Contours
The definition of radially ordered correlation functions becomes N o v e m b e r 0 7 , 2 0 1 4 If the two fields are fermions, a minus sign is added in front of the second expression.Since all fields within correlation functions must be radially ordered, so must be the l.h.s. of any OPE if it is to have an operator meaning.
We now relate OPEs to commutation relations.Let () azand () bz be two holomorphic fields and consider the integral We take 0   as () bwshould be the only field present between the two contours as there can be arbitrary number of fields between () azand () bw.This makes the commutator an equal-time commutator.In practice, the integral in 3.14 can be evaluated by substituting the OPE of () azand () bw, of which on the term 1/ ( ) zw  contributes, by theorem of residues.
The commutator [ , ] ABof two operators, each the integral of a holomorphic field is obtained by integrating 3.16 over w : where the integral over $z$ is taken around w and the integral over w is around the origin.Also We have managed to find a relation between OPEs to commutators and this allows us to translate into operator language the symmetry statements contained in the OPEs.which means that the operator Q  is the generator of conformal transformation.

Virasoro Algebra and Conformal Generators
We expand the energy-momentum tensor according to 3.7: We also expand the infinitesimal conformal change LL  is proportional to the Hamiltonian of the system.These quantum generators obey the following algebra which is similar to the classical Witt algebra except for the appearance of central charge of the theory.This is called the Virasoro algebra.Notice that only the global subgroup (2, ) SL C is not effected by the central charge.

The Hilbert Space and Verma Module
The vacuum state 0 must be invariant under global conformal transformation.This means that it must be annihilated This means that excited states can be obtained by successive applications of these operators on the asymptotic state h .This results in an infinite tower of states and all the states obtained in this way are called the 'descendants' of state h .
From this initial primary state, the tower fans out: The whole set of states is called the 'Verma module'.They are the irreducible representations of the Virasoro algebra.This means that if we know the spectrum of primary states, then we know the spectrum of the whole theory.We are not guaranteed however that all the above states are linearly independent.That depends on the structure of the Virasoro algebra for given values of h and c .A linear combination of states that vanishes is known as a null state, and the representation of the Virasoro algebra with highest weight is constructed from the above Verma module by removing all null states (and their descendants).

DISCUSSION
In this review, we investigated the elementary structure of Conformal Field Theory in 2d which is a recipe for further studies of critical behavior of various systems in statistical mechanics and quantum field theory.We briefly reviewed CFT in d dimensions which plays a prominent role for example in the well-known duality AdS/CFT in string theory where the CFT lives on the AdS boundary.
We studied the generators of conformal transformations and derived their commutation relations.The commutation group in $d$ dimensions was identified with the non-compact group ( 1,1) SO d  .The notion of a "quasi primary field" was developed which are covariant under global conformal transformations.Using the constraints of conformal symmetry, we derived the forms of 2  , 3  and 4  point correlation functions.We further derived the Ward identities associated with the translation, rotation and scale invariance of the theory.
Main goal of our study was to understand the development of the structure of a conformal field theory in 2 dimensions. 2 dimension case is special as the conformal symmetry becomes a local symmetry of theory and we have an infinite set of mapping from the 2d complex plane onto itself.The earlier derived group of global conformal symmetry turns out to be a subset of the 2d CFT algebra.We developed the essential language of holomorphic and anti-holomorphic coordinates which is a very helpful tool in the study of 2d CFT.We defined the notion of "primary fields" and calculated their correlation functions.We also introduced the concept of short-distance product of operators and applied this language to specific example of free bosons.Transformation properties of the energy-momentum tensor were explored for this example and the idea of central charge c was explained in brief.
In the last section, we mapped our theory from the cylinder to a complex plane which helped us gain an insight into the process of radial quantization and radial ordering.Then carrying out the mode expansion of fields led to development of the "Virasoro algebra".Once we had the algebra and the generators in Hilbert Space, we could develop the whole tower of descendant fields given a primary field.This infinite tower of states generated from a given asymptotic state form a representation of the Virasoro algebra which is the well-known "Verma module".
The last section ends with an introduction to the Verma module which provides the basic tool to study particularly simple conformal field theories called 'minimal models'.

C: Delta Function in complex coordinates
In terms of holomorphic and anti-holomorphic coordinates, delta function () x  can be written in following way To justify this identity, let us consider a vector F  whose divergence is integrated within a region M of the complex plane bounded by a contour M  .Applying the Gauss's theorem this gives us ordinary translation independent of x .To get the properties of b  , we insert bx b  is a sum of antisymmetric part and a pure trace: signifies a scale transformation while the antisymmetric part is an infinitesimal rigid rotation.Substitution of the quadratic term into eq 1.5 yields c b b b

JJ
 and a,b take values { 1,0,1,..., } d  .These new set of operators obey the has Minkowski signature, ( 1,1,...,1)  and is constructed by adding two dimensions of signature ( 1, 1)  to the d Euclidean dimensions.This shows the isomorphism between the conformal group in d dimensions and the group ( 1,1) SO d  .Note that the d dimensional Euclidean group along with dilations forms a subgroup ( ) (1,1) SO d SO  of the full conformal group.Note that if we work in d dimension space-time with Minkowski signature, the conformal group will be ( , 2) SO d .
the determinant of the metric, the above equation is equivalent to  is the Jacobian of conformal transformation.This gives the following relation between the Jacobian and the scale factor: transforming in above form is called a "quasi-primary" field.N o v e m b e r 0 7 , 2 0 1 4

C
is a constant coefficient.Recall that the function should also be invariant under special conformal transformation.Under such a transformation

aG
is the generator of symmetry transformation.We use this equation to write down the Ward identity for translation invariance.The energy momentum tensor is the current associated with translation invariance and the corresponding generator is

3 )
Now under a conformal transformation ' xx  , the metric tensor transforms as N o v e m b e r 0 7 , 2 0 1 4 allows non-zero n a only for 1 n  .To check the behavior of () vzas z , we carry out a transformation at 0 w  demands that 0 n a  only for 1 n  .Hence we find that only conformal transformations generated by nn al for 0, 1 n  are defined globally.These transformations form a subgroup of the whole conformal group which are well-defined and invertible over the Riemann sphere (i.e. the whole complex plane plus the point at infinity).The globally defined generators are 1 0 1 1 0 1 { , } { , } l l l l l l    and they form the special conformal group.From 2.10, we can identify the 1 l  and 1 l as generators of translations,  as generators of scale transformation and rotation and 11 , llas generators of special conformal transformations.A complete set of such global conformal transformations can be written as b,c,d are complex numbers.These mappings are called the projective transformations and they form the (2, ) SL C group .To each global conformal transformation we can associate the matrix, SL C language, we can write the transformations as characterizing properties of physical states.If we work in a basis of eigenstates of the two operators 0 l and 0 l , and denote their eigenvalues by h and h respectively (here h and h are meant to indicate independent (real) quantities, not complex conjugates of one another), they are known as the conformal weights of the state.Since 0 0 ll  and 0 0 () i l l  generate dilatations and rotations respectively, the scaling dimension  and the planar spin s of the state are given by hh    and s h h .Or we N o v e m b e r 0 7 , 2 0 1 4can say for given a field with scaling dimensions  and spin s , we write the holomorphic conformal dimension h the map zw  is close to the identity, i.e. and  very small, the variation of the quasi-primary field is ,  .This results in the following form of the two-point function 24) N o v e m b e r 0 7 , 2 0 1 4

G
    to take the form[3] can always be mapped to three reference points ,or the global conformal transformations, which will reduce the above equation to for higher dimensions, conformal invariance doesn't fix the precise form of four-point function and beyond.The four-point function can be written down as a real function of ratios of the four variables

2 (X
scaling while (1/ 2)     is the local rotation angle.Integrating both sides of the equation above, the three Ward identities can be encapsulated into under local conformal transformation.

( 2 .
43) N o v e m b e r 0 7 , 2 0 1 4 have any interesting property under conformal transformation and for this reason the field ( , ) zz  is not really the object of interest in this theory.Instead we separate the holomorphic and the anti-holomorphic parts of above equation by taking derivative with respect to holomorphic and anti-holomorphic coordinates, z  and z  ,and in the process we will discover important aspects of the theory 2.50) Using Wick's theorem and 2.46, this is


is a primary field with conformal dimension 1 h  which is true as  has spin 0 s  and scaling dimension 0  and the derivative operator  has spin s  and 1  .We can also find the OPE of the energy- momentum tensor with itself 2 42 ( ) ( ) 4 : ( ) ( ) :: ( ) ( ) :


are called the central charges.They are perhaps the most important numbers characterizing a CFT.

.
If we instead considered D non-interacting free scalar fields, we would get c c D   .This gives us a hint: c and c  are somehow measuring the number of degrees of freedom in the CFT.Note that c not necessarily an integer i.e.

.
However it turns out, because of the presence of the anomalous term in the TT OPE, the above equation is corrected to derivative.This additional term appearing during the transformation of

Figure 3 . 1 :
Figure 3.1: Conformal mapping from cylinder to complex plane Time reversal becomes () i z z e      on the complex plane.N o v e m b e r 0 7 , 2 0 1 4

,
-defined inner product with the in  .Using the above defined formula for Hermitian conjugation, we get the inner product to be † form of two-point function as shown in 2.23, the above expression is independent of  .This justifies the presence of the prefactors in defining the Hermitian conjugate for if they were absent, the inner product out in would not have been well defined as  .N o v e m b e r 0 7 , 2 0 1 4 two expression for the Hermitian conjugate of modes to match, we must have † if the "in" and "out" are to be well-defined, the vacuum state must satisfy

2 C
contour circles counterclockwise around w .This expression has an operator meaning within correlation function as long as it is radially ordered.Accordingly we split the contour into two fixed time circles going in opposite directions.Then the above integral resembles a commutator: are the fixed time contour of radii respectively equal to || w   and || w   ,  being infinitesimal and the operator A is the integral over fixed time of the field () az:

Let () z ( 3 .
be the holomorphic component of an infinitesimal conformal change in the coordinates.We define the conformal 21) N o v e m b e r 0 7 , 2 0 1 4

1 L  , 0 L and 1 L
write the conformal charge 3.19 in terms of these modes as displays that the mode operators n L and n L of the energy momentum operator are the generators of the local conformal transformations on the Hilbert space, exactly like the n l and n l of 2.12are generators of conformal mappings on the space of functions.Similarly are the generators of (2 ) SL C .In particular the operator fact time translation in radial quantization.Thus 0 0

1 L
which in turn can be recovered by requiring the regularity of ( ) 0 Tz at $z=0$.Evidently only the terms with 1 n  are allowed.So we learn that invariance of vacuum 0 with respect to the global conformal group.It also implies the vanishing of the vacuum expectation value of the energy momentum tensor: of primary fields on vacuum state creates asymptotic states which are eigenstates of the Hamiltonian.To show this, let us first find the commutator of n L with a primary field ( , ) zz  of conformal dimensions ( , ) hh .N o v e m b e r 0 Now to find the ladder operators which on application give the excited states above the asymptotic state , hh , we compute the commutator of the modes of a holomorphic field () w reduces the conformal dimension of the state by m .Similarly m   acts as a raising operator.The generators m L  , ( 0) m  also increase the conformal dimension as we have from 3.25.37) N o v e m b e r 0 7 , 2 0 1 4 Atth N level in this tower, there will be () PN fields with conformal dimension hN  , where () PN is the number of partitions of N into positive integer parts.() PN is given in terms of generating function

2 )
These theories are characterized by a Hilbert space made of a finite number of representations of Virasoro algebra and are used to describe discrete statistical models (e.g.Ising model) at their critical points.N o v e m b e r 0 We assume the functional integration measure is invariant under local transformation ( . .[ '] [ ]) i e d d    .Expanding the equation above to first order in 3 satisfies for any infinitesimal function () a wx, so we can write, using eqs.B.3 and B.4, the Ward identity for where  is an outward-directed differential of circumference, orthogonal to the boundary M  .Let us use a counterclockwise differential ds  , parallel to M  circles counterclockwise.We now consider a holomorphic function () fzand check if the above defined delta function works fine by integrating it against () fzwithin a neighborhood M of the origin: