PoS(Modave VIII)001 dimensions http://pos.sissa.it/ d 2, including an introduction to the operator = d † ∗ [email protected] formalism and a discussion of the trace anomaly. Speaker. FNRS Research Follow and finish with the analysis of the special case An elementary introduction to Conformal Fieldmetries Theory in is classical provided. and quantum We field start theoriesby by and Noether’s reviewing the theorem. sym- associated existence Next of we conserved describe currents general facts on conformal invariance in ∗ † Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c Eighth Modave Summer School in Mathematical26th Physics august - 1st september 2012 Modave, Belgium

Antonin Rovai Université Libre de Bruxelles and InternationalE-mail: Solvay Institues Introduction to Conformal Field Theory PoS(Modave VIII)001 3 4 6 8 2 3 30 32 26 27 29 26 23 19 19 10 11 11 15 16 16 Antonin Rovai 2 dimensions d Operator Product Expansion Virasoro algebra Radial order and mode expansion 3.3.2 3.3.3 Trace anomaly 3.3.1 Operator Formalism Ward identities Identification of the transformations and analysis Definitions Noether’s theorem The energy-momentum tensor Consequences for the quantum theory General considerations and algebra Action of conformal transformations on fields Energy-momentum tensor and conformal invariance Conformal invariance in quantum field theory and Ward identities These lectures notes were written for the eighth edition of the Modave Summer School in 3.4 3.3 3.2 3.1 1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4 . Conformal invariance in . Conformal invariance in two dimensions . Symmetries and Conservation laws Foreword Mathematical Physics. During this edition, thereField were Theory several basics, lectures and [1, thus 2] self-consistency relyingthis of vast on the subject. Conformal school The required present an lectures elementary werethe introduction scheduled gong to at the show very on beginning of the thetaken first school, day from right after and very in basics theatmosphere to morning of advanced of the the research charming second topics village day.and throughout of I People Modave, the hope were Belgium. this week, then school This smoothly overall will eighth continue in edition to the exist was for relaxed a many real more success, years to come. References 2 3 1 Introduction Contents Foreword Introduction to Conformal Field Theory PoS(Modave VIII)001 - d , the last d ). Vector and Antonin Rovai ,..., 2. We introduce 1 3.3.1 with basic results = = conformal transfor- 1 d µ , we introduce and study 2 with ) µ x = ( for an equality valid by definition. x ≡ . Conformal invariance is a vast topic, and 3 we focus on the special case of 3 1 and we use the symbol = dimensions. We discuss finite transformations as well as the algebra ¯ h d local scale tranformations = c . Space-time points are denoted by d R 4 super-Yang-Mills in four dimensions. = being Euclidean time (except at the end of chapter 3, from section N d x , that may be seen as Symmetries lie at the heart of our modern conception of physics. It is therefore very important For (much) more details on Conformal Field Theory, we refer the interested reader to the Our conventions are the following: unless explicitly stated, we work in the euclidean These lecture notes consist of an elementary introduction to Conformal Field Theory (CFT). The plan of these notes is as follows. We start very slowly in section to understand how we formulate theon symmetry properties observables. of a given In theorypreparing and this the their ground section consequences for we the analysis quickly of review conformal invariance. some elementary notions along these lines, 1. Symmetries and Conservation laws abundant literature. The standardFrancesco, reference Mathieu and for Sénéchal a [3], modern onlecture which treatment notes most by is of Ginsparg the the [4]. very present For[5]. lectures nice a Note is more also book based. old-fashioned that by Polchinski’s presentation, See book Di see also [6]a the the and quick lectures Tong’s lecture introduction by notes to Coleman [7] conformal on invariance Stringa in Theory nice both two introduction include dimensions. to more See advanced also topics. the chapter 15 in [8] for primary fields, study Ward identitiestensor, and operator formalism the and associated operator product consequences expansion.of for We end the the these trace lectures energy-momentum anomaly. with a discussion dimensional flat space component form indices are lowered anduse raised natural using units the in which flat metric, and we sum over repeated indices. We concerning symmetries in field theory, both at thewill classical be and quite at general the quantum and level. severalmomentum The examples discussion tensor, will which be play presented, a with crucial special rôle emphasis in on the the following. energy- In section This class of theories have the property that they are invariant under so called Introduction to Conformal Field Theory Introduction mations in these notes we will beance able to was cover first only the used very tofrom basic study elements. the Historically, statistical divergence conformal of systems invari- the atof correlation a fundamental length. critical interactions: Since point, it then, where is thea scale use at two-dimensional invariance of the CFT; follows CFTs heart in has of twothe spread string resulting dimensions, to theory, effective when the the gravitational field quantum world-sheet action theoryunderstood gravity is example being the is of itself Liouville coupled a action gauge/gravity to which dualityside, a that is namely matter we a know CFT, CFT; of finally, involves the a best CFT on the field theory conformal transformations in of the conformal group, and we exploreand some simple three-point consequences functions. of Finally conformal invariance in on section two- PoS(Modave VIII)001 , 0 F S (1.7) (1.2) (1.4) (1.5) (1.6) (1.8) (1.9) (1.1) (1.3) 7→ . 0 S S = is fixed by S Φ Antonin Rovai the transformation µ , together with some , which are functions d V Φ R ∈ . x ) a + x ( 0 , , ) . , the transformation is Φ . x ) )] ( ϕ ) x ν Φ λρ , ( x , . For a vector field ) = µ δ ( of the rotation group, we write the corre- , 0 V ν ν . The dynamics of the fields Φ ∂ ν )) x [ x defined by x a ϕ d , L = ( µ x ν R ] ν 0 R ( Φ + µ ρ L µ R Φ ( Φ ν [ Φ x R ) = R ( S R x L · = = = ) = ) = λ F 4 ) = 0 x x x µ R µ µ x d · x · 0 ( ) R ( 0 d x x , and the transformation is a symmetry if ) = R R ] 0 · Φ ϕ ( ( µν Z 0 x as µ that we consider are scalars under translation, that is, R Φ δ ( 0 [ is some invertible function of ( 0 Φ S F V d Φ Φ ] = )) = R x Φ [ ( ] = defined by 0 ∈ S 0 Φ 0 or by the action Φ ( x [ Φ 0 ) F S Φ 7→ µ ∂ . For example, for a scalar field Φ , Φ is such that , where Φ ) ( R x corresponding to rotations is characterised by the representation that we ( 0 L x F . Under such a transformation, the action will generally be modified: are simply defined by . Most of the fields transforming in any representation d F 7→ Φ are given by R x ∈ a defined by the equation 0 where we use the common notation is sponding transformation function where the matrix The function and so on forspinor tensors representations of of the various rotation ranks.however group, continue that to In is, write representations the “representation up quantumrepresentation of to of rotations” theory, a the instead we phase. rotation of group.” are We “representation shall also of spinor For interested a in field Rotations choose for the field where reduces to the identity: Translations S We consider a classical theory for some fields, collectively denoted as Let us consider some examples. 2. 1. for some function transformation of the fields Introduction to Conformal Field Theory 1.1 Definitions on a space-time manifold that wea shall Lagrangian take density to be flat Consider a map with PoS(Modave VIII)001 the x (1.16) (1.11) (1.12) (1.14) (1.15) (1.10) (1.13) λ = 0 x Antonin Rovai . )) . is given by x ] ( . ϕ ∆ R [ Φ S µ ∂ 6= is given by ∆ . In this case the jacobian of the , −  1 Φ 2 )) ϕ L 0 , − µ , ϕ x ) for a given field will always coincide λ of ( 2 ϕ x , then under the dilatation , 0 2 ( ) ∆ ϕ∂ m ϕ Φ · x µ Φ 2 0 ( µ ∂ ∆ 2 x λ ∂ Φ − 2 1 , 2 − λ ∆ ) + λ 0 − d x ϕ = ( λ ) = 0 0 µ ( = 5 ) = x ϕ Φ x µ ∆ ( L ϕ∂ ∂ λ µ x , ( L 0 d ∂ ϕ 0 d 1 2 ( Φ x scaling dimension d  Z , namely the Lorentz transformations. As far as these lec- and hence d L d 1 that we shall use throughout these notes). It should however x d ) for the scalar field µν λ Z d λ = d η = by ¯ h = ] = 0 Z 1.14 0 | = Φ x Φ (or dilatations) read c [ ∂ ] = 0 0 / S is called the 0 ϕ x [ 0 ∆ ) is given by ∂ S | is invariant if and only if the scaling dimension S 1.12 0. We define is the linear operator representing the transformation 6= R L λ 2 to the lagrangian ( / 2 ϕ 2 where the number and the action transformation is that is, Scale transformations Note that in Minkowskian space-time,serve the we Minkowski are metric more interestedtures in are transformations concerned, the that differences pre- not between be the important. rotation group and the Lorentz group will where In the case of a free theory for a massless scalar field m 3. Mass terms are thus prohibitedconstants. in dilatation invariant theories, as well as dimensionful coupling Scaling dimension versus mass dimension As far as classical field theory is concerned, the value of transformed action ( Introduction to Conformal Field Theory with the opposite ofdimensionless its (in units mass where dimension, itself determined by the requirement that the action is be clear that the requirementis of dimensionless, scale since invariance under isother a not quantities dilatation, (like a coupling only consequence constants of fieldsterm or the and masses) fact are coordinates held that are fixed. the changed, For action whereas instance, if all we add a mass PoS(Modave VIII)001 of the (1.22) (1.17) (1.18) (1.20) (1.21) (1.23) (1.25) (1.26) (1.19) (1.24) and is µ is in this a νµ a ω for rotations − Antonin Rovai µν = L ), the effect of an µν ω by 1.9 a is characterised contin- 0 x as in ( , 7→ ) L x x ), reads . . . ( , , ) ), the generators 0 ] Φ µν ν )) a x S generators G Φ 1.20 x = [ · a G . ( µρ 1.25 a a µ ν µν Φ ) + F F δ x ω ( S δω i µ . δ ν δ , δω δω F − ∂ µ µ a − µν δ ν x ρ ∂ ) we thus find that − ω i ω x x a ) and ( δ i δω 2 − Φ ω ) = + − 1.4 a µν µ x , µ ν − δ = ( ω ∂ 1.24 x ( 6 ∂ ) + a µ µ Φ µν Φ + 1 2 µ x x P = ( δ x x − δ ( δω µ = i Φ ) 0 ] = = = x x . Using ( = 0 Φ ( = ν µ [ µ 0 νρ µ x x x R a Φ Φ )) = µν δ for an infinitesimal translation are the components a L δ and define by equation ( δω representing the rotation algebra, the numerical factors being δω x = L a ( ) is introduced for later convenience. µ iG µ ω ) = νµ P Φ ( x S ω is understood. We define the ( 1.19 F : − a Φ . We can then consider a transformation “close to identity,” that is, µ a ω = δ ω defining the infinitesimal transformation, and thus the index µν ) are thus given by a S 1.20 transforming under a general representation ) then yields the following variation: Φ 1.17 in the definition ( i and defined in ( case a space-time index given by infinitesimal vector The generator, that we write Formula ( infinitesimal rotation is of the form for some operators introduced for future convenience. Using ( For a field We now identify the parameters and the generators in our examples. Let us consider a continuous transformation, that is, the map 2. A infinitesimal rotation is characterised by an antisymmetric matrix 1. Translations: the parameters we can write where summation on the index The factor of Introduction to Conformal Field Theory 1.2 Noether’s theorem uously by some parameters for and hence PoS(Modave VIII)001 ) ε is is D by + a x µ a 1 ∂ 1.33 Q ) then (1.35) (1.33) (1.30) (1.31) (1.28) (1.29) (1.32) (1.27) (1.34) / 0 = x in ( λ ∂ a 1.18 . ω = Antonin Rovai ! J ). If we call a F δ δω 1.11 a . We write where ω λ . Equation ( 2 µ ) · µ are arbitrary (but small) ∂ J x a
( a F = Φ ω O δ δω , µ L + ) ) ∂ is conserved, the charge δε ∂ x . J Φ / ( 2 µ a ∂ µ tr µ canonical current densities j L j ) 0 Φ 0. Since the functions ∂ x ) ∂ ( ω + + , ε ( δ = ∂ 1 ) Φ ∆ O S x by the formula ρ , − ( δ − . ∂ + a , ∆ a ) = µ ν a ω 1 ρ d a J Q x )  ∂ νµ a x + x a µ δ δω µ B . δ + δω x j ( ∂ x x ν 1 0 a µ · 1 a ) = (  ∂ j δ δω ( − 0 = x ω d a = x µ ) then gives µ + L d ( 7 ∂ ∂ 0 ω = ∂ µ a Σ µ µ · a ν j j x
Z Φ  δ x d µ 1.18 iD φ µ 0, completing the proof. d 0. Note that several currents may give rise to the same , the transformation parameter is ∂ = = − µ ∂ L x ∂ a µ a = Z = Φ ∂ )) = ˜ λ j + ) yields Q ). When the equations of motion are satisfied, the action is ν x d µ a 1 ∂ = j ( ∂ , we find x = ) and ( 0 0 ) d µ S Φ x = which is a consequence of the definition ( − x 1.27 / ( ∂ 1.29 Φ δ

a 0 ∆ µ F L 1.17 7→ L x 7→ Q x ∂ − ∂ a ( x ∂ µ x ∂ x

∂ = δ δω , we find after some algebra the following formula for the variation  a a . Using the formula det ) are conserved, δε ) ω ω = / x µ ( µ a F ∂ j , a we associate a conserved charge 1.29 ) are given by ( δ ω

constant hyper surface. When the current µ a Φ µ j a j ( x = d S , and equations ( d d ε − x ) Z Φ ( 0 ) = S is the Φ ( ≡ Σ where we used the convenient notation for small yields where we used the dilatation generator, then ( S ) δ To any current Φ 3. For scale transformations ( S the jacobian matrix for the map where the currents δ To show this, it is usefulspace-time functions to assume for a moment that the parameters Working to first order in are arbitrary, this is possible only if by definition stationary under arbitrary variations and thus charge. For example, if we define For any continuous transformation, we can define the associated When the transformation under consideration issatisfied, a the symmetry currents and ( when the equations of motion are Dropping some total derivative terms, we find that also conserved in the sense that d Introduction to Conformal Field Theory where PoS(Modave VIII)001 µ F and and (1.36) (1.38) (1.40) (1.41) (1.37) (1.39) (1.42) 1.1 ρ , defined ν µ c is called the T Antonin Rovai µν B . Notice that the T . , .  vanishes. Terms such Φ Σ µρ S ) over Φ id a ν (rotations) L B ∂ (dilatations) i ∂ ( ∂ ∂ , = is not symmetric under the exchange + . L d , ν ν a µ Φ Φ ν . µν ) is now replaced by a space-time index x c B a suitable improvement term, following δ µ T ν µν νρ is conserved as well. Moreover, the charge d ρµν µρ ∂ − B S S c B µ a µν T 1.29 ) ) ˜ c j ρ transforming as scalars under translations, Φ T x T Φ Φ ∂ are themselves antisymmetric. Moreover, the ν − 1 ρ µ ∂ in ( L Φ L ρ − + is antisymmetric under the exchange of ∂ ∂ ) x ∂ ∂ d 8 µν ( µ ( a j Φ d S ∂ ∂ µν µν c canonical energy-momentum tensor Σ µ B i L ∆Φ ρµν T Z ∂ T + ) ∂ ( B + = Φ = ∂ = Φ ν µ implies that x L µ . We will make use of such terms to define a symmetric ∂ νρ µν = ). µ ∂ a P B ( S j µρ µνρ c T ν ∂ j ) , as the integral of ) and the formulas derived in the examples of sections T µ a c defined by Φ 1.40 + − T Q µ L ν : ρ ∂ 1.29 ∂ x ( x µ ρµν ν ∂ µ B µν c c  T T i ). The resulting symmetric energy-momentum tensor 2 = = = , and is given by are conserved since µ D 1.35 j indicates that this is the improvement terms µνρ c j µν µρν B coincides with B T µ a ˜ j , then conservation of νµ a ) associated to Lorentz invariance can be written as . In a Lorentz invariant theory however, it is however always possible to define a B ν − are called 1.38 in the general definition of the current = ), thus carries two space-time indices. νµ a a with B µν a For future reference, let us quickly write the currents associated to Lorentz and dilatation In general, the canonical energy-momentum tensor For many reasons, the energy-momentum (or stress) tensor plays a crucial rôle in quantum The associated conserved charges also carry a space-time index, and are the components of ν B 1.36 , we find the following expressions for the currents: µ ∂ , since translations are parameterised by a space-time vector. The associated current µ where the subscript index where the improvement term Belinfante stress tensor The new currents thanks to the fact thatcurrent ( the Lorentz generators invariance. From the definition ( 1.2 reduce to the identity and thus in ( of symmetric energy-momentum tensor by adding to field theory, and in our caseto this translation will invariance be of shortly the obvious. theory. It is For defined fields as the current corresponding the energy-momentum vector P 1.3 The energy-momentum tensor the discussion below ( Introduction to Conformal Field Theory for as associated with stress-energy tensor, see equation ( PoS(Modave VIII)001 ] g ; ). Φ [ (1.44) (1.49) (1.45) (1.47) (1.48) (1.43) (1.46) S : 1.43 has been ). Evalu- µν B ) yields µν T 1.44 Antonin Rovai T . 0 when restricted , and let 1.47 g S = 0 to show (  ) into ( . = µ ) µ ε µ ε ν µν ε B ν , ) that is used to define the ∂ ν T ∂ ∂ µ , + − ∂ δ ν + 1.49 ν ε ν ε µ ε µ ∂ µ ∂ ( 0 implies 0 and ∂ − ( µν 0 imply the symmetry of · = µν T = ), appears as a source term in Einstein’s δ ] = = 2 1 ), as any term that vanishes on flat space may be g = , g µν µνρ . µν

x T j , − g ] ] µν 1.32 Φ T d B . [ 1.46 ) g µ δ d T Φ S x ; µν [ ∂ (curved space-time) µ ( µν νµ δ Z g S B Φ is symmetric only when the equations of motion ∂ ε g [ T δ 2 1 S δ ] δ 9 + µν g ] = = δ − = x ; B g δ 2 µν

T ; = Φ 0 and µν ] g [ − B 7→ g Φ ν S δ T is given by ( [ ; = ε x = δ S µν S µ Φ g [ g ∂ δ µν 2 S δ µνρ √ j µν T δ µ  ∂ x T is the identity matrix that we write x ≡ − d d g ) d d g ( Z Z − µν = T δ = = S g on flat space, the requirement

δ ] is not uniquely defined by the condition ( S g ] is by definition invariant under the infinitesimal diffeomorphism ( ; δ g ] ; Φ g [ Φ ; [ S ) suggest the following definition for the energy-momentum tensor in curved space- where the metric S Φ δ [ 1 S 1.48 The action We now motivate an alternative definition of the momentum tensor that is automatically sym- 1 the infinitesimal variation of the action the factor involving the determinant of thethan a metric density). being Loosely introduced speaking, in the order energy-momentumunder to tensor an have captures a the arbitrary variation tensor variation of (rather of the theory the metric. Notice that it is formula ( added to it. Now consider the generalization of this theory on a curved space-time of metric ating its variation Plugging the variation of the metric around flat space energy-momentum tensor in general relativity,equations. where times: metric and generalises to curved space-times. Assume the energy-momentum tensor An alternative definition of the energy-momentum tensor made symmetric. Under an arbitrary infinitesimal change of coordinates be a diffeomorphism invariant actionon such flat that space it reduces to our original action Formula ( Introduction to Conformal Field Theory As a consequence, the constraints are satisfied, because we used the conservation laws Let us remark that the Belinfante stress tensor The action PoS(Modave VIII)001 = νµ a µ a B ˜ j . We − (1.50) (1.52) (1.51) by = µ a j , are called µν a i , ). When the B x Antonin Rovai ) is lifted to an i . x i anomalous a ) 1.50 ). Loosely speak- n ). Indeed, thanks − ω with x x ( ( µν a d Φ 1.30 1.52 B δ Ward identities ). Such terms, that are i ν ··· , ) ∂ ) n i i x x 1.52 )) ( ( , we can replace n Φ x ν Φ ( with the associated transforma- a n 0 ··· at which the fields are evaluated. G ) x Φ ) and i x ( ) for a continuous symmetry. Using n ( x ··· F µ ( a 7→ fail to be conserved when inserted in ) 1 ω Φ x µ a 1.50 ··· i x a ,..., j ) ( 1 n G )) Φ 1 x h ( = x ) ··· ( i i , we find the so-called Φ ) . In this section (and in most of these lectures), 1 x a , is evaluated at one of the points 1 i ) should be adapted because the classical action Φ x ω x ··· − µ ( 3.4 ( a j ) x 10 F 1 Φ ( if and only if h h x 1.49 d 1 ( n δ )) = = ∑ i Φ 1 x i ) ) n ( = ) ∑ i ) as a statement between operators. However, if one wish x x 0 n i Φ ( ( x ( a µ a ( − j under the exchange on F n ω 1.49 h = x Φ µ µν a d i ∂ ) = ) d B 0 ··· n x D x ) without any further changes. ) d x ( Z 0 1 ( 0 d i x ), has no consequence on the Ward identities ( Φ Φ ( contain all the points Z − 1 1.52 are fields (or operators) of the theory evaluated at some space-time points d ··· = = Φ ) 1.35 , then h R ) n i ) with 1 x ) x 0 x ( n ⊂ ( ( n x a Φ ( D Φ Φ ω as in ( ) Φ 7→ x ) are the quantum versions of the classical conservation law ( µ a ( . j ··· Φ µ a d ,..., . j ) , ) 0 into equation ( R 1 h 1 1.52 x x x µ ∈ ( ( x mean that the quantized theory also has this symmetry, in the sense of ( µν a ∂ 1 n 7→ Φ ∂ B x h Φ ν x δ ∂ It should be kept in mind that if the classical theory has some symmetry, then in general this To define the energy-momentum tensor in the quantum theory on a general curved space-time, Similarly to the classical case, the addition of an improvement term We now study symmetries in quantum field theories. Since observables are typically expressed Let us write the infinitesimal version of the condition ( + ,..., 1 µ a j x does not where Equations ( where the domain Since this is valid for any value of the parameters correlation functions because of the terms on the right hand side of ( to the current to keep the spirit of this definition, equation ( one could simply use the definition ( ing, one can say that in the quantum theory, thenon-zero currents (and singular) only whencontact terms the current Definition of the energy-momentum tensor in the quantum theory tion of fields as a condition onunder correlators. To be specific, the quantum theory is said to be invariant symmetry is broken quantum mechanically, weshall say encounter an that example the of classical anomaly in symmetry section is the path-integral formalism forarbitrary example, function it is a simple matter to show that if to the antisymmetry property of in terms of correlation functionssymmetry of of the a fields quantum theory (or under operators) some of transformation the theory, it is natural to define a Introduction to Conformal Field Theory 1.4 Consequences for the quantum theory we assume that the symmetries that we consider are not anomalous. PoS(Modave VIII)001 ) will (2.1) ] 1.53 (1.54) (1.55) (1.56) (1.53) g ) from [ Z changed, 1.49 , ] not Antonin Rovai g ; is some strictly Φ [ S Λ − , and it will also be e  3.4 , and explore their basic , that may be curved or µν . Therefore, if one wish d g g is metric-dependent, as a g . For this reason, such that δ ] ) are quite analogous, but it g Φ g ) [ Φ µν D x Z ( D 0 , 1.49 x ) g T x ( √ 7→ x x µν ) and ( d · . g d ] ) ] g , ; x g , ] Z ( [ Φ is perturbed: one from the usual variation ] µν 1.56 g [ of the theory, and we recover ( 1 2 [ g Λ S g g [ W Z S − δ Z δ + e 1 g ) = g ln 0 2 to be specified later and where  x √ Φ − ( 0 g 11 D − Φ Φ ρλ Z ] = g D = g x 7→ [ i

Z ] = λ ν Φ 0 W µν g x = x [ T ] ∂ ∂ Z h g x δ

-dimensional space-time with metric + dimensions ρ d g µ is a smooth, invertible map 0 ; ), the subtle quantum effects contained in the path-integral ( d x x Φ ), the formal path-integral measure [ ∂ ∂ S − 1.56 e 1.53 g ) when we consider the trace anomaly in section is equivalent to δ goes to the classical action + g µν 1.56 W Φ T D Z , and a new one from the variation of the measure ] g [ ] = by S g δ is an arbitrary, small perturbation of the metric. If we define as usual the generating W + g g conformal transformation δ [ . As a side remark, let us mention that conformal transformations are sometimes defined as ). Z g Let us insist on the fact that under a conformal transformation, the metric is In this section we define conformal transformations in any dimension Consider a field theory on a doesn’t contain all the quantum effects of the theory. To state this more precisely, let us consider 7→ 1.56 together with some field transformation positive function. where functional have been taken into account.generating Of functional course, if we consider( the classical limit of our theory, then the g 2. Conformal invariance in consequence of diffeomorphism invariance of the partition function then the definition of We will use equation ( very useful in other lectures of thisshould school now [1]. be Formula clear ( that in ( properties. 2.1 General considerations and algebra not. A Introduction to Conformal Field Theory S the path-integral expression for the partition function receive essentially two contributions when the metric of the action As indicated explicitly in ( to keep the ideaperturbation that of the the metric, energy-momentum it tensor is very captures natural the to define response it of through the the equation theory under a PoS(Modave VIII)001 . 2 µν = δ (2.6) (2.5) (2.8) (2.2) (2.4) (2.7) (2.9) (2.3) d = . Let us µν g ). Antonin Rovai 2.1 ). The case , we find the con- as it corresponds to Λ 3 from now on. 2.1 2 − ≥ 1 conformal group d for flat space, = 0 f x . f 7→ is also a constant, and hence ρ , ∂ x ε ρ of the conformal transformation is x µν , 2 | µν ν δ x ∂ . Setting x . , µν ) ∂ − f δ x / , 0 f . ( µν f µ µνρ . µ x ε 0 δ c ∂ µ ρ x ε ) yields f ∂ ) ∂ ), that ∂ | µ · + + = d B x ∂ ν = . Let us analyse these transformations case by νρ f 2.3 = 2.6 x − ) on the map δ 2 2 + d µ µ 2 7→ ε ∂ ε 12 µν µρν + A ν 2.1 ) c b x f 2 ∂ 1 ρν ) = = ( ν ∂ + 2), and we focus on the case = x ) = + ∂ − µ ( µ x + ε ν f = d ( a 2 ε ( µρ ν f µνρ d ∂ µ ε δ c ∂ 2 2 ) = ρµ = x ∂ ρ ( ε µ with ε 2 µν ) we find ∂ µνρ 2 . This implies, using ( yields 2.2 c µ , 0 is solved by B µν 1. Otherwise, the Jacobian µν = δ changed under a conformal transformation, while for diffeomorphisms it b f , and , this gives µ ) = 2 a µν not x A µν ( ∂ δ Λ on equation ( 1 is trivial: any coordinate transformation obviously satisfies ( to this equation and combining with ( ρ ∂ µ = ∂ d , as can be seen by evaluating the determinant of both sides of equation ( 2 / d − The equation In the Minkowskian case, the conformal group contain the Poincaré group. We will now study the consequences of ( Consider an infinitesimal transformation ) 2 x ( Applying Contracting with for some constants is particular and will beresults studied derived in below will detail also in apply the to next section (although, as we will see then, some whose contraction with which implies The case straint for some constants is. Since we measure distancesdistances before between and points after change, a because conformal theinvariant transformation points under with have a the moved. diffeomorphism. same On Moreover, metric, the atomatically other diffeomorphism-invariant conformally hand, theory invariant: distances is conformal are transformations clearly are not not au- a subset of diffeomorphisms. Introduction to Conformal Field Theory diffeomorphisms that leave the metricsince unchanged the up metric to is a factor. This formulation is misleading the special case If we apply case. The set of conformal transformationsalready obviously note form that a this group group called should the have the Euclidean groupΛ as a subgroup, PoS(Modave VIII)001 , x ) α (2.10) (2.12) (2.11) (2.14) + 1 we have ( and again 0 x µ 7→ b Antonin Rovai ). x 7→ x 1.6 with this new point 2 R . ∼ = plus its antisymmetric part ) C µν , ( b ρ c special conformal transformation . µν (SCT) (2.13d) δ and . µν d − µν m Riemann Sphere / µ δ µ (Rotations) (2.13c) , c f + µ (Dilatations) (2.13b) µ b (Translations) (2.13a) νρ b 0, one should add to space-time an extra point , = µν 2 δ δ 2 − = ) and hence is a conformal transformation. But x ρ = + νµ 2 2 µ µ ρ d b f ν 2 b x x x b 13 2.1 c x , then the various constraints imply + µ + = = as its symmetric part µρ b x λ µ µν δ · 2 λ µ µ 0 − b 2, the resulting space-time 0 µν µν c is a matrix that satisfies the constraint ( b a ν x x = µ b b 2 x ν = x = µ + ν µ . This is a new transformation that is special to conformal satisfies ( − is a constant: these are just translations. x µ d µ µ 2 R ρµν 1 2 c λ R x µ x c x ε µ / = = = = c x µ µ µ µ − 0 0 0 0 x x x x → ν x x ν 0. Then c 0. In this case we find µ 0. If we set = , while the last term corresponds to a rotation. x = d 2 = / µ µνρ are numbers and µν µνρ c µ ) = . Hence µ c b b x b ( µν = µ m ε 0 and α and 0 and 0 and ≡ µ = ] = = a , µν µ µ µν [ λ It is always possible to decompose b and so transformations, and is therefore(SCT called for a short). (infinitesimal) It is not difficult to seewhere that the first term consists of an infinitesimal dilatation a a b To get some intuition about an SCT, remark that under such a transformation Note that the inversion 2. 3. 1. We have now explored all possibilitiesfind for the conformal finite transformations transformations, at we the must infinitesimalwe exponentiate level. just the found. To different infinitesimal Although transformation this that SCT). is The straightforward result in is principle, it can be tedious (in particular for the “at infinity” has the topology of a sphere and is called the for this transformation to be defined at the origin where so a SCT is nothing but the composition of an inversion, a translation characterised by “at infinity” such that theany origin point is may mapped be onto mapped it.space-time to point. Moreover, this by In point composition two at with dimensions infinity, a and translation, hence it has nothing special: it is an extra Introduction to Conformal Field Theory an inversion. PoS(Modave VIII)001 (2.19) (2.16f) (2.16a) (2.16c) (2.16e) (2.17c) (2.18a) (2.18c) (2.17a) (2.16b) (2.16d) (2.17d) (2.18b) (2.17b) . We now µ . ) K Antonin Rovai . 1 or + 2.2 D ,..., . 1
+ , . µρ 1 ) L when we consider Euclidean − ac , J ) νλ 1 δ 1 bd , we easily find the generators of − defined as ˜ + η − ( (SCT) (2.15d) d − 1.2 νλ L bd ,..., J 1 . ,..., (Rotations) (2.15c) µρ ) ) ac 1 (Dilatations) (2.15b) + δ , µ µ ˜ , η (Translations) (2.15a) . 0 . , 1 K K − , ) ) reads − 1 , 1 µν − µ − +
( νρ − 1 δ
∂ ad − µ µ µ L 2 J µ is of signature
P P 2.16 ) x K = = = P ( ( = bc µν µ ˜ µλ 14 1 η ˜ µν 2 1 2 1 − η ρν L D ∂ 00 δ ρν µν − L , ˜ δ ν diagonal matrix with δ η ˜ ∂ 1 η + x + = = = = · ) ,... − − − − 0 x 2 b µ µ bc do not involve any of the generators ˜ , − , , η ν , µλ J ν D µν 1 µ 0 1 ν . We will later on consider the action of these generators ∂ a J + K P x L J − − ∂ ad · µ µν µν 2 J µ J d ˜ µ µ ( ∂ η L δ ρµ ρµ νρ ( 1.1 i i ix x ( δ δ δ iK iP ( i i i − − − × with − 2 i i i − ) } and = = = = 2 ] = ab µ µ µ ] = ] = ] = ] = ] = ] = D + J cd ) are valid for fields invariant under conformal transformations, i.e. P µν P ν µ µ K J { d L , µν µν P ρλ P ( K , , L L L , ab µ , , , 2.15 is of Minkowski signature J D , the conformal algebra ( D [ [ ρ ρ K [ ˜ [ η µν P ab K [ [ J L as the [ ) ab and ˜ η ˜ ( η 0 in the notations of section = The algebra satisfied by these generators is Following the general discussion on generators in section In term of F The generators given in ( We also define Remark that the matrix conformal transformations: It is now manifest that thethe Euclidean commutation algebra is relations a of subalgebra of the above conformal algebra, since for on fields transforming non-trivially under conformal transformations, see section Introduction to Conformal Field Theory Algebra consider redefinitions of the generators that willConsider allow the us new to generators identify the conformal group explicitly. space-time, while for Minkowskian space-time PoS(Modave VIII)001 2 / ) 2 (2.27) (2.23) (2.24) (2.25) (2.20) (2.26) (2.21) + (2.22e) (2.22c) (2.22a) (2.22d) (2.22b) d )( 1 Antonin Rovai + under a general d ( ) x ( . , while in Lorentzian with the definition of φ ) ) . ) ··· x 1 must be proportional to x ( ,
( 1 ˜ Φ D ] + µρ Φ ,
, + S A ) µ , d ] x on µν ∂ ( νλ ( 2 A S δ ˜ , D Φ ] ix SO − A µν + , . S νλ B ∂ . ∆ S · i . ) [[[ x ) + x − µρ 1 µ ( x 3! δ ( φ ix = 0 invariant, that is, the subgroup generated d is obtained by conjugation with the transla- 2 − Φ , ˜ / ] + D
x ∆
= − A νρ µ − , µ S x

] ∂ , ˜ 0 µν K ν , A ) x x S x µλ , 15 ) x ∂ . These generators must satisfy ρν ν ∂ δ ( that belongs to an irreducible representation of the x B ˜ δ x −

( K [[ ) Φ + ν Φ − x − 1 2
∂ ( µ ˜ ˜ ν ) = µ µλ D D 0 and ˜ ∂ Φ x , K i S µ ] + x quasi-primary fields + x ( µ i − A 0 µν νρ ρµ ˜ 2 , ∂ K S φ δ δ i , we deduce that · , , B , + 1 with x i i 0 − 0 ) = ) = ˜ i 0 µ D x x + [ ˜ , φ . The Poincaré generators act as ( ( − K s B 0 ] = ] = ] = ] = ] = Φ Φ ν µ → µ ˜ = ˜ µν µν ρλ K µν K P φ S S S , ) = ) = ) = , A L , , , µ x x x ˜ representation matrices of the rotation group. Consider first the subgroup D ˜ ρ ( ( ( ˜ commutes with all the rotation generators . In either case, a generic conformal transformation has [ D Be K ˜ of spin ) µν [ s [ K A ˜ Φ Φ Φ [ 2 S D ˜ [ , µ − Φ K D e d K ( . The result may be easily derived using the Hausdorff formula: P · SO x i e are the spin- µν S are denoted respectively by To conclude, let us write the finite transformation for a spinless field We now turn to the question of how the fieldsConsider transform a under field a general conformal transfor- Φ The action of the generators attion any operator space-time point Fields transforming this way are called Explicitly, rotation group. Since the identity matrix by Schur’s lemma. By comparing the action of conformal transformation: space-time it is We now restrict our attention to a field scaling dimension given in section of conformal transformations that leave the point Introduction to Conformal Field Theory We thus conclude that the conformal group in Euclidean space is where by dilatations, rotations and SCTs.on The matrices representing the action of these transformations mation, that is, we look for representations of the conformal group. parameters. 2.2 Action of conformal transformations on fields PoS(Modave VIII)001 (2.33) (2.29) (2.32) (2.31) (2.30) (2.28) , i κσ Antonin Rovai δ ) . κσ ( 2.1 σ  ρν δ λ µ , δ − is also symmetric and satisfies ) . λρµν µν λρ Φ µν ( δ X ) m . ρ σ T λρ ∂ µρ ), we have that under a conformal trans- ε · λ δ µν iS 0, ∂ , we may construct an improvement term  δ ∂ . . 2 1 + σ 1.48 = 1 µ µ ν + µ ] ∆ x µ ) + 1 − ν m ρλ d µ µρ νρ T x T )[ ( m δ d ρµν 16 + ( σ T = d µν B ( ) µ ρ D = λ µ Z j X Φ ∂ as the combination δ L ρ µ 1 d µ D j by + ∂ ∂ ∂ − Φ ( = ) µν ∂ S c for some tensor ρν T ( δ λρµν is given by is not arbitrary as we have seen in section = σ X ε = µ λ µ · µν λ µ V σ m ∂ µν of a field λ δ T m ∂ µ T − ) ). Note that since µν ( σ 1.41 virial V λρ δ h 2 2 − d can be written as µ = is defined in ( V B Let us mention that for two-dimensional quantum field theory and under some broad hypothe- Let us now turn to the aspects of a quantum conformal field theory. As reviewed in section Using the definition of energy-momentum tensor ( Under certain circumstances it is possible to find improvement terms that renderLet the us define stress the λρµν X , all information about a continuous symmetry is encoded in the Ward identities, so let us derive ses (including unitarity), rotation and scale invariance imply conformal invariance [9]. 2.4 Conformal invariance in quantum field theory and Ward identities Then if such that the resultingconformal. modified Explicitly, if energy-momentum we tensor define is traceless, and hence the theory is We then see explicitly that inwe get this a case, new scale expression for invariance the implies dilatation conformal current, invariance. valid when Moreover, the equations of motion are satisfied: them for conformal transformations.translation invariance. As a warm up, we start by writing them for rotation and 1 where tensor traceless. In order to dowill that see, we this need may the not theory be to enough. be rotation and scale invariant, but as we Introduction to Conformal Field Theory 2.3 Energy-momentum tensor and conformal invariance formation, then the modified stress tensor Hence if the stress tensortrue is in traceless, general, then for the the theory function is conformal. The converse, however, is not PoS(Modave VIII)001 (2.41) (2.40) (2.39) (2.37) (2.36) (2.38) (2.35) (2.34) , i ) , n i Antonin Rovai x , ) . ( n i i n ) x ) n ( n Φ , n x x i ( ( ) Φ n ··· n n ) . Assuming the stress Φ x Φ 1 ··· ( x ) ( ··· Φ 1 ··· µνρ 1 ) x j ) 1 ( 1 Φ ··· 1 x h x ) ( (  1 Φ 1 1 h x Φ ( νρ i Φ h µν i ) then implies h Φ iS ) S ν i h i i ) i ∂ − ∆ ∆ . ) ). If we further assume that the stress x 2.39 i ν ) i ν i + are the generators of the Lorentz group x ∂ i − x x ρ i ∂ x − 2.37 x µν · i i − µρ ( i x ) , S d x ( T − x 2 ν ( δ d x ρ x d i 1 − )( ( δ i ν n ∂ δ 2 and = µ 1 ρ , up to global multiplicative constants. Consider ∑ i x ν i 1 φ n x i = T is µ i x n ∑ . Equation ( ) i 17 = i ∑ − i  x 1 − µ µν D i ) = − x x Φ j i ( ( T − = x µ D 1 d = j = i φ = δ = i ) − h i 1 ) n i ) x n n x = ) n ( ∑ i x ( n ) we find d x µνρ i ( n j x ( n δ ( n Φ = 1 . For translations, the Ward identity reads Φ n i Φ Φ i = 2.36 completely fixed ∑ i ) ··· Φ n ··· ) ··· ··· x ) 1 ) ) ( 1 x 1 1 n x ( x x ( ) and ( 1 Φ ( ( 1 1 Φ Φ Φ ) ··· Φ ) )) 2.35 x ) ) x x ( 1 ν ( ( µ x x µ ( µν νµ 1 T µρ T h T Φ h T ν µ − x − ∂ ) ν µ ρ x translates into a similar statement in terms of correlation functions up to the addition of has been symmetrized, we have x ( T denote the derivative with respect to h is the scaling dimension of the field νµ µν µν µν µ i µ i T ∂ T ∂ T ∆ T ( ( h h = We now consider consequences of conformal invariance on two- and three-point functions, Consider the canonical current associated to rotational invariance µ ∂ µν the two-point function and show that they turn out to be which is the quantum analog of the classical statement thatConstraints the on stress two- tensor and is three-point traceless. functions where contact terms, as can be seen on the right hand side of ( which is the quantum analogthere of are no the anomalies statement for that theT the considered classical stress invariances, tensor the is classical property symmetric: of as symmetry usual when Combining equations ( tensor is traceless, then the dilatation current yielding the following Ward identity, Introduction to Conformal Field Theory tensor where in the representation of the field The corresponding Ward identity is PoS(Modave VIII)001 ), we (2.42) (2.43) (2.44) (2.49) (2.46) (2.47) (2.48) (2.50) (2.45) ) implies 2.42 2.42 Antonin Rovai , the quantum 0 x , ) 7→ c . ) yields , x b in formula ( , . x a i 2.43 ( ) λ 0 2 x ( 7→ 2 x φ . Our final result is therefore ) 2 0 1 · ∆ x , . 2 ( . This yields the following formula ) and 0 otherwise ∆ 0 6= 1 | λ + 2 2 φ 1 1 / = h x . ∆ ∆ , ∆ i 1 d | i ) into formula ( ) / 2 ) 12 − ∆ = s 2 2 3 = x 1 , ) by taking one of the two operators to be C x ∆ 1 λ x

x s 2 ( ( | 0 − all permutations in ∆ 2.45 ∆ f 3 ( x x 1 12 + f 2 φ if + ∂ 2.41 1 x ∂ ) ∆ | C ∆

2 + s = c 13 d x 1 ) / 1 x = 18 ( i ∆ 1 1 ∆ ) imply ) 2 i ∆ x 2 λ b 23 2 abc )

φ | ), it is easy to check that the constraint ( ) = ( 123 x 0 x 2 ) 2 s 0 iff x ( 1 x C x ( x 12 a 12 2 2.27 ( x f ∂ ) = ∂ x C = ) applied to translations, rotations and dilatations, we find 2 ( φ s −

) 1 φ ( i 6 . Under a conformal transformation 2.13d 1 ) must be such that 1 ) ) φ f = } x = x 1 i h x | i 1.50 ( x ( i ) i Φ 1 ( ) 3 2.43 { φ = 1 φ 2 ) must take the form x h ) at the special point h φ x i ( ) is automatically zero for h ( ) 3 2 2 φ φ x ) 2.49 2.44 ) ( 2 2.41 1 2 x x φ ( ) together with ( ( ) 2 1 1 φ defined in ( φ x ) h ( 1 f 1.50 1 of one real variable. Focusing on a dilatation x φ ( f h 1 is an unknown constant. Plugging ( are scalar operators that we assume to be quasi-primary and not necessarily ap- φ ) h 2 1 ( φ f ≡ and 12 1 φ C , Let us now turn to the three-point function, f From rotation and translation invariance, we find that for some function We now consider constraint ( Using the constraints obtained from ( invariance condition ( that the two-point function ( the identity) is non-zero only if the scaling dimension of the operator is zero, that the three-point function ( for If we consider now an SCT defined in ( In particular, a one-point function (obtained from ( Let us mention that itzero is scaling possible dimension. to show that in a unitary CFT, identity is the only operator with pearing in the collection of fields Introduction to Conformal Field Theory where find that the function where PoS(Modave VIII)001 + 1 (3.1) ∆ (2.53) (2.54) (2.51) (2.52) = c , ) + c , b Antonin Rovai b , + a a ( , 2 ∆ − 1 ) for an SCT characterised ∆ . 3 + 3 ∆ ∆ 13 1.50 2 x 1 = ∆ 2 / − c all permutations in c 3 , we find that ) satisfy the constraint ∆ 123 + 3 3 . + . + c γ C 2 i ∆ b 2 our space-time coordinates. Recall that , 1 x ∆ 23 is γ µν b ) 2 x ( , 1 ) δ b , 3 2 a z z 2, because the set of conformal transforma- c 13 ). As we shall see, there are however much and / ∆ λ 2 , ( ν x b + 0 2 z − µ ) ∆ w = z 2 µ i 2.9 3 ∆ b 23 2 ∂ ) the constraint ( ( w ∆ ∂ x , γ d ) requires x 1 + 2 µ ρ = µ 1 γ 19 a 12 ∆ 7→ b z ∆ 12 ( w x b 2.49 2 2 x ∂ µ 2.51 ∂ / z + a − = ) a = 1 2 i γ ) 1 = 3 ρλ , γ ) and ( i x ( δ 1 γ ( ) in two dimensions. 3 3 ∆ in terms of ∆ 3 φ 2 2.50 by ) c γ ) 2.1 , i 2 2 = γ b ∆ x abc 2 , ( 123 c ( γ 2 a remains valid when 1 C + φ ∆ 1 2 ) γ a ) yields 1 . Applying now to ( | x j ) for = ( x 1 i φ ) − h 2.13d 3 2.53 i x x ( 3 φ ≡ | ) as in ( 2 i j x are unknown numbers, the three numbers µ x ( b is an undetermined coefficient. ) 2 φ abc ) and ( 123 123 1 3 C x C ( ∆ Let us slightly change our notations and call As a consequence of scale invariance, there is no physically meaningful notion of distance in a We now turn to the study of conformal field theories in two dimensions. Actually, much of 1 φ + h 2 3.1 Identification of the transformations and analysis the definition of a conformal transformation tions in two dimensions containsmore those solutions satisfying to ( the constraints ( conformal field theory. It is thenassume impossible that to interactions define are an negligible S-matrixsee when in this wave a is packets standard by are fashion, remembering separated that whereassumed inversions enough. we to are tend now Another to symmetries way zero of at to the infinity, then theory,cannot they and be should if be defined interactions zero in are everywhere. a The conceptphysically standard of relevant fashion; observables quantities for to our consider. purposes, we shall assume that correlators are the 3. Conformal invariance in two dimensions what we derived in section by a vector Solving the system ( where Comment about the non-existence of an S-matrix in non-trivial CFTs Introduction to Conformal Field Theory where ∆ where we defined the quantities Compatibility of the two formula ( PoS(Modave VIII)001 and (3.8) (3.7) (3.2) (3.3) (3.4) (3.5) 1 (3.6a) (3.6c) (3.6b) iz + 0 z Antonin Rovai = . We have the ¯ z z = ∗ is holomorphic” (or z f , 2 is considered as independent  , . z ) 1 1 . . 1 z ! w ) · , i , ∂ ¯ z ∂ i ∂ ! ) 2 1 1 0 ∂ 0 0 ¯ z respectively). The set of conformal z z w read +  w − ) − ∂ ∂ 0 − ∂ 0 2 2 0 ¯ 1 ∂ i z z + : either we set ∂ 0 z ( 2 − ∂ )

( ( 2 f − = ( z δ i

i 1 2 ( 1  2 = 0 = 1 µ 1 0 z = = 1 (or w = z 0 w w 1 = ∂ − 1 1 ¯ z ) ∂ z ∂ w ∂ 1 z ∂ ∂ − δ . ∂ z ( ∂ ε  0 f , 20 = = , , by , ! 1 1 1  1 ) , ) ! 2 z z w 0 w 1 1 1 2 / 1 ∂ and its inverse µν ∂ , z ∂ ∂ ∂ w z / w i ε ) i ) , ∂ 0 ∂ ( 1 ¯ ∂ z ∂ − ¯ z 2 0 z − w 2 0 µν ∂ / 0 1  + ∂ 0 / ∂ = δ = 1 0 i z ∂ + + ( ( 0 0 0

− z 0 + 2 z 1 1 2 2 z w = ( ∂ w 1

0  = ∂ ∂ ∂ ∂ z δ w 0 = = = 0 = δ ∂ z ∂ z w 0 0 z ε ∂ ∂ 0 ∂ ∂ 0 z w  ∂ ∂ exist. In the rest of these notes, when we say that “ )) ¯ z ( ¯ w , ) z ( and extend space-time to a complex space-time, where ¯ w 1 ( iz − 0 z . We recover our original space-time by imposing the reality condition z = ¯ transformations in two dimensions isand hence anti-holomorphic infinite-dimensional, maps as of it the consists Riemann of sphere. all holomorphic Note that we did notand yet the imposed Cauchy-Riemann that relations thefunctions are coordinate of transformation course is everywhere valid“anti-holomorphic”) well only we defined, when actually evaluated mean at meromorphic(or points (or anti-holomorphic) anti-meromorphic), where on the that the whole is, complexifiedwhere holomorphic space-time the except function maybe may at have some poles, isolated and points we write We also define the antisymmetric tensor In these coordinates, the metric usual complex analysis relations Introduction to Conformal Field Theory Explicitely, this is equivalent to There are two ways one can solve these equations for or we can choose These a precisely theholomorphic Cauchy-Riemann function! relations It is for thenz (respectively) very an convenient to holomorphic define or theof complex an variables anti- PoS(Modave VIII)001 ) (3.9) (3.12) (3.11) (3.14) (3.10) 3.13a (3.13a) (3.13c) (3.13b) ) of the , which is w 3.13 (that are not ∂ Antonin Rovai n n is holomorphic − c 1 ε w . 1 are defined at the = ) ¯ z n ( . Since we are in two ` ¯ ε ) ≥ − ¯ , where ) z ¯ ) z , n ¯ z , z ( z ( ). Therefore, there are only ¯ ( n ε φ ¯ ` φ ¯ + z with = ¯ z ∂ . n . ¯ z ` φ } − ∂ 7→ , ) and thus the algebra ( 1 ) 1 n z z
¯ − ` + ¯ , ¯ ( , we find that ` , , n φ ) with some coefficient ¯ z , ε ¯ z n z , m m 1 ) ¯ / ` ¯ ( ¯ ¯ ε z ` 1 + + − 1 n ε , , n n ¯ + c ¯ z 0 ` ` n ¯ = ( = ` + ) ) z + { z n n φ m m ¯ z w ` c φ ∂ n ∞ 7→ − − 0, and hence under our infinitesimal conformal ` . − − z n n n ∞ 21 ∑ 0 = c and = n , commute. The algebra defined by the relation ( ) = z ∆ ∞ admits, at least in a non-empty open set, the following ¯ z ] = ( ] = ] = ( n ∂ } − ¯ , ∞ ` ε 1 ) = m m m 1 is ∑ z ¯ ¯ = z ` ` + ( n − ( ,` , , n φ φ n n n z ε (and similarly for the ,` ¯ and = ` ` ` 1 [ [ [ ). Consider a scalar field − n − n n ` ` ) 1 (similar conclusions apply for ,` c ¯ = z 0 δφ , ` n ≤ z { ` ( n 0 φ ) for the generators, we see that only ) = ¯ z , z 3.12 ( are defined so that 0 only for n ¯ ` δφ = , for the generators n w ` are its Laurent coefficients, and similarly for n c is antiholomoprhic. We assume that In this section we are interested in conformal transformations that are globally defined on the Given the form ( We consider an infinitesimal transformation ¯ Witt algebra ε It is easy to show that they satisfy the following commutation relations: Note that the two sets of generators a priori related to the coefficients transformation we have and hence and where dimensions, the scaling dimension of Laurent series around zero: The generators whole complex Riemann sphere. If suchcalled transformations “true leave symmetries” the or action “global untouched, symmetries,”currents then as that they they are are are everywhere the well only defined. ones that provide conserved is a generators of conformal transformationcopies in of two the Witt dimensions algebra. at the classical levelGlobal consists conformal of transformations two origin. Moreover, one shouldthe require latter them belongs to to be the also Riemann well sphere. defined at Setting 6 the generators point that at correspond infinity,Riemann to since sphere: conformal transformations that are well defined on the complex well defined at Introduction to Conformal Field Theory Generators PoS(Modave VIII)001 ) 2 d ). 2 , Z d c / 2 as , , ) 2 b (3.19) (3.17) (3.18) c = , C 3.13a (3.15c) (3.16a) (3.16c) (3.15a) , (3.15b) (3.16b) , a 2 d 2 b ( , , and there- 1. 2 ) SL a holomorphic Antonin Rovai 1 = , 1 bc and + 1 − d d ( , 1 ad c SO , 1 and such that 1 b = , 1 a cb 2. This means that all results , z , − = β . α z , d ad − , z + ! 1. It is easy to check that the result of 1 s 1 1 z λ b d = 2 − with . We define for this field its 1 7→ 7→ 7→ 1 ∆ , . c ∆ a d bc , as for instance the transformation law for z z z 1 , , 0 ) , ) is a subalgebra of the Witt algebra ( ` c = 1 − 1 − b , d , 2 ` ` , ¯ ! h b 1 , 2 2 2 + , ad + , , we focus on the holomorphic part and similar z z b d 3.15 a + n ] = ] = ] = ) and preserves the relation ε ε ε 3, the conformal group is cz az 22 ` 1 0 1 2 2 d being in addition infinitesimal. Combining the three c a − − ( + + + , ≥ ,` ε z z z → 1 s

,` ,` 3.17 d ` 1 0 SO z [ 2 + ` ` = 7→ 7→ 7→ [ [ by ∆ z z z ! ¯ h = are such that 1. Algebra ( , factor coming from the fact that flipping the sign of h c d a b , it can be shown that the resulting group for global conformal d 0 : : : ∗ , 2

, z c and scaling dimension 1 0 1 1 Z , ` ` s − b = − ` , z a = . Recall that for ), these generators satisfy the following algebra: ) n 1 , of spin 3 3.13 ( Φ from invariance under are complex parameters, , with is similar to the one for the SO n 2 β ¯ ` n ¯ ` and anti-holomorphic weight λ , and α , ε Consider a field Similar results apply for anti-holomorphic global conformal transformations. If we reduce to and similarly for results will apply for the anti-holomorphic part): where above transformations, we find thatthe a form general global holomorphic conformal transformation is of where the complex numbers the composition of two such transformations (parametrized, say, by well. Transformation laws for fields with arbitrary spin and primary fields weight h transformations is scalar fields, Ward identities and constraints on two- and three-point functions apply for real space-time by imposing ¯ Here are the infinitesimal andalgebra finite for transformations the corresponding to these generators (since the fore we have shown thatderived in this section result is actually also valid for Introduction to Conformal Field Theory As a consequence of ( corresponds to a transformation parametrized by Global holomorphic conformal transformationswith hence matrix form multiplication, a the groupreturns isomorphic the same to conformal transformation ( PoS(Modave VIII)001 Φ 7→ that z ) are 3 of the (3.23) (3.21) (3.20) (3.22c) (3.22a) (3.22b) s 2.40 is called a Φ Antonin Rovai ) and ( , , . i i i ) ) 2.37 ) n n n x x x ( ), ( ( ( n n n Φ Φ Φ 2.36 ··· ··· ··· . If, in addition, the field . Then it can be shown ) ) ) ) . 1 ¯ 1 z ). Using the standard formula 1 ) x ( ¯ x x z ( ¯ ( ( , w 1 3.6 1 z 1 ( Φ Φ Φ h → h Φ h i , ν i i z ¯ h s , ¯ ∆ ∂ z with 1 ) , ¯ − i ) ) ) i i ∂ x z  ∂ε Φ x x ( 1 ε h π z w − , being two by two antisymmetric matrices, w δ − − d¯ d ¯ x defined in ( − = x x in the following. µν ( + ( (  ) 2 ¯ → z S Φ ¯ z 2 2 z 1 h ∂ δ Φ z , quasi-primary fields ¯ δ δ − ∂ z 1 23 ε∂ 1 1 ( n = = 1  n n π ∑ 7→ i = = − ∑ ∑ i i i ¯ z ∂ w Φ d d − − − = ) =  x = = = Φ ( ε that i i i 2 δ ) ) ) . The proportionality factor is nothing but the spin δ ) = n n n Φ ¯ x x x w µν ( ( ( , ε n n n w ( Φ Φ Φ 0 Φ ··· ··· ··· . We also write conformal transformation (not necessarily global), then z ) ) ) . With this observation, the Ward identities ( ∂ 1 1 1 x x x µν any ( ( ( ε 1 1 1 conformal transformation s Φ Φ Φ ) ) ) = x x x ) for ( ( ( ), valid for scalar fields only, generalizes to µ µν µ µν µν S global 3.20 T T T h h h 2.27 . Obviously, a primary field is always quasi-primary, but the converse is not true. As µ µν ∂ ε is a shorthand for , we have for a primary field ) ∂ z ( To convince yourself, you might want to start by showing this formula for an infinitesimal transformation. For future reference, we note that under a purely holomorphic conformal transformation We will now write the Ward identities for translation, rotation and scale invariance in two 3 ε + We now specialize to our complex coordinates z we will shortly see, the energy-momentumthat tensor is of a not generic primary. CFT is an example of quasi-primary transforms as ( primary field where are automatically proportional to (in any coordinates) equivalent to dimensions. Note that the generators of rotations 3.2 Ward identities Introduction to Conformal Field Theory and consider a the formula ( Again, fields transforming this way are termed representation: PoS(Modave VIII)001 , is ) ¯ z fails , ) (3.26) (3.27) (3.28) (3.25) (3.29) z z (3.24c) (3.24a) ( (3.24d) (3.24b) ( ) for the ¯ T , T 0 . everywhere of primaries 3.19 , , i = Antonin Rovai i i i ) ) ) ) n i , , n n n ) , , x i  i x ¯ ¯ n ( ) i i w w i ) ( ¯ n ) ) n , , i i w ) n n n ¯ n n ) ) , Φ n ¯ w Φ ¯ ¯ w n n n ¯ , w w w w w , ¯ ¯ , , n ( ( w w w , n ··· n n ··· n n , , ( n w ) w ) n n n ( w w 1 i i w Φ Φ ( 1 ( ( n w w ) ) ( n x Φ x n n ( ( n n n ( ( Φ ¯ ¯ n n z z ··· ··· Φ Φ Φ 1 , , Φ Φ ··· ) ) Φ Φ n n h 1 1 ) Φ ··· z z ··· ¯ ¯ h 1 ··· ··· ) ( ( ··· w w ) ¯ ) ) 1 ··· ··· , , n n w ) ) with the definition ( 1  1 1 ¯ 1 1 , ) ) , i 1 ¯ w ¯ ¯ Φ Φ ¯ w 1 1 z 1 s ¯ , w w w w ¯ z w , ¯ ¯ , , 1 ( ( ν w w w , 1 T 1 1 3.21 1 1 , , ε ( ··· ··· 1 w π w 1 1 1 ) ) ( µ w w w Φ Φ ( 2 ( ( 1 1 1 ∂ w w 2 ( h h 1 Φ ¯ ¯ z z 1 1 i i ( ( 1 h − Φ , , ¯ i 1 1 h h Φ µν h Φ Φ 1 1 i i Φ w h i h h z z ε i Φ Φ = ¯ h i i ∂ ¯ w w ( ( s ∆ z ¯ i 2 w w z zz ¯ 1 1 ) ) 1 1 . Using ( T + i i ) ∂ ∂ T T − − ) w i . One should not forget, however, that i i i Φ Φ h h ) w w ¯ x 1 z z ¯ ¯ z z i ¯ w ∆ ¯ ¯ z zz zz ( ¯ w w − z z h ( 24 − − T T ε ε 1 1 z ∂ ∂ − π∂ h h π∂ · − − z z T 1 1 2 , z 2 2 2 2 ( ( ¯ +  n n z z = = ( ∂ ∑ ∑ 1 2 2 i i ¯ z z zz x ) yields n by + + + + = ∂ ∂ δ δ T ∑ i + − − + 1 1 1 1 i i i i ) π i ¯ n n n n ) ) ) ) 7→ z = = = = ∑ ∑ ∑ ∑ i i i i 2 ∂ , n n n n = = x z i − · ¯ ¯ ¯ ¯ w w w w i i − 3.24d ( ) − − − − , , , , ε ) ) n T n n n n n n ¯ =  = = = = w ¯ ¯ w w w w w w , 1 ( ( ( ( , , T n n = n n n n n n ∑ i w ) and ( . The quantity between brackets is hence holomorphic w w Φ Φ Φ Φ ( − ¯ ( ( n T n n ··· ··· ··· ··· Φ = Φ Φ ) ) ) ) i 3.24c 1 1 1 1 ) ··· ¯ ¯ ¯ ¯ n ··· ··· w w w w ) , , , , x ) ) 1 ( 1 1 1 1 1 1 ¯ w n ¯ ¯ w w w w w w , ( ( ( ( , , Φ 1 1 1 1 1 1 1 varies according to w . w w Φ Φ Φ Φ ) ( ··· ) ¯ ¯ ¯ z z z ( ( n 1 ¯ z ¯ ¯ ) zz z z z 1 1 ( x 1 T T T T Φ ( ¯ h h h h Φ Φ x ) T ¯ z z z ¯ ( z 2 2 ¯ Φ z zz 1 , T T z − π∂ π∂ h h Φ ( 2 2 h π π T ε ,..., 2 2 h δ ) We now derive a formula giving the variation of a correlation function of primaries under an  1 ¯ z x ∂ ( Combining the formulas ( replaced by Transformation of correlation functions of primaries infinitesimal conformal transformation and therefore, introducing we finally find with a similar expression for and in our notations we may replace to be holomorphic when inserted at coincident point in a correlation function. Similarly, Φ holomorphic and anti-holomorphic weights, the correlation function Introduction to Conformal Field Theory we find the formulas PoS(Modave VIII)001 ) and (3.35) (3.33) (3.34) (3.30) (3.31) (3.32) ). Using , . 3.28 i i ) ) 3.6 n n in a correlation ¯ Antonin Rovai x w , ( , i n µν n ) + n T Φ w . i x ( ) i ( n ··· ) Φ n ¯ n ) ) is trivial to perform, w Φ 1 x , ··· ( x n ) ( ··· Φ w 1 ) are primaries, it may be 1 3.30 ( ) ¯ w , 1 Φ , Φ ··· h i x 1 ) ) ( 3.33 .  1 n 1 i w ··· ) ) relating the variation of a cor- s ( x x . ¯ z ) Φ ), thanks to equation ( ( ( ) ν ( 1 h Φ ¯ n z ε ¯ ¯ ) Φ ( T  w ¯ z µ in its symmetric and antisymmetric ) Φ ) ¯ ) 1.51 , ε ( i ¯ x z ∂ 2 3.33 1 ν ¯ ( ( T w + ··· ε w ¯ ν ε h ( µν ¯ ( z ) µ ε ) ε 1 ¯ ), the integral ( ) z ∂ Φ ∂ε ( x x ) ): 7→ ( ) = ¯ + ( ε z ¯ z 1 i + z ( z ( i 3.31 ∆ ¯ µν j Φ T d¯ w 3.33 h ε T ∂ )) ) · h 2 ) I z x i µ i ( 25 ( ∂ w ∂ ε π 1 ( , µν + x 2 ). z ε ) i 2 d T z  ∂ d at which the operators are inserted. This integral is not ) ( 1 C · i D 2.37 x n I T = appearing in the equation ( ∑ ( i ε x Z ) i ) ν )  z and we explicitly wrote that the variation depends on two i i − π ε 1 ( ) = x ) i ( 2 ε ( i n D x µ = ) 3.30 x − ∂ Φ n i ( − h we finally find ) x n ) = . Using the general formula ( = n x ( x z ¯ ¯ ( Φ ε x T 2 ( i Φ 2 j ( ) d n n δ D ··· , we find ¯ 1 ··· w Φ Z ) ) and and n , ) = 1 z ∑ i n 1 ( ε x T ··· x . ( ε w − ) ( 1 ( ) and decomposing the term with 1 contains all points + Φ Φ = x 3.3 Φ ) of z h ( D )) ε 1 3.30 ··· δ x 7→ Φ ( ) is the boundary of -functions in the right hand side of ( 3.27 h z 1 ε δ ¯ µν associated with holomorphic and anti-holomorphic transformations respectively are w D δ , T ) ∂ 1 ¯ ) z ( x w ¯ j ) we end up with = ( ( ν Φ C ε we find for the integrand of ( h ( ¯ 3.29 ε and If your are confused by the presence of an antisymmetric piece, recall that symmetry of For primaries, we may actually push further formula ( µ 4 , 4 ε ∂ ) h δ z ( j and similarly for an anti-holomorphic transformation ¯ (independent) functions where While we assumed that the fields zero only because of the singularitiesthe of derivative the in correlations ( function at coincident points. Performing straightforwardly read-off from the Ward identity ( where the domain part, Thanks to the relator under a symmetry transformation and the corresponding conserved current, the currents performing the contour integration usingtransformation Cauchy’s theorem. In the case of a purely holomorphic function is violated by contact terms (if any), see ( We now apply Gauss’s theorem and specialize to the complex coordinates defined in ( shown that the same formula holds fordescribed non-primaries. in This section will be clearer in the operator formalism Introduction to Conformal Field Theory On the other hand let us consider the following integral, and ( the definitions ( PoS(Modave VIII)001 of ] B , (3.39) (3.38) (3.37) (3.36) A [ Antonin Rovai is a circle also radial direction − o , C that we define with )] , | w . . In two-dimensional ( w S b | − , , , e A | | w w | | the integral taken on a small ) = [ , it should be kept in mind that < > w z is a commuting-number valued | | C ( , R z z H a ) 2 | | . , as ) z radial ordering ) ) ( Φ are anticommuting-number valued. if if w z w ( ( 2 ( z b or b and we have used the implicit presence b Φ d | z b radial quantization ) ) ) 1 and the integrals are taken over a small o z d z w I w ( z Φ | ( − o and ( 1 C 2 and ) = I z a Φ z Φ 1 ) , and assume they may be represented as ( d B ) − w a w z B Φ ) C ( ( 2 1 I w 26 ( Φ Φ w and , b d ) ± ) o A z z C ( ( ( I z a z a d d ] = o + )] = o B I C w , I ( A 2 = [ Φ A ) = ) z w ( ( 1 b can be expressed, using ) Φ [ z B ( on the left hand side. We therefore conclude that the commutator of complex Riemann sphere, we have R R z a are some operators depending on w d and ) w z A C ( I b with R and is a circle centred at the origin with radius slightly bigger that ) z + o ( a C Let us consider two arbitrary operators Even if this was implicit in all our formulas, one should not forget that there is, in any correlator Quantum field theories are more frequently studied in the path integral formalism, as it is for where centred at the origin but with radiusof slightly the smaller radial that operator the two operators where circle around the origin of the complex plane. If we denote by it is present in any expression we consider. where the plus sign is selected if one ofAlthough we the will operators not write explicitly the radial ordering operator circle around a point operator, while we chose the minus sign if both we considered so far, aleft time-ordering to operator right. In that radial sorts quantization, the time ordering fields is in replaced decreasing by time order from CFT field theories however, it turns outintegral that formalism, the is operator much formalism, more which convenient. isare In equivalent very particular, to conveniently transformations the described of path by fields the andIn Operator operators the Product Expansion operator that formalism, we a willfields classical introduce to shortly. operators theory on is this quantized Hilbertticular, by space a defining and coordinate (“time”) giving a is their Hilbert singled equal-time space,direction out. commutation for relations. promoting In a Euclidean In time-like space-time par- coordinate. however, there We(like is shall string no hence preferred choose theory it for arbitrarily, example) although this inin is other the actually contexts complex the plane. natural The choice: resulting time quantization will is be called the 3.3.1 Radial order and mode expansion instance very convenient toformal establish expression Feynman of rules correlators or in describe terms semi-classical of effects path integrals from weighted the by Introduction to Conformal Field Theory 3.3 Operator Formalism the operator PoS(Modave VIII)001 ), + z and ) 1.34 (3.44) (3.45) (3.43) (3.41) (3.42) (3.40) z 7→ ( z a ). In order Antonin Rovai 3.11 , defined by ( ε . ) . Q ¯ z ) ( w ¯ T ( diverges as some negative 1 , Φ i + ) , ) ) n z 2 introduced in ( ¯ z z ( n w ( + ( ¯ z T n L m . Therefore, the associated varia- T ¯ ¯ b ) z d¯ ` ) ) ) z o , z z ∞ z ( C n ( ( ( ε − I ` ε ∞ ε a j ∑ i = z h z n d π 1 d o n 2 o . C , we first need to introduce the concept of ε C n I n 2 ) = I = ¯ ¯ i z L L + i ( , n n n π 1 z ¯ ¯ π ε 1 m L T 2 by decomposing the energy-momentum opera- 2 ∞ L ∞ 5 − − ∞ − ∞ are the conserved charges associated to conformal ∑ ∑ 27 = = ) = n n m z , , ( ¯ L 2 ) are the conserved charges corresponding to conformal ε = , )] = ) for the current n z + n ) = n ( ε n L w z L z j L z ( ( T Q d 1 ε ∞ 3.34 o Φ + , C − ∞ n ε I ∑ = i n Q z z [ π 1 d mode expansion 2 − o ) = C z are the quantum analogues of I = ( . . Moreover, as we have seen in the previous section (see for example i ) = ε m T π w 1 ¯ w L Q 2 whose interpretation will be given in a moment. These relations may be ( , . This observation will be our motivation to define the operator product n → 3.3.2 Φ m essentially depends on the singular values of the product of the operators = L w ε ¯ z L as ] ) that the operators n δ , n B L ε → , L z A 3.45 [ when as | ) )) such expectation values often appear in a contour integral and thus by Cauchy’s w as w parameterising the transformation, and we finally find that − ¯ ( T n z b | 3.33 ε reads Φ and ε be some operators. Typically, the expectation value with This terminology is more natural in polar coordinates, but we shall nevertheless use it. We will now show that the operators We now introduce the concept of operator product expansion, or OPE for short. Let We introduce the following δ T . In the radial quantization scheme, the associated conserved charge ) 5 ) ) z w z ( ( ( for some We deduce from ( We now decompose power of formula ( theorem, the result of such typical integrations will be completely determined by the singular part tion b ε reads where we used the explicit expression ( invariance (for holomorphic transformations).formations, A and similar so result the holds forto determine anti-holomorphic the trans- algebra satisfied by the operators operator product expansion. 3.3.2 Operator Product Expansion Introduction to Conformal Field Theory We thus see that tors for some operators easily inverted using Cauchy’s theorem: transformations. Let us consider an infinitesimal holomorphic conformal transformation a expansion in subsection PoS(Modave VIII)001 Φ ) is (3.52) (3.53) (3.51) (3.48) (3.49) (3.50) (3.55) (3.46) (3.54) (3.47) 3.54 3. When ≥ . n i Antonin Rovai ) is of the form ¯ is given by (we z , ) z Φ with ( w . 1 n ( }
b i − ) ) ) ) ¯ Φ z ¯ z z w , ¯ ( . , , that we can infer from T , z z a ) ? Under an infinitesimal { ( ) ( − Φ ¯ ¯ z w ¯ z w Φ Φ , Φ , ( , ) w ) w ) ¯ ¯ w ) + ( α w ( − w ¯ w z ¯ ( h ( , Φ z ¯ Φ , . z T − w T + ( h ∂ z ∂ ), we can look at the case where 2 Φ Φ ¯ ∂ w , ) α ¯ } h h w , is defined as the sum of the singular ¯ ) 0 h α n + , ) + ) Φ 1 w ) = = − n ¯ ¯ 2 2 yields ( w w w ) T z ) 2 2 ) w n , ( d ¯ z w } } ) + ( n w } h{ w w ¯ z − α ( } ( I ¯ Φ Φ , − α ) + z with a quasi-primary field ) = ( − ab i T z ¯ OPE is ( Φ z ¯ T T ¯ ab h ( z w { 2 π ( + 1 2 { { T , ( , { ) = 2 ) i Φ ∞ h z w 1 ) is primary. ¯ ( ( TT − w ∂ ( ¯ + z N N = ¯ ), and we write + z ∑ h ε ∑ , = with a primary operator 28 n 4 Φ Φ i − n z ¯ α ) ) ( 2 , , z T ¯ ∼ z ) 1 2 w ( , 3.46 , we have ) } Φ Φ / w z z ) = ) + c − ¯ ( h + ∂ ∂ ¯ w z Φ ) with w α z ( − , Φ ( ( T z b z with a quasi-primary operator = = ) b ( + ) { ( ) 1 1 z z w z ∼ 3.33 T z Φ ( } } ( ∼ ··· ( ∼ ) a ∂ ) T a Φ Φ z 7→ ) ) + w ¯ h ¯ ¯ ) implies that w ¯ z ( T T ). It can be shown that in this case, thanks to unitarity and using z w α , w , { { , Operator Product Expansion T z w α ) ( w ( z − ( 2 3.48 3.54 ( w Φ } Φ ) d T ) z Φ . The z ) = ( . As a consequence, the I ¯ ( n T z 4 i T , } T − z h{ are thus given by π 1 ) ( ab 2 α n w Φ { } , where the dots denote extra terms that diverge like − − δ − ¯ Φ T in formula ( z = = ) should always be thought of as inserted in some correlator, possibly with other T ( i ): { T ) is itself a quasi-primary (of weights ¯ z , 3.47 T z 3.28 ( Φ h δ Since The first example of OPE is the OPE of In conclusion, the most general form of the OPE of The operators What can we say aboutglobal conformal the transformation OPE of On the other hand the Ward identity ( for some operators terms in the series of the right hand side of ( and similarly for is taken to be a dimensional argument, the onlyproportional to possible terms in addition to those already shown in ( restrict to holomorphic operators for convenience) where by definition the rightoperators, hand ( side is the OPE of the left hand side. As an equalityequation ( between fields inserted. Introduction to Conformal Field Theory of the integrand. To get some control on this, we assume that the product all these extra terms vanish, ( PoS(Modave VIII)001 ) and 3.57 (3.59) (3.56) (3.57) (3.58) (3.60a) (3.60c) (3.60b) that matter in Antonin Rovai Virasoro algebra ). As in the classical case, , . up to a phase and is defined by , 0 0 , , ) n n , that , ¯ ¯ w ¯ w . + + 3.13 T (  m m 2 − } T . δ δ ¯ z  z ) ) ¯ a primary field. It is a nice exercise ∂ ; T 2 1 1 of the CFT. The origin of this name z z w + d d − − { ε∂ not / / 2 2 2 ) ) c − n n w w is 12 ¯ ¯ ( ( w 2 w d ( n n d T − ¯ ∂ε − T ¯ ) c c  ¯ z 0, T 2 12 12 ). The result reads z ( ( 2 3 2 = + + T ), we find that under any holomorphic conformal + − − c n n  3.39 29 4 ε + + 3 2 is called the central charge. The appearance of a ) 3 central charge coming from the infinitesimal transformation ( z z corresponding to holomorphic and anti-holomorphic m m − ¯ 3.55 2 0 ∂ w d d ¯ c L L m /  / / ) ) T ¯ ¯ c c − L 12 z w w w m m ¯ z 3 d d d ( 7→ and ( − d − − and ,  T n n T ∼ = and commuting with all other operators; this term is called 0 m ) T } ≡ L ¯ c w ) = z ε ( ; ] = ( ] = ] = ( δ using formula ( w ¯ ( T m m m w 0 ) m ¯ ¯ L L L { ¯ z ) with the classical algebra given in ( ¯ T , , , L we have ( n n n , ¯ ¯ T n L L L ) [ [ [ Schwarzian derivative of w with respect to z z L replaced by 3.60 ( ε Φ ), we see that except when + z , called the (holomorphic) c ) for the holomorphic part. The Virasoro algebra differs from the Witt algebra 3.21 7→ ) with , given by z ) , and this is the reason why is a number, called the anti-holomorphic central charge. z is called the c ( 3.60a 3.43 } w z ; 7→ w { z Now that we have introduced the concept of OPE, it is straightforward to find the algebra Using ( We similarly have, considering the anti-holomorphic operator central term Let us compare algebra ( defined by ( satisfied by the operators by an extra term, proportional to to show that the finite transformation 3.3.3 Virasoro algebra the two different sets of generators transformations (respectively) close to commuting subalgebras, known as the a central extension, when going fromCentral the extensions classical generically to appear the when quantum theconsidered, theory, as algebra should a for not consequence the be of operators surprising. the in fact the that quantum it theory is is only representations where again ¯ Comparing with ( is, for the quantum theory. It would beclassical however much field more theories, surprising and to this find can central actually extensions happen in (see purely e.g. [10]). Introduction to Conformal Field Theory for some number will be explained in the next subsection. transformation where PoS(Modave VIII)001 , 2 − trace (3.65) (3.66) (3.61) (3.63) (3.64) (3.62) . Thus we g ). For a detailed Antonin Rovai . To this end, a 3.61 , T of the metric ∂ ) has dimension on length g π ε ( 2 i R µ + µ T h ∂ε ) under the composition of the fol- T 1 π , 3.64 + . ∂ε , ε , R . 3 + z R z ∂ R ∂ 1 ¯ z aR ∂ z a 2 π 2 a g = c = − and start with the operator relation

a 2 24 i 0 30 = ) . The jacobian of this map is x x = µ ¯ ¯ z ) z = = z ∂ , z µ ∂ ¯ z zz z ( T

z T ¯ z T ( T ε ) yields T ε z h ∇ δ + ∇ π z . Using the conservation law (valid upon taking the expec- 1 ¯ 3.62 z 2 ∇ 0. 7→ ). − ¯ z . Our goal is now to find the precise value of z z = a g is proportional to the Ricci scalar = i 6 i 3.57 zz = µ µ T z µ µ of the energy-momentum tensor varies according to ε ∇ T δ T h zz h to equation ( T z 0, we deduce that ∇ = ) under an infinitesimal conformal transformation composed with a suitably µν T 3.61 µ ∇ that is the failure for the trace of the energy-momentum tensor to have a vanishing 7 6 , and the component where we used equation ( The trace anomaly is also knowA as word the of conformal caution: anomaly. the arguments given here does not constitute a complete proof of equation ( We use the usual complex coordinates Let us start by some general considerations. The quantity When a two-dimensional CFT is deformed to a curved space-time, one should expect that 6 7 1. A conformal transformation lowing two transformations: proof, we refer the reader to the literature, see e.g. [3]. must have for some dimensionless constant we will compare to first orderderivative around of) flat ( space the transformationchosen properties diffeomorphism. of both sides of (the where we used the fact that tation value) We now consider the variation of both sides of equation ( conformal invariance is broken since thethe space-time curvature curvature radius. introduces explicitly In a this length section scale, we make this statementvacuum expectation more value, precise by computing the Introduction to Conformal Field Theory 3.4 Trace anomaly anomaly it must be local andthe scalar only under possibility rotations. is that As we have no dimensionful parameter in the theory, Applying the operator PoS(Modave VIII)001 R δ (3.73) (3.72) (3.74) (3.71) (3.75) (3.67) (3.68) (3.70) (3.69) Antonin Rovai , the curved lapla- , we find that the ) ¯ z ω is simply , z zz ( T of the energy-momentum zz T . Our final result is thus ) . In the case of the infinitesimal , π g , T
24 ∂ ( σ / π 2 g ε . . . c 2 we find, for the net variation of the right ε ∇ . on the Ricci scalar is straightforward to 3 . ¯ 2 = . ∂ε − . ∂ε ε ¯ z ω g 3 2 3 a R g ∂ − ¯ σ ∂∂ ∂ z ) ∂ ∇ ∂ε 2 ) π a ∂ π e c g π T 2 c ) the following expression for the variation ω − ( ∂ε c 24 2 1 π 12 = R 24 = − + ˜ 31 ∂ε = and in complex coordinates e g − and working to first order in 1 = R 3.71 g = 4 R i ( z = µ ∂ zz ), we find µν √ 7→ − zz µν = T µ δ T 7→ zz δ z tot g 2 ω = T T δ ω g tot ∇ e h ε ) = ∇ 2 3.74 a 2 δ ˜ R g e − tot = ( , under which the component δ δ ) is given by − δ ) of the same transformation. The net effect on the coordinates = z ˜ gR µν ( R g ε ) and ( µν 3.64 p g − z 3.69 7→ ) and working at first order in z 3.72 ), we find using formula ( ), 3.70 ) into ( 3.64 3.73 is the covariant derivative compatible with the metric g reads ∇ 2 tensor varies according to where we simply used thediffeomorphisms. fact that the energy-momentum tensor is a tensor under arbitrary ∇ Many thanks to Blagoje Oblak for his careful reading of the first version of these notes. The 2. A diffeomorphism Plugging ( Moreover, in the conformal gauge cian variation of the left hand side of ( Finally, comparing equation ( work out (see e.g. in [11] for a proof of this formula generalised to any dimension) and reads author is a Research Fellow of the Belgian Fonds de la Recherche Scientifique-FNRS. where of the Ricci scalar: hand side of ( Acknowledgements Introduction to Conformal Field Theory Under the composition of the two above transformations, the net effect on Going to the conformal gauge The effect of an arbitrary Weyl rescaling Weyl rescaling ( Let us now determine the effectis on the identity, while the net effect on the metric is an infinitesimal Weyl rescaling, PoS(Modave VIII)001 , 303 (1986) 104 Antonin Rovai Nucl. Phys. B , Springer, New York , Univ. Pr., Cambridge . , Univ. Pr., Cambridge UK . , Commun. Math. Phys. , Unpublished. hep-th/9108028 , Conformal field theory Taylor & Francis, Boca Raton 2003. 32 . , USA: Cambridge U. Press 1985. Central Charges in the Canonical Realization of Asymptotic a- and c- theorems arXiv:0908.0333 [hep-th] , An introduction to universality and renormalization group techniques Applied Conformal Field Theory Geometry, topology and physics, Scale And Conformal Invariance In Quantum Field Theory, String theory. Vol. 2: Superstring theory and beyond String theory. Vol. 1: An introduction to the bosonic string Aspects of Symmetry String Theory (1988) 226. Symmetries: An Example from Three-Dimensional Gravity 1998. 207. UK, 1998. 1997. arXiv:1210.2262 [hep-th] [9] J. Polchinski, [5] S. Coleman, [8] J. Polchinski, [6] J. Polchinski, [7] D. Tong, [4] P. H. Ginsparg, [2] A. Sfondrini, [3] P. Di Francesco, P. Mathieu and D. Sénéchal, [1] M. Moskovic and D. Redigolo, [10] J. D. Brown and M. Henneaux, [11] M. Nakahara, Introduction to Conformal Field Theory References