Bootstrapping CFTds

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Supervisors:

Dhr. Prof. Dr. J. de Boer Dr. D.G. Robbins

Examiner: Dr. D.M. Hofman

Institute for Theoretical Physics in Amsterdam

Theoretical Physics

Master Thesis

Bootstrapping CFTds

by

Fernando G. Rej´

on-Barrera

10392165

August 2014 60 ECTS

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In recent years, bootstrapping correlators of scalars has produced new insights into the space of allowed CFTs in 3D, 4D and 5D. However, this represents a small fraction of the full set of consistency conditions that CFTs must satisfy. Since scalars can only exchange traceless symmetric operators, these correlators do not constraint the spectrum of other Lorentz representations. In this work we revise and refine the techniques needed for bootstrapping unitary conformal field theories in arbitrary d > 2 dimensions for four-point functions of tensors. We concentrate on the four-four-point correlator of two identical scalars and two identical vectors, which is the simplest case that supports the exchange of operators with mixed symmetry. For this case we give an algorithm as well as the explicit input for producing conformal blocks in arbitrary dimensions.

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I want to thank my advisor Daniel for guiding me throughout the project. I enjoyed working along him and learned a lot during this year. I would also like to thank Prof. H. Osborn and Prof. S. El-Showk for discussions. I am grateful to the Consejo Nacional de Ciencia y Tecnolog´ıa (CONACYT) for the financial support they gave me in order to pursue my master studies abroad. Finally, thanks to Gis for her unconditional support.

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Introduction

Conformal field theories (CFTs) are quantum field theories (QFTs) with an extended local symmetry that preserves angles. CFTs arise in many areas of theoretical physics: Condensed Matter, interactions beyond the Standard Model, String Theory, Quantum Gravity, etc. The relevance of CFTs comes from the fact that they are fixed points in the RG flow of QFTs, for example in the IR or UV limits, as well as phase transitions in condensed matter systems. In two dimensions, several exactly solvable models exist thanks to an underlying infinite dimensional Virasoro algebra. However, in higher dimensions there are no known comparable methods.

One approach to classify CFTs in higher dimensions is known as the conformal boot-strap program[1]. Due to the implications of conformal symmetry all operators of the theory can be classified, giving rise to an orthogonal basis for the Hilbert space of the theory. In addition, all correlators between operators can be constructed from a set of free parameters known as CFT data. The usual approach for studying QFTs is to write a Lagrangian for the classical theory and calculate quantum correlation functions through a quantization method (e.g. path integral). However in CFTs, since correlators are con-structed from the CFT data, a general non-Lagrangian formalism [2] is more natural. Therefore each particular CFT is labeled by its CFT data, which may or may not have a Lagrangian representation. The bootstrap program idea states that conformal symmetry and a few principles that all QFTs satisfy (e.g., unitarity, crossing/exchange symmetry) are enough to classify all CFT data, i.e. we can obtain the subspace of theories that are consistent quantum theories.

The bootstrap program idea has been studied since the 70’s [3–17], producing suc-cessful results in 2D [18] in the 80’s. In higher dimensions, one of the main ingredients for the bootstrap program that was missing are functions called conformal blocks. It was not until the 2000’s that useful expressions were obtained [19–21]. Then in 2008, the first numerical bootstrap program result appeared in [22], which revived the interest in the bootstrap program. In recent years, there has been a large effort for understanding the space of allowed CFTs. In particular for theories in 3D [23–30], 4D [22, 31–36], and 5D [37].

In these studies, the consistency conditions arising from 4-point functions of scalars with global symmetries were analyzed. However, this represents only a small fraction of

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the set of consistency conditions that CFTs must satisfy. In general, one can have correla-tors between operacorrela-tors with spin in different representations. These cases are interesting because looking at, for example, conserved currents or 2-tensors might provide fundamen-tal constraints on central charges or representations of global symmetries. Also, studying conserved 2-tensors might be useful for obtaining a bound on the largest space-time di-mension in which non-trivial consistent theories can exist.

The main difficulty in extending the current studies to include operators with spin is the calculation of the corresponding conformal blocks. The first results in this direction were obtained in [38]. There, a method for constructing conformal blocks with spin by differentiating scalar blocks was given. However, it only works for the exchange of traceless symmetric operators. Then in [39] a generalization of the shadow formalism, first proposed in [8], was studied. The author was able to show that conformal blocks for the exchange of operators in arbitrary representations can be constructed via combinations of derivatives of scalar blocks. However, the actual computations are still challenging, specially when high spin operators are included.

In this thesis we will review the techniques involved in the bootstrap program for corre-lators with spin in arbitrary space-time dimensions. We use similar ideas as those proposed in [39], however we refine some of the techniques and show the explicit constructions that are needed for automatizing the computation of conformal blocks. The bootstrap pro-gram can be split into two parts. The first one concerns the extraction of the consistency conditions arising from conformal symmetry, unitarity and crossing/exchange symmetry. The second part consists in finding the subset of all possible CFT data that is consistent with the conditions found in the first part.

The thesis is organized as follows. The first part of the bootstrap program is covered in chapters2-5, while the second part is reviewed in chapter6. In chapter2the implications of conformal symmetry in quantum field theories are studied. These include the classification of the operator spectrum of the theory via the spin representation and the conformal scaling dimension, and a converging operator product expansion. General references for this part are [40–43]. Implications of unitary representations [16, 44] are also reviewed. In chapter 3 we construct 2-, 3-, and 4-point correlation functions by using conformal symmetry alone. This is easily done in embedding space which was first introduced by Dirac [45]. More recent treatments of this topic include [46–48]. A new formula for 3-point function of arbitrary spins in a symmetric representation is given in this chapter. Chapter 4 is the main focus of this work. We construct conformal blocks of correlators with spin in arbitrary dimensions. We mainly follow [39] and review all the results that are needed for achieving this. Additionally, an algorithm for generating symbolic expressions for conformal blocks is given, along with the input for the particular cases: three scalars and a vector hsssvi, and two scalars and two vectors hsvsvi. In chapter 5 we study the implications of exchange and crossing symmetry in 3- and 4-point correlators. We show the explicit consistency conditions for particular cases. Chapter 6 reviews the current numerical techniques used in the second part of the bootstrap program. These were first introduced in [22] and later refined in [35]. Finally, we give some concluding remarks in chapter 7. In appendixA we show some identities regarding SO(d) projectors which are crucial for the computation of conformal blocks.

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Table 1.1: Table of conventions.

Symbol Description

T¯a Ta1,a2,···a` where the specific value of ` will be clear form the context

xij, ∆ij xi− xj, ∆i− ∆j respectively

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Conformal Symmetry

Conformal symmetry is a generalization to the usual Poincaré symmetry that quantum field theories possess. In general, a conformal transformation is one that preserves angles. That is, transformations generated by Poincaré and scale transformations, as well as spe-cial conformal transformations. Although conformal invariance in field theories implies Poincaré+scale invariance, the converse is not always true1.

The richer algebraic structure of the conformal group compared to the Poincar´e group, imposes stronger restrictions on the characteristics of the field theory. In order to under-stand those constraints, we will review in this chapter the group and representation theo-retical aspects of the conformal group. We will see that conformal symmetry classifies all the operators by the quantum numbers associated with (pseudo)rotations and dilations, and also form a complete basis of the Hilbert space. This allows for a stronger version of the well known operator product expansion from QFT. In addition, unitary CFTs will have a lower bound on their spectrum of operators. In later chapters we will combine these constraints along with others, in order to construct a set of non-trivial consistency conditions that are required for the bootstrap program.

2.1 Conformal Group

Consider space-time as a (pseudo-)Riemannian manifold (M, η) where M has dimension d and η is a smooth symmetric and non-degenerate tensor field. In particular we are interested in flat space-time which corresponds to Rd−1,1 ≡ (Rd_{, η}d−1,1_{) for the case of}

Minkowski space-time or Rd,0 ≡ (Rd_{, η}d,0_{) for Euclidean space-time. Throughout this}

work we will use Euclidean signature, unless otherwise specified. Nonetheless, both points of view are related by a Wick rotation.

A conformal transformation is defined as an invertible mapping

xµ→ x0µ, (2.1)

1_{In two dimensions it is a well known fact that a scale invariant QFT is also conformal [}₄₁_{]. In four}

dimensions, arguments were given in [49] for scale invariance and unitarity implying conformal invariance. However, no proof is known in higher dimensions.

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such that the metric is preserved up to a position-dependent scaling of the metric

ηµν → Λ(x)ηµν. (2.2)

Case d = 2

Two dimensions is special in the sense that eq. (2.2) is equivalent to the statement that the mapping xµ→ f (x) satisfies the Cauchy- Riemann equations for holomorphic or anti-holomorphic functions. Therefore by mapping the real coordinates (x1_{, x}2_{) to complex}

coordinates (z, ¯z), one concludes that these transformations are the infinite dimensional set of complex analytic maps. However, this is a local result and the analytic maps need not be everywhere defined nor invertible, hence these are called local conformal transformations and do not extend to a globally defined group (although there are infinite dimensional Lie algebras associated with these transformations, e.g. the Witt algebra).

On the other hand, global conformal transformations are all the analytic maps that are invertible and everywhere defined. These maps are also called M¨obius transformations:

f (z) = az + b

cz + d, ad − bc = 1 (2.3)

where a, b, c, d ∈ C. They form the group P SL(2, C) which is isomorphic to SO(3, 1). Case d ≥ 3

For d ≥ 3 consider the infinitesimal version of (2.1)

xµ→ xµ+ µ(x), (2.4)

which implies that

ηµν → ηµν− (∂µν+ ∂νµ). (2.5)

By imposing (2.2) we get that ∂µν+ ∂νµ∝ ηµν, where the proportionality factor is fixed

by taking the trace. Hence,

∂µν+ ∂νµ= 2 d∂ρ ρ_η µν. (2.6) Manipulation of (2.6) leads to (d − 1)∂2∂µµ= 0, (2.7)

which implies that the most general is quadratic in x. It then can be proven that µ(x) = αµ+ ωµ_νxν+ γxµ+ 2β · xxµ− βµx2, (2.8) where αµ, βµ, γ and ωµν = −ωνµ are the parameters of the transformation.

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The first two terms correspond to Poincar´e or Euclidean transformations, the third term corresponds to dilations and the last two to the special conformal transformation (SCT). The finite form of the transformations are given by the following expressions

translation : xµ→ xµ+ aµ (2.9) rotation : xµ→ Λµ_νxµ (2.10) dilation : xµ→ λxµ _(2.11) SCT : xµ→ x µ_{− b}µ_x2 1 − 2b · x + b2_x2. (2.12)

2.2 Conformal Lie algebra

Let us write the elements of the group as eiθaTa_{, where T}

a are the generators of the Lie

algebra. Here the parameters θa are the same ones that appear in the conformal Killing

vector (2.8) written in compact form2:

θa= {αµ, ωµν, γ, βµ}. (2.13)

Similarly,

Ta= {Pµ, Mµν, D, Kµ} (2.14)

generate translations, rotations, dilations and SCT’s respectively. Thus by comparing (2.8) with (eiθaTa _{− 1)x}µ _{≈ iθ}

aTaxµ we can extract a representation of Ta acting on the

coordinates:

Pµ= −i∂µ, Mµν = i(xµ∂ν− xν∂µ),

D = −ixµ∂µ, Kµ= −i(2xµxν∂ν− x2∂µ). (2.15)

From here we obtain the algebra of the conformal generators: [D, Pµ] = iPµ, [D, Kµ] = −iKµ,

[Kµ, Pν] = 2i(ηµνD − Mµν), [Kρ, Mµν] = i(ηρµKν − ηρνKµ),

[Pρ, Mµν] = i(ηρµPν− ηρνPµ),

[Mµν, Mρσ] = i(ηνρMµσ+ ηµσMνρ− ηµρMνσ− ηνσMµρ), (2.16)

the rest of them being zero. We can map these generators to Jab

Jµ,ν = Mµν, J−1,µ = 1 2(Kµ− Pµ), J0,µ= 1 2(Kµ+ Pµ), J−1,0= D, (2.17)

where a, b ∈ {−1, 0, . . . , d} and ηab= diag(−1, 1, · · · 1). The commutation relations of the

generators J are

[Jab, Jcd] = i(ηbcJad+ ηadJbc− ηacJbd− ηbdJac), (2.18)

which correspond to the algebra of SO(d + 1, 1).

2

Notice that because ω is antisymmetric, the contraction θaTa is equivalent to 1₂ωµνMµν. The factor 1

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2.3 Field representations of the conformal group

Consider a tensor field φ¯a(x) of SO(d), mapping M into some tangent space T . Then a

transformation of the coordinates given by an element g of the conformal group, xµ → x0µ= gxµ produces a change in the field given by

φ¯a(x) → φ0¯a(x) = L(g, x) ¯ b ¯ aφ¯b(g −1_x), _(2.19)

where L is a matrix that acts on the field indices. For x = 0 one can show that

L(gg0, 0)¯b_¯_a= L(g, 0)_a_¯c¯L(g0, 0)¯b_c_¯, (2.20) for transformations g, g0 that leave x = 0 invariant. Therefore L(g, 0)¯b_¯_a forms a represen-tation of the little group. The infinitesimal generators of this subgroup are D, Mµν and

Kµ. From the commutation relations (2.16) it is easy to see that these generators form

an algebra that is isomorphic to a Poincar´e algebra plus dilations, where the translations are generated by Kµ. Let Σµν, ˜D and κµ be the generators of this algebra. Because

we are interested in fields with definite tensor-spin structure, we demand that Σµν is an

irreducible finite-dimensional representation. This implies that ˜D ∝ I as it commutes with Σµν(Schur’s lemma) and thus (2.16) forces κµ= 0.

In terms of the Hilbert space of the CFT, fields become operators which generate states from the vacuum, which is annihilated by the conformal group generators. The field transformations are implemented by

φa¯(x) → e−iθaτ (Ta)φ¯a(x)eiθaτ (Ta), (2.21)

where τ is the representation that acts on φ and T are the generators of the conformal algebra defined in (2.14). Therefore from the previous discussion we have for x = 0,

[Mµν, φ¯a(0)] = i(Σµν) ¯_b ¯ aφ¯b(0) [D, φ¯a(0)] = i∆φ¯a(0) [Kµ, φ¯a(0)] = 0, (2.22)

where the number ∆ is the scaling dimension of φ. Fields satisfying (2.22) are called primaries3, which are labeled by their spin representation and the scaling dimension.

The field representation of the full conformal group can be induced from the little group by choosing a basis of T in which Pµacts trivially on the index structure,

[Pµ, φ¯a(x)] = i∂µφ(x). (2.23) Hence [Ta, φ¯a(x)] = [Ta, e−ix µ_P µ_φ ¯ a(0)eix µ_P µ_{] = e}−ixµPµ_[{eixµPµ_T ae−ix µ_P µ_{}, φ} ¯ a(0)]eix µ_P µ_{, (2.24)}

where the term in braces is to be evaluated via the Baker–Campbell–Hausdorff formula: eXY e−X = Y + [X, Y ] + 1

2![X, [X, Y ]] + 1

3![X, [X, [X, Y ]]] + . . . (2.25)

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From the conformal algebra (2.16), one can convince oneself that such evaluation is a finite sum. Therefore evaluating (2.24) for each generator leads to

[Pµ, φ¯a(x)] = i∂µφa¯(x), [Mµν, φ¯a(x)] = i δ¯_ab_¯(xν∂µ− xµ∂ν) + (Σµν) ¯_b ¯ a φ¯_b(x), [D, φ¯a(x)] = i(xµ∂µ+ ∆)φa¯(x), [Kµ, φ¯a(x)] = i δ¯_ab_¯(2xµxν∂ν+ 2∆xµ− x2∂µ) − 2xν(Σµν) ¯ b ¯ a φ¯_b(x). (2.26)

Consider a state generated by the action of the primary φ¯a(0) on the vacuum. By

(2.22) and the commutators (2.16), Dφa¯(0)|0i = i∆φ¯a(0)|0i,

DKν1· · · KνnPµ1· · · Pµmφ¯a(0)|0i = i(∆ + m − n)Kν1· · · KνnPµ1· · · Pµmφa¯(0)|0i. (2.27)

Therefore, φ¯a(0)|0i are eigenstates of the dilation operator with eigenvalue i∆.

Further-more, the generator Pµ(Kµ) raises(lowers) the dimension ∆ by 1. Notice that because

Kµ annihilates primary states, the action of Pµ on φ(0)|0i generates a tower of states of

dimensions ∆ + n, n ∈ Z+ called descendants.

For operators at x 6= 0 the resulting state will not be an eigenstate of D due to the x dependence in (2.26). Nonetheless, they will be a superposition of states with definite scaling dimension, φ¯a(x)|0i = X m (ixµPµ)m m! φ¯a(0)|0i. (2.28)

This mapping from operators to states is known as the state-operator correspondence. The states generated from all primaries and descendants will then form a complete basis for the Hilbert space.

The finite form of the transformation of a tensor field of spin ` and dimension ∆ is given by φ¯a(x) → ∂x ∂x0 −∆+`_d ∂xb1 ∂x0a1 · · · ∂xb` ∂x0a`φ¯b(x 0_). _(2.29)

To see this from the action of the group generators on the field (2.26), consider the in-finitesimal version of (2.29) under xµ → xµ₊µ_{(x) with}µ_{(x) given by (}_2.8_{). Then for}

each generator we have

Pµ φ¯a(x) → φa¯(x + α) ≈ φ¯a(x) + αµ∂µφa¯(x) (2.30) Mµν φ¯a(x) → (δab11− ω b1 a1) · · · φ¯b((δ µ ν + ωµν)xν) ≈ φ¯a(x) +1 2ωµν δ_a¯b_¯(xν∂µ− xµ∂ν) + (Σµν) ¯ b ¯ a φ¯_b(x), (2.31)

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where Σ is given by the tensor representation of SO(d) (Σµν) ¯_b ¯ a= (ηνb1δµa1 − η b1 µδνa1)δ b2 a2. . . δ b` a`+ . . . + δb1 a1. . . δ b`−1 a`−1(η b` ν δµa`− η b` µδνa`). (2.32) D φ¯a(x) → (1 + γ)∆φ¯a((1 + γ)xµ) ≈ φ¯a+ γ(∆ + xµ∂µ)φ¯a(x) (2.33) Kµ φ¯a(x) → (1 − 2dβ · x)− ∆+` d δb1 a1(1 − 2β · x) − 2(βa1x b1_{− β}b1_x a1) · · · × φ¯_b(xµ+ 2β · xxµ− βµx2) ≈ φ¯a(x) + βµ δ¯b_¯_a(2xµxν∂ν + 2∆xµ− x2∂µ) − 2xν(Σµν) ¯_b ¯ a φ¯_b(x) (2.34)

This is clearly the same result as the one we would get from (2.26), by the infinitesimal version of the implementation (2.21),

φ¯a(x) → φa¯(x) − iθα[Tα, φa¯(x)]. (2.35)

This shows that φ¯a(x) transforms as a tensor density of weight −(∆ + `)/d.

2.4 Unitarity bounds

A fundamental property of a unitary theory is that all states |ψi in the Hilbert space have positive norm

hψ|ψi > 0. (2.36)

Note that QFT’s are naturally described in Minkowski signature, where space-time is foliated by d − 1 dimensional surfaces. Therefore when passing from Minkowski to Eu-clidean signature, not all the generators will remain Hermitean after the Wick rotation. For example, the “time” evolution operator E that moves states between surfaces be-comes anti-Hermitian in the Euclideanized version, E → iE. In particular, for CFTs, the Minkowski version of (2.17) corresponds to SO(d, 2) where the generators are Hermitian

˜

J_ab† = ˜Jab. Then a Wick rotation, ˜J−1,0→ J−1,0 = i ˜J−1,0, ˜J−1,µ → J−1,µ = i ˜J−1,µ implies

that the Euclidean generators, are given by

Mµν = ˜Jµ,ν, Kµ= i ˜J−1,µ+ ˜J0,µ,

Pµ= −i ˜J−1,µ+ ˜J0,µ, D = i ˜J−1,0. (2.37)

The Hermiticity property of ˜J translates into

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Notice that the dilation operator, which in our case is the Hamiltonian of the theory, becomes anti-Hermitian as expected.

Therefore when studying the constraints arising from the positivity condition (2.36) one has to be careful in picking the right set of Hilbert space operators depending on which signature one is studying. It can be shown that the restrictions arising from unitarity, bounds the operator dimensions ∆ from below [44]:

∆ ≥ d

2 − 1, for ` = 0,

∆ ≥ |hi| + d − i − 1, for ` > 0,

(2.39)

where hi is the i-th highest weight of the SO(d) representation {h1, · · · , hbd/2c} and i is

the smallest value for which hi ≥ |hi+1| + 1.

In this thesis we will focus on fields transforming as single valued (i.e. not spinors) tensor irreps of SO(d). These irreps are traceless and are related to Young tableaux with the highest weight. For simplicity we will assume that the number of rows in the Young pattern [λ] is is less than d/2. That way we do not have to worry about representations splitting into anti-/self-dual parts. In this case hi is just the number of blocks in the i-th

row of the Young pattern.

For example, in d > 2, the unitarity bound for a completely symmetric tensor repre-sentation of spin ` > 0 · · · is

∆ ≥ ` + d − 2, (2.40)

whereas in d > 4, for a mixed symmetric pattern of ` ≥ 2 indices · · ·

∆ ≥ ` + d − 3, for ` ≥ 3,

∆ ≥ d − 2, for ` = 2. (2.41)

2.5 Operator product expansion

The operator product expansion (OPE) is an concept from QFT which says that the product of two local operators φ1(x1), φ2(x2) can be written as an expansion over local

operators when x1 and x2 approach each other,

lim x12→0 φ1(x1)φ2(x2) = X i P_i12(x12)φi(x2). (2.42)

This is to be understood as an expression that holds inside correlators

hφ1(x1)φ2(x2) · · · i, (2.43)

where · · · represents other operator insertions.

However, the OPE statement for theories with conformal symmetry is stronger; it con-verges at finite separation x12, provided that |x12| < |x2k| for all other operator insertions

k 6= 1, 2 in the correlator. By radially quantizing around x2 (see [43] for more details), the

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Thus we can generate a state φ1(x1)φ2(x2)|0i on the surface and insert a complete basis

of eigenstates with definite scaling dimension ∆ φ1(x1)φ2(x2)|0i =

X

n

Cn(x12)On(x2)|0i. (2.44)

As seen in a previous section, the operators of definite scaling dimensions are primaries and descendants, with the latter being generated by the action of Pµ= −i∂µon primaries.

Therefore the expansion (2.44) can be repackaged into φ1(x1)φ2(x2)|0i =

X

O

P¯c(x12, ∂2)O¯c(x2)|0i, (2.45)

where the sum is now over all primaries (with spins indicated by ¯c), with each coefficient P generating all the corresponding descendants. The convergence of the expansion (2.45) follows from properties of complete bases in Hilbert spaces. Then we can map (2.45) back to the form of an OPE by the state-operator correspondence

φ1(x1)φ2(x2) =

X

O

P (x12, ∂2)O(x2). (2.46)

Generalization of the OPE to operators with spin is straightforward: φ1_a_¯(x1)φ¯2_b(x2) =

X

O

P_¯_a¯¯c_b(x12, ∂2)Oc¯(x2). (2.47)

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Correlation functions

In a non-Lagrangian formalism, the theory is determined by its spectrum of operators and the set of all correlation functions among them. From group/representation theory we concluded in the last chapter that the spectrum of operators of a CFT is parametrized by their scaling dimension and spin representation. Furthermore, for unitary theories the operator dimensions are bounded from bellow.

In this chapter, the correlators between primary operators will be obtained by im-posing conformal symmetry. We will use a d + 2 dimensional formalism which provides a simpler framework for analyzing tensor fields transforming under the conformal group. This will result in a set of building blocks for constructing correlation functions. Using this technology, we give closed form formulas for 2- and 3-point functions of arbitrary spin operators, whereas 4-point functions are constructed in a case-by-case basis from a finite basis of building blocks. Finally, we give some remarks on constraints arising from correlators that include conserved operators.

We will see that imposing conformal symmetry on the correlators is not enough to completely fix them. In particular, 3-point functions are fixed up to a set of constants, while 4-point correlators depend on arbitrary functions of two invariant cross-ratios. How-ever, in later sections, it will be shown that these arbitrary functions can be written in terms of the operator spectrum and the 3-point function constants, by using the OPE.

3.1 Embedding space

As seen in the previous section, the conformal group can be mapped to SO(d + 1, 1) via (2.17). In this case, the group acts linearly on a d + 2 dimensional embedding space, Md+1,1 ≡ (Rd+2, η ≡ η(d+1,1)). Working with light cone coordinates on Md+1,1; XA = (X+, X−, Xµ), the inner product of two vectors X, Y is defined as

X · Y = −Xd+2Yd+2+ Xd+1Yd+1+ η(d,0)_µν XµYν = −1 2(X +_Y−_{+ X}−_Y+_{) + η}(d,0) µν XµYν, (3.1) where X±= (Xd+2± Xd+1_). _(3.2) 12

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The indices µ run over 1, 2, . . . , d, the indices a over 1, 2, . . . , d, d + 1, d + 2 and the indices A over +, −, 1, 2, . . . , d. In order to recover the d dimensional physical space, we have to reduce 2 extra dimensions. Imposing the condition X2 = 0 leads to an SO(d + 1, 1) invariant subspace of dimension d + 1, called the null cone. The null cone condition is satisfied when XA= X+ 1,X µ_X µ (X+₎2, Xµ X+ . (3.3)

We can recover the coordinates xµ _{∈ R}d,0 _{by fixing a gauge X}+ _{(reducing form d + 1 to}

d). This gives the mapping

xµ= X

µ

X+, (3.4)

where the gauge choice X+ = 1 is called the Euclidean section. First, we have to check that indeed a SO(d + 1, 1) transformation on the null cone provides a consistent map in physical space. However, under a transformation U ∈ SO(d + 1, 1), the gauge will change X+ → (U X)+_{. Thus in order for U to induce a map between physical points in the}

same gauge, we need a further rescaling U X → _{(U X)}X++U X. With this construction, we

are identifying physical space Rd,0 with the projective null-cone (X2 = 0, X ∼ λX with λ ∈ R \ {0}).

In fact, this projectivisation is what gives the nonlinear action of the conformal group in Rd,0, from a linear SO(d + 1, 1) transformation on embedding space. More precisely, the embedding metric ds2 = (dX)2 changes under the rescaling operation X → λ(X)X by

ds02= (d(λX)a)2 = (Xa∂bλdXb+ λdXa)2 = λ(X)2ds2, (3.5)

where we used the null cone properties X2 _{= 0, X · dX = 0. In terms of the mapping}

(3.4), this will correspond to the definition of a conformal transformation (2.2), provided that ds2 is flat. In light-cone coordinates

ds2= η(d,0)_µν dXµdXν − dX+_dX−_, _(3.6)

which will be flat provided dX+ = 0. Thus for simplicity, we take the Euclidean section X+ = 1. With this setup, the relation between embedding space and physical coordinates becomes

XA= (1, x2, xµ) (3.7)

3.1.1 Tensor fields in embedding space

A tensor field T¯a(X) of rank ` in d + 2 dimensions transforms as

T¯a(X) → T¯a0(U X) = Uba11. . . U

b`

a`T¯b(U X). (3.8)

Therefore the idea is to find a projector P_µ¯a_¯(x; X) such that when applying transformations on embedding space, the projection t,

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transforms as a tensor of weight −(∆ + `)/d. Such projector can be written in light cone coordinates as P_µA_¯¯(x; X) = ∂X A1 ∂xµ1 . . . ∂XA` ∂xµ` , (3.10)

where the derivatives are given by the map (3.7) ∂XA

∂xν = (0, 2xν, η µ

ν). (3.11)

To see that (3.10) has the correct form, it is enough to show that under X → X0 = U X, U ∈ SO(d + 1, 1) followed by a X dependent scale transformation X0 → X00_{= λX}0_,

the tensor P¯a ¯ µ(x; X)T¯a(X) transforms appropriately. P_µ_¯¯a(x; X)T¯a(X)−→ PU µ¯a¯(x; X)Uba11. . . U b` a`T¯b(X 0 ) = P_µ¯a_¯(x; X0)T¯a(X0) λ −→ λD+`P_µ_¯¯a(x; X0)T¯a(X0) = λD+` ∂ ˜xν1 ∂xµ1 · · · ∂ ˜xν1 ∂xµ1P ¯ a ¯ ν(˜x; X 0 )T¯a(X0) = λD+`∂ ˜x ν1 ∂xµ1 · · · ∂ ˜xν1 ∂xµ1tν¯(˜x). (3.12)

In the first line we used ∂X_∂xµa

U

→ ∂X_∂xµa and ∂X a

∂xµUb_a= ∂X 0b

∂xµ. In the second line, P_µa(x, X0)

λ → λDPa µ(x, X00) = λD(X0a∂µλ + λ∂µX0a) and Ta¯(X0) λ → λCT¯a(X00) = T¯a(X0). In order to

eliminate the derivative with respect to λ we also assumed that T is transversal in each index,

T¯a(λX) = λ−CT¯a(X), XaiT¯a(X) = 0 ∀ i. (3.13)

As seen in the previous section, λ = _{(U X)}1 + which can be checked to be

∂ ˜_∂xx −1 d _{for a general}

conformal transformation x → ˜x. Thus consistency requires D = ∆. When t is a scalar the mapping is simply t(x) = T (X). Thus under a dilation t(x) → s∆t(sx), which gives the interpretation C = ∆.

Notice from (3.11),

∂XA

∂xν XA= 0. (3.14)

This means that the physical tensor tµ¯(x) is unchanged if we shift T_A¯(X) by an amount

proportional to any XAi, called pure gauge. The tracelessness and transversality of the

d + 2 dimensional tensor is equivalent to the tracelessness of the d dimensional tensor. For example ηµ1µ2_t ¯ µ(x) = KA1A2 ∂XA3 ∂xµ3 . . . ∂XA` ∂xµ` TA¯(X), (3.15) where KAB ≡ ηµν∂X A ∂xµ ∂XB ∂xν = η AB_{+ X}A_X_˜B_{+ X}B_X_˜A_, _X_˜A_{= (0, 2, 0).} _(3.16)

Other properties like permutation symmetry of the indices is also carried over to physical space. Therefore tensors belonging to irreps of SO(d+1, 1) will project to tensors belonging to irreps of SO(d).

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1. Defined on the cone X2 = 0

2. Homogeneous of degree −∆: T_A¯(λX) = λ−∆T_A¯(X)

3. Transverse on each index: XAi_T_¯

A(X) = 0, for i = 0, . . . , `

projects under (3.10), a tensor tµ¯(x) of SO(d) that has the correct transformation

prop-erties under the conformal group.

3.2 Conformal correlators

In embedded space, n-point tensor correlators have a tensor structure in SO(d + 1, 1) with rank n, and depend on a set of coordinates Xi. Hence they must satisfy the 3 properties

in each Xi, in order to project SO(d + 1, 1) tensors with the right transformation law. In

addition, the form of the correlators has to be invariant under SO(d+1, 1) transformations. This means that for tensor fields T_Ai_¯

i(Xi) of degree −∆i, the n-point correlator has to

satisfy hT_A1¯₁(X1) . . . T_Ai¯_i(λXi) . . . T_An¯_n(Xn)i = λ−∆i_hT1 ¯ A1(X1) . . . T i ¯ Ai(Xi) . . . T n ¯ An(Xn)i, i = 0, . . . , n, (3.17) XA i j i hT 1 ¯ A1(X1) . . . T n ¯ An(Xn)i = 0, i = 0, . . . , n, j = 0, . . . , `i, (3.18) and hT1 ¯ A1(X1) . . . T n ¯ An(Xn)i = UB11 A1 1 . . . UB 1 `1 A1 `1 . . .UB1n An n. . . U Bn `n An `n hT1 ¯ B1(U X1) . . . T n ¯ Bn(U Xn)i. (3.19) 3.2.1 Scalar fields

For scalars, (3.19) implies that the correlator has to be made out of contractions of Xi, i.e.

all combinations Xi· Xj. In addition, (3.17) requires that the pairs appear in powers, such

that the sum of powers of all pairs in which Xi appears is ∆i. The number of different

pairs for i = 0, . . . , n is n(n−1)₂ , therefore the required configuration of pairs is unique only when n ≤ 3. For 4-point correlators, the number of different pairs is 6, which makes it possible to define configurations of pairs that are invariant under rescaling of any Xi. For

example, U = X12X34 X13X24 , V = X14X23 X13X24 , (3.20) where Xij ≡ −2Xi· Xj. (3.21)

As a result, the form of the 4-point correlator is defined up to an arbitrary function of U and V .

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Using these results, 2-point functions are given by hΦi(X1)Φj(X2)i =

δij

(X12)∆i

, (3.22)

where we have fixed the normalization constant. For 3-point functions, they will be fixed up to an arbitrary constant hΦ₁(X1)Φ2(X2)Φ3(X3)i = λ123 (X12) 1 2(∆1+∆2−∆3)(X₂₃) 1 2(∆2+∆3−∆1)(X₁₃) 1 2(∆3+∆1−∆2) , (3.23) while the 4-point function is fixed up to an arbitrary function of U and V 1

hΦ1(X1)Φ2(X2)Φ3(X3)Φ4(X4)i = X24 X14 1₂(∆1−∆2)_X 14 X13 1₂(∆3−∆4) _{f (U, V )} (X12) 1 2(∆1+∆2)(X₃₄) 1 2(∆3+∆4) . (3.24)

To project the correlators we use Xi· Xj = − 1 2(X + i X − j + X − i X + j ) + XiµX µ j = −1 2(xij) 2_, _x ij ≡ xi− xj, (3.25) or Xij = x2ij, (3.26) as well as φi(xi) = Φi(Xi). (3.27) 3.2.2 Tensor fields

To satisfy (3.19) in non-scalar correlators we need, as in the case of scalars, all combi-nations Xi· Xj. In addition, the tensor structure must be invariant under the combined

transformation X → U X and the action of UA_B on the tensor indices, while (3.18) implies that the tensor structures must also be transversal to each Xi when contracted with its

corresponding index.

Now, the tensor field X_Aii

n transforms

2 _{under the combined action of U ∈ SO(d + 1, 1)}

as X_Aii n → U B Ai n(U C B XCi) = XAii

n. However, it is a pure gauge and thus we can ignore all 1

Notice that the arbitrary functions f (U, V ) can have a parametric dependence on the dimensions of the external operators.

2

Here we are following [46] where the combined transformation is considered, i.e. acting on the coordi-nates as well as the tensor structure. In this case when we say that X is invariant, it has the same meaning as X being covariant from a point of view in which the transformation is applied to the tensor structure only.

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the structures proportional to any X_Aii

n. The simplest SO(d + 1, 1) invariant vector field

is of the form Xi

Ajm

with i 6= j. Consider the combination K(ik,j) Ajm = αXi Ajm+ βX k Ajm, (3.28)

where i 6= j, k 6= j, and the coefficients α and β are SO(d + 1, 1)-invariant scalar functions of the coordinates. Requiring that K(ik,j)

Ajm is transverse with respect to X

Ajm j fixes β in terms of α: β = −αX i_{· X}j Xj_{· X}k. (3.29)

The function α is arbitrary so we can fix it in such a way that K(ik,j)

Ajm is scale invariant with respect to i, j, k K(ik,j) Ajm = Xjk XijXik 1₂ X_Aij m− Xij XjkXik 1₂ X_Akj m. (3.30)

Notice that K(ik,j)

Ajm

is antisymmetric in i, k.

In order to construct such tensor K, we have assumed that there are at least three different coordinates Xi, Xj, Xk. Therefore the tensor structures of 2-point functions cannot be constructed from them. Now, assuming there are four different coordinates, one could ask whether there is an independent combination of three of them αX_Ail +

βX_Ajl+ γX_Akl that is scale invariant and transverse with respect to XA l

l . However, it is

easy to see that such combination is given by a linear sum of K_A(ik,l)l and K

(jk,l)

Al , so it is

not independent.

The rank-2 SO(d + 1, 1) invariant tensors are η_Ai nA j m and X k Ai nX l Ajm

, where the latter is not a pure gauge if k 6= i and l 6= j. Notice that transversal, scale invariant combinations of X_Aki

nX

l

Ajm can be written as linear sums of K

(pq,i) Ai

n K

(rs,j) Ajm

for appropriate p, q, r, s. However, by including the metric, there is one combination that cannot be expressed in this way, namely for i 6= j M(ij) Ai nA j m = η_Ai nA j m+ 2 X_Aji nX i Ajm Xij , (3.31)

For i = j, the transversality conditions requires that η is accompanied by terms propor-tional to pure gauges,

˜ M_Ai mAin = ηAimAin+ 2 X k X_Aii mX k Ai n+ X k Ai mX i Ai n Xik , (3.32)

which means that this tensor structure will project to η in physical space. Notice that both (3.31) and (3.32) require at least two different coordinates and so they can appear in 2-point functions.

By combining the building blocks (3.30), (3.31) and (3.32) we can generate all possible tensor structures that appear in correlators, where the structure ˜M is used to subtract

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traces. Notice that due to our arrangement of the space-time dimension in section2.4, all the structures will be parity-invariant. For a discussion on structures with parity see [48]. To project correlators to physical space, we use (3.26) for scalar combinations, and the projections of (3.30), (3.31), and (3.32) given by

K_A(ij,k)k m → k(ij,k) ak m = x2 jk x2_ikx2_ij !1₂ (xik)ak m− x2_ik x2_jkx2_ij !1₂ (xjk)ak m, M(ij) Ai nA j m → m(ij) ai na j m = η_ai najm− 2 (xij)ai n(xij)ajm x2_ij . ˜ M_Ai mAin → ηaimain (3.33)

We now list some properties of these building blocks that will be useful for future computations.

Linear dependence

The structures k(ik,j)a are parametrized by three numbers i, k, j where the last one

repre-sents the insertion coordinate of the operator carrying the index a. They are antisymmetric under the exchange i ↔ k, thus two k structures differing by a permutation of the first two parameters are related to each other. Furthermore, two k structures with the same third parameter that have one more parameter in common are related by

k(ik,j)_a = x2 ilx2kj x2 ikx2lj !1₂ k_a(il,j)+ x2_ijx2_lk x2 ikx2lj !1₂ k(lk,j)_a . (3.34)

On the other hand, m(ij)_ab structures are parametrized by two numbers i, j which represent the insertion coordinates of the operators carrying each index a, b. These structures are symmetric with respect to the exchange i ↔ j, as well as a ↔ b.

Contractions

All the possible contractions of building blocks k and m are given bellow k(ij,k)· k(mn,k)= 1 2 (x 2 ijx2mn) 1 2 x2_inj m_{i n}_k− x2_im_{j n} i m k+ x 2 jm _{i n} j m k− x 2 jn _{i m} j n k , k(ij,k)_a m(kl)a_b = x2 ilx2kj x2_ijx2_lk !1₂ k(ik,l)_b + x2 ikx2jl x2_ijx2_lk !1₂ k_b(kj,l),

k(ij,k)_a m(kj)a_b = k(ik,j)_b ,

m(ij)_ab m(jk)b_c= m(ik)_ac − 2k(kj,i) a k(ij,k)c , m(ij)_ab m(ij)a_c= ηbc, (3.35) where _{a b} c d e≡ x2 aex2be x2 cex2de 1₂ . (3.36)

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From the first line of (3.35) it is clear that (k(ij,k))2 = 1. Notice also that the R.H.S of the second line of (3.35) does not have the same form as the R.H.S of (3.34).

Derivatives

Derivatives of the tensor structures are given by ∂ ∂xb_kk (ik,j) a = x2_ij x2_ikx2_jk !1₂ k(ik,j)_a k(ij,k)_b − m(jk)_ab , ∂ ∂xa_j0k (ik,j) a = 1 (x 2 ijx2ikx2jk) 1 2 " x2_ikk_a(ik,j)0 k_a(ik,j)+ 2 x2_ijx2_ik x2_jk !1₂ k_a(ik,j)0 (xkj)a − η_aa0(x2_jk− x2_ij) # , ∂ ∂xa_i0m (ij) ab = − 2 x2 ij ηaa0(x_ij)_b+ (x_ij)_am(ij) a0_b . (3.37)

The derivative of a tensor does not transform correctly under conformal transformations, and thus the R.H.S of the above expressions cannot be part of the n−point tensor struc-tures as they stand.

Now we will build correlation functions of tensor fields. Each field ti will be in an irreducible representation Ri of SO(d), obtained by imposing the corresponding index

permutation symmetry and subtracting traces via projectors Πi, constructed from the

metric tensor. For simplicity we will give the correlators in physical space.

The tensor structure of the 2-point correlator of a spin-r tensor fields ti_a₁_...a_r(x1) and

a spin-s tensor tj_b

1...bs(x2) can only include m tensors which means that the correlator is

zero unless r = s. The tensor structure is thus

Ξ_¯_{c, ¯}_d(x1, x2) = m_c(1,2)₁_d₁ · · · m(1,2)_c_r_d_r (3.38)

There is only one scalar structure x2₁₂, therefore in embedding space the homogeneity condition for each coordinate can only be satisfied when i = j, i.e., ∆i = ∆j or when

the 2-point function is zero for i 6= j. Notice that both the scalar and tensor structures are invariant under x1 ↔ x2. Therefore the 2-point function must be invariant under

x1 ↔ x2 and also under the exchange of the external indices ¯a ↔ ¯b, i.e. hta¯i(x1)t¯i_b(x2)i =

hti ¯

a(x2)ti¯_b(x1)i = hti¯_b(x1)tia¯(x2)i. Now suppose that ti¯aand t¯i_bare in different representations,

say given by Π and Π0 respectively, then using the fact that Π2 = Π and ΠΠ0 = 0,

hti ¯ a(x1)ti¯_b(x2)i = Π¯ac¯hti¯c(x1)ti¯_b(x2)i ¯ a↔¯b x1↔x2 = Π¯¯c_bhtia¯(x1)ti¯c(x2)i = 0. (3.39)

Therefore they must be in the same representation for the 2-point function to be non-zero. Notice that this symmetry also implies that it is enough to project one set of indices,

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The 2-point function is then fixed to (by choosing the appropriate numerical normalization) hti_a 1...ar(x1)t j b1...bs(x2)i = δrsδijδRi,R0jΠ ¯ c ¯_bΞ¯a,¯c(x1, x2) (x2 12)∆i , (3.41)

where Ri and R0j are the SO(d) representations of ti and tj respectively.

The 3-point function of arbitrary tensor fields can be obtained similarly to the scalar field case, ht_¯_a1(x1)t¯_b2(x2)t3¯c(x3)i = Π1 ¯ d ¯ aΠ2e¯_b¯Π3 ¯ f ¯ c(λt1_t2_t3)_pΘp_¯ d,¯e, ¯f(x1, x2, x3) (x2₁₂)12(∆1+∆2−∆3)(x2 23) 1 2(∆3−∆12)(x2 13) 1 2(∆3+∆12) , (3.42)

where Πi are the projectors to the representations of tiand Θ is the tensor structure made

out of the building blocks

k(23,1)_a_i , k(13,2)_b j , k (12,3) ck , m(12)_a ibj, m (13) aick, m (23) bjck, (3.43)

for i = 1, . . . `1, j = 1, . . . `2, k = 1, . . . `3. Notice that in this case, there might be more

than one independent allowed structure, therefore the 3-point function will be fixed up to several numerical constants λp. In (3.43) we are picking an arbitrary basis for representing

the tensor structures Θ, therefore one must be careful when comparing with other results in the literature. For general spins `1, `2, `3, the tensor structure can be written as

Θ_¯_a¯_b¯_c(x1, x2, x3) = `2 X i=0 min(`1,`3−`2+2i) X j=0 min(i,j) X k=max(0,`2−`3+j−i) ci,j,k × `1 Y n1=j+1 k_a(23,1) n1 i Y n2=k+1 k_b(13,2) n2 `3 Y n3=`2−i+j−k+1 k(12,3)_c n3 k Y n4=1 m(12)_a n4bn4 j−k Y n5=1 m(13)_a n5+kcn5+`2−i `2 Y n6=i+1 m(23)_b n6cn6−i. (3.44) Here, c represents the 3-point function constants λ after choosing a linearization scheme.

For the simple cases we treat here, we will label each structure manually. Notice that not all the tensor structures in (3.44) will remain after applying the projectors Πi, Πj, Πk,

thus the tensor structures will depend on the representations Ri, Rj, Rk.

Four-point functions can be constructed in a similar way, ht1_¯_a(x1)t2¯_b(x2)t3¯c(x3)t4_d¯(x4)i = x2 24 x2₁₄ 1₂∆12 x2 14 x2₁₃ 1₂∆34 _Π₁¯e ¯ aΠ2 ¯ f ¯ bΠ3 ¯ g ¯ cΠ4 ¯ h ¯ dΩ p ¯ e, ¯f ,¯g,¯h(x1, x2, x3, x4)fp(u, v) (x2₁₂)12(∆1+∆2)(x2 34) 1 2(∆3+∆4) , (3.45) where fp are arbitrary functions of u and v, and Ω encodes the linearly independent tensor

structures, with the index p numbering them. The cross-ratios u, v are the projections of U, V given by u = x 2 12x234 x2₁₃x2₂₄, v = x2₁₄x2₂₃ x2₁₃x2₂₄. (3.46)

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Notice that now, for each set of operator indices, there are are 3` compatible k structures, e.g. for t1, we have k(23,1)ai , k

(24,1) ai , k

(34,1)

ai . However, by (3.34) they are not all linearly

independent, so we have freedom of choosing a linearly independent basis of two vectors. The building blocks for constructing 4-point function tensor structures Ωp are, in our case

k(24,1)_a_i , k(34,1)_a_i , k(14,2)_b j , k (34,2) bj , k(12,3)_c_k , k(24,3)_c_k , k(12,4)_d l , k (23,4) dl , m(12)_a ibj, m (13) aick, m (14) aidl, m(23)_b jck, m (24) bjdl, m(34)_c kdl. (3.47)

3.3 Conserved operators

Classically, continuous symmetries of the physical theory imply the existence of conserved currents _∂x∂aja(x) = 0. At the quantum level, the conservation statement is carried over to

currents inside correlation functions _∂x∂ah· · · ja(x) · · · i = 0 provided there are no quantum

anomalies and that x does not overlap with the other operator insertions.

For conformal field theories, the conservation of operators will impose constraints on the operator dimensions, the 3-point functions, and the arbitrary functions from 4-point correlators. For example, consider a rank ` tensor T¯a with dimension ∆ in a symmetric

traceless representation given by Π. Conservation inside a 2-point function implies ∂ ∂x1a1 hT_¯_a(x1)T¯_b(x2)i = Π¯c¯aΠ ¯ d ¯_b ∂ ∂x1a1 m(12)_c 1d1· · · m (12) c`d` (x2₁₂)∆ = Π¯d_b¯ (−2) (x2₁₂)∆+1 (d + ` − 2 − ∆)(x12)d1m (12) a2d2· · · m (12) a`d` + (ηd1d2(x12)a2 + ηd1a2(x12)d2 − ηa2d2(x12)d1) m (12) a3d3· · · m (12) a`d`+ · · · + (ηd1d`(x12)a`+ ηd1a`(x12)d`− ηa`d`(x12)d1) m (12) a2d2· · · m (12) a`−1d`−1 = 0, (3.48) in other words, the dimension of T saturates the unitary bound ∆ = d + ` − 2 (the other terms are canceled after applying Π).

For 3-point functions, we will look at the case of a conserved vector vb with dimension

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∆T and spin ` in an arbitrary traceless SO(d) representation: ∂ ∂x2b hφ(x1)vb(x2)Tc¯(x3)i = Π ¯ d ¯ c ∂ ∂x2b λ1m(23)bd1 k (12,3) d2 · · · k (12,3) d` + λ2k (13,2) b k (12,3) d1 · · · k (12,3) d` (x2 12)α1(x223)α2(x213)α3 = [λ1(d + ∆φ− ∆T + ` − 2) + λ2(∆φ− ∆T)]Π3 ¯ d ¯ c k(12,3)_d 1 · · · k (12,3) d` (x2₁₂)α1+1₂_(x2 23) α2+1₂_(x2 13) α3−1₂ = 0, (3.49) where α1 = ∆φ+d−1−∆₂ T, α2 = ∆T −∆φ+d−1 2 , α3 = ∆T+∆φ−d+1

2 . Therefore we obtain the

following constraint

λ1(d + ∆φ− ∆T + ` − 2) + λ2(∆φ− ∆T) = 0. (3.50)

Similarly, conservation inside a 4-point function will produce a set of differential equa-tions that the arbitrary funcequa-tions of the cross-ratios u, v must satisfy. This might produce new independent constraints on the arbitrary functions, however we will not explore the consequences of such expression in this thesis. We refer the reader to [50] where conserved vectors and 2-tensors in 4-point correlators are analyzed.

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Conformal blocks

As seen in previous chapters, conformal symmetry allows us to classify operators by their spin and scaling dimension, and also fixes the 2-, 3- and 4-point functions up to some arbi-trary parameters. For 3-point functions these parameters are numerical constants while for 4-point correlators they are arbitrary functions of invariant cross-ratios. At first sight, it might look like in order to write higher point correlators we need more unknown informa-tion, thus the idea of solving all n−point functions from crossing/exchange symmetry and unitarity alone might not seem so promising. However, it turns out that the conformally invariant OPE allows a recursive construction of correlators from 2-point functions.

In particular, for 4-point correlators, this means that we can fix the arbitrary functions of cross-ratios in terms of the spectrum of the theory and 3-point function constants via an expansion over some computable functions called conformal blocks. The same idea can be used, in principle, to write any n−point function provided we know all the lower correlators.

This chapter concerns the computation of conformal blocks for 4-point correlators of operators with spin in arbitrary space-time dimensions. As we will see in the following chapters, not only are they useful for fixing the “arbitrary” functions from 4-point correla-tors, but they also give rise to the non-trivial consistency conditions that are fundamental for carving out the space of consistent CFTs, through the bootstrap program.

4.1 Conformal partial waves

Inserting the OPE in a n+1-point correlator gives an infinite expansion in terms of n-point functions hφ1 ¯ a1(x1)φ 2 ¯ a2(x2) Y i>2 φi_¯_a_i(xi)i = X O P_a_¯¯b₁_a_¯₂(x12, ∂2)hO¯_b(x2) Y i>2 φi_a_¯_i(xi)i. (4.1)

However, for CFTs the OPE converges at finite separation |x12| when the other operator

insertions xk, k 6= 1, 2 satisfy x212 < x22k. In that case we can construct any n−point

correlator recursively if we know the functions P . For n = 3 we have, by virtue of the

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orthogonality of the 2-point function, ht1_¯_a 1(x1)t 2 ¯ a2(x2)t 3 ¯ a3(x3)i = P ¯_b ¯ a1a¯2(x12, ∂2)ht 3 ¯ b(x2)t 3 ¯ a3(x3)i. (4.2)

Thus demanding equality of both sides fixes the coefficients of the function P . This has been done in the past when t1,2 are scalars in [3,4,6].

From the previous chapter we know that the 3-point function is fixed up to several constants (one for each tensor structure) hence we can write the OPE in a more suitable way, ti_¯_a_i(xi)tj¯aj(xj) = X O (λti_tj_O)_rC_a_¯(r)¯b i¯aj(x12, ∂2)O¯b(x2), (4.3)

where the index r runs over the independent tensor structures from htitjOi. It clear now that by recursively applying the OPE, the n−point correlators will depend only on the spectrum of operators O (spins and scaling dimensions) and the 3-point function constants (provided we fix all the coefficients of C). For that reason, the spectrum and 3-point function constants are called CFT data.

Let us now concentrate in the case n = 4. Consider the four-point function of tensor fields of SO(d), ti_¯_a_i(xi) of dimensions ∆i and spins `i and insert the OPE in the channel

(12)(34)1, ht1 ¯ a1(x1)t 2 ¯ a2(x2)t 3 ¯ a3(x3)t 4 ¯ a4(x4)i = X O (λ_t1_t2_O)_r(λ_t3_t4_O)_sWOa¯rs1¯a2¯a3¯a4(x1, x2, x3, x4), (4.4)

where the functions W are called conformal partial waves and are defined by WOrs¯a1a¯2a¯3a¯4(x1, x2, x3, x4) = C (r)¯b ¯ a1¯a2(x12, ∂2)C (s)¯c ¯ a3¯a4(x34, ∂4)hO¯b(x2)O¯c(x4)i. (4.5)

Comparing with the actual form of the 4-point correlator (3.45), we can factor out the scalar and tensor structures from the partial wave, setting

WOrs¯a1¯a2¯a3¯a4 = x2 24 x2₁₄ 1₂(∆12) x2 14 x2₁₃ 1₂(∆34) _Ωp ¯ a1¯a2a¯3a¯4gO rs p (u, v) (x2₁₂)12(∆1+∆2)(x2 34) 1 2(∆3+∆4) , (4.6)

where we included the projectors inside Ω. The functions g are called conformal blocks and depend on the conformal cross-ratios u, v. In other words, the arbitrary functions f in the 4-point correlators are given by the conformal block contributions of all primaries

fp(u, v) =

X

O

(λt1_t2_O)_r(λ_t3_t4_O)_sg_Ors_p (u, v) (4.7)

Thus if we can compute them, then the 4-point functions will be completely determined up to the CFT data. The computation methods will be discussed in the following sections.

1_{The choice of channel is completely arbitrary. In fact the equivalence between channels is a crucial}

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4.1.1 Casimir differential equation

One way of computing conformal blocks is by solving the Casimir differential equation. To demonstrate the method of setting up the differential equation, we consider the case of a 4-point function of scalars, exchanging a traceless symmetric tensor.

The Casimir operator is given by CI =

1 2J

AB_J

AB, (4.8)

where J are the SO(d, 2) generators obtained by the mapping (2.16) to the conformal group generators. Hence

CI= −D2− KµPµ+ idD +

1 2MµνM

µν_. _(4.9)

By the commuting properties of the Casimir, we can use the field representation of the conformal generators at x = 0 (2.22) in order to get an eigenvalue equation

CIO∆,`= C∆,`O∆,`, (4.10)

where O are primary fields, and the eigenvalue is given by C∆,`= ∆(∆ − d) +

1 2ΣµνΣ

µν_. _(4.11)

For symmetric traceless operators, the Casimir ΣµνΣµν = 2`(` + d − 2) [51, 9.4C].

On the other hand, the Ward identities for conformal invariance imply that for n−point correlators,

n

X

i=1

J_AB(i)hφ1(x1) · · · φn(xn)i = 0, (4.12)

where J_AB(i) acts on φi(xi). Furthermore, conformal invariance of the OPE (4.3) implies

that the action of J(1)+ J(2) on a 2-point correlator hφ1(x1)φ2(x2)i can be replaced by a

generator J(O,2) acting on O(x2). Thus for the conformal partial waves (4.5), which are

obtained by inserting two OPEs, the statement (4.12) implies that

J_AB(O,2)+ J_AB(O,4)WO = 0, (4.13)

for each O. In particular, for the operator (J(1)+ J(2))AB(J(1)+ J(2))AB we have

1 2(J (1)_{+ J}(2)₎AB_(J(1)_{+ J}(2)₎ ABWO= 1 2(J O,2₎AB_(JO,2₎ ABWO= C∆,`WO, (4.14)

where the last identity comes from the conformal invariance of the partial waves (4.13). Therefore using (4.6) and the representation of the generators acting on the coordinates (2.15) produces a second order partial differential equation for the conformal blocks

Dg∆,`(u, v) =

∆(∆ − d) + `(` + d − 2)

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with D =A(u, v) ∂ 2 ∂u2 + B(u, v) ∂ ∂u ∂ ∂v+ C(u, v) ∂2 ∂v2 + D(u, v) ∂ ∂u+ E(u, v) ∂ ∂v + F (u, v), (4.16)

A(u, v) = u2(1 − u + v), B(u, v) = −2uv(u − v + 1), C(u, v) = v((v − 1)2− u(v + 1)), D(u, v) = −u

d − 2 + ∆34− ∆12 2 + 1 (u − v + 1) , E(u, v) = ∆34− ∆12 2 + 1 ((v − 1)2− u(v + 1)), F (u, v) = ∆12∆34 4 (u − v + 1). (4.17) The boundary conditioons are given by the leading order approximation obtained by first sending u → 0 and then v → 1. Using

D(um(1 − v)n)

u→0

v→1_{∼ (n(2m + n − 1) − m(d − 2m))u}m_{(1 − v)}n_, _(4.18)

we get two possible boundary conditions g∆,`(u, v) ∝ u ∆−` 2 (1 − v)`(1 + O(u, 1 − v)), gd−∆,`(u, v) ∝ u d−∆−` 2 (1 − v)`(1 + O(u, 1 − v)). (4.19)

Therefore when calculating conformal blocks we impose the first boundary condition. The second one corresponds to a non-local operator called shadow (see next section).

This method has been investigated in [20, 21], producing recurrence relations for conformal blocks associated with 4-point functions of scalars, in any dimension. However, for correlators with spin it produces a coupled system of second order differential equations. Although it might be interesting to study whether there are particular cases in which simple solutions exist, for the present thesis we use an alternative method. This will be developed in the rest of the chapter.

4.2 Shadow formalism

The method we will follow to compute conformal blocks of tensor fields relies on the concept of shadow operators. The idea is to introduce an operator ˜O with scaling dimension d − ∆ which is known as the shadow of the operator O of dimension ∆. We define such operator as a non-local quantity ˜ O¯a(x1) = Z ddx0hOa¯(x1)O¯_b(x0)i|∆→d−∆O ¯_b (x0), (4.20)

where ∆ → d − ∆ represents a formal substitution inside the 2-point correlator. By construction we can define a conformal projector of dimension zero given by

PO= NO

Z

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where the normalization constant NO is fixed by requiring it has a trivial action in a

correlator with O:

h· · · POO(x)i = h· · · O(x)i (4.22)

We will prove this statement for the particular 3-point functions we will need for computing conformal blocks. Consider a 3-point correlator of operators with arbitrary spins and scaling dimensions. Inserting the shadow projector implies

(λ_t1_t2_O)_pht1_¯_a(x₁)t_¯2 b(x2)O¯c(x3)i p = NO(λ_t1_t2O˜)p Z ddx0ht1¯a(x1)t¯2_b(x2) ˜O_d¯(x0)iphO ¯ d_(x 0)Oc¯(x3)i, (4.23)

where in the notation h· · · ip we have split the 3-point functions by their linearly inde-pendent tensor structures. Note that in the R.H.S, the 3-point function constants will, in general, not be the same as the ones in the L.H.S. In fact, they will be related by a linear transformation (λ_t1_t2_O˜)p = (M_t−11_t2_O)

q

p(λt1_t2_O)_q.

In order to prove (4.23) for the cases we need, it is enough to find the unique mixing matrix (C_t−11_t2_O)

q

p ≡ NO(M_t−11_t2_O)

q

p by solving the integral and comparing with the L.H.S.

4.2.1 Mixing matrix

Inserting the 2- and 3-point function forms in (4.23) for ht1_¯_a₁t_a2_¯₂t3_¯_a₃i gives Π1¯c¯a11Π2 ¯ c2 ¯ a2Π3 ¯ c3 ¯ a3 (C123)pqΘq¯c1c¯2¯c3(x1, x2, x3) (x2₁₂)d2−α3(x2 23) d 2−α1(x2 13) d 2−α2 = Π1ca¯¯11Π2 ¯ c2 ¯ a2 Z ddx0 Θ_¯_cp₁_¯_c₂_c_¯₃(x1, x2, x0)Π3¯c¯e3Ξe¯¯a3(x0, x3) (x2₁₀)α1_(x2 20)α2(x230)α3 , (4.24) where Πi are the projectors to the representation of ti, Θp are the three-point function

tensor structures (3.44), Ξ is the two-point function tensor structure given by Ξe¯¯a3(x0, x3) = m (03) e1_a1 3 · · · m(03) e`_a` 3 , (4.25) and α1= 1 2(d + ∆12− ∆3), α2= 1 2(d − ∆12− ∆3), α3 = ∆3. (4.26) Notice thatP

iαi = d. We will now compute (C123)pq for several cases.

hSSSi

The simplest case is when the three operators are scalars. To compute the integral we will use the Feynman-Schwinger trick

n Y i=1 A−ai i = Γ (Pn i=1ai) Qn j=1Γ(aj) n Y k=2 Z ∞ 0 dukua_kk−1 ! A1+ n X m=2 umAm !−Pn_l=1al , (4.27)

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as well as the following integral which can be solved by changing variables to spherical coordinates, Z ddx0 X i αix2i0 !−d = π d 2Γ d 2 Γ (d)   X i<j αiαjx2ij   −d 2 . (4.28)

The integral at hand becomes Z ddx0 1 (x2 10)α1(x220)α2(x230)α3 = π d 2Γ d 2 Γ(α1)Γ(α2)Γ(α3) Z ∞ 0 ds s α2−1 (sx2 12) d 2 Z ∞ 0 dt t α3−1 1 + tx213+sx223 sx2 12 d 2 = π d 2Γ d 2 Γ(α1)Γ(α2)Γ(α3) 1 (x2₁₂)d2−α3(x2 23) d 2−α1(x2 13) d 2−α2 Z ∞ 0 ds0 s 0d 2−α1−1 (1 + s0₎α3 Z ∞ 0 dt0 t 0α3−1 (1 + t0₎d2 = π d 2Γ d 2 − α1 Γ d 2 − α2 Γ d 2 − α3 Γ(α1)Γ(α2)Γ(α3) 1 (x2₁₂)d2−α3(x2 23) d 2−α1(x2 13) d 2−α2 ! = CSSS(αi) (x2 12) d 2−α3(x2 23) d 2−α1(x2 13) d 2−α2 , (4.29)

From line 2 to 3 we made the appropriate change of variables, and from 3 to 4 we used the beta function representation

Γ(a)Γ(b) Γ(a + b) = Z ∞ 0 dx x a−1 (1 + x)a+b. (4.30) Therefore CSSS(αi) = π d 2 Γ d₂ − α₁ Γ d 2 − α2 Γ d 2 − α3 Γ(α1)Γ(α2)Γ(α3) . (4.31) hSST i

Next consider the case of two scalars and a rank-` tensor. There is only one tensor structure given by

Θc¯(x1, x2, xj) = k_c(12,j)1 · · · k

(12,j)

c` , (4.32)

thus the only compatible representation is when T is in a symmetric traceless irreducible representation. A trick that one can use to obtain the mixing matrix without actually computing the integral, is to contract the free indices of the three-point function with the tensor k_a(12,3)1 3 · · · k(12,3) a` 3 , i.e. Ξe¯¯a3(x0, x3)k (12,3) a1 3 · · · k(12,3) a` 3 = y_e1· · · y_e`, ye = [13]0[31]2k (13,0) e − [32]1[23]0k (23,0) e . (4.33)

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Here we have introduced notation similar to (3.36), [a_b]_e≡ x2 ae x2_be 1₂ , (4.34)

Then the contraction of the three-point function structure with this vector y is given by the following identity concerning the symmetric traceless projector2 Π[`]:

xa1· · · x_a`Π[`]¯ a ¯ byb 1 · · · yb` = `!(x 2_y2₎`₂ 2` d 2 − 1 ` C d 2−1 ` x · y (x2_y2₎1₂ ! , (4.35) where (a)n≡ Γ(a + n) Γ(a) (4.36)

are the Pochhammer symbols, and C are the Gegenbauer polynomials satisfying the fol-lowing recursion relation

C d 2−1 ` (t) = (d + 2` − 4) ` tC d 2−1 `−1 (t) − (d + ` − 4) ` C d 2−1 `−2 (t), (4.37) as well as C d 2−1 ` (1) = (d − 2)` `! . (4.38)

Therefore the integral (4.24) becomes Z ddx0 C d 2−1 ` (t) (x2 10)α1(x220)α2(x230)α3 = (d − 2)` `! CSST(`, α1, α2, α3) (x2₁₂)d2−α3(x2 23) d 2−α1(x2 13) d 2−α2 , t = k(12,0)c[1 3]0[31]2k (13,0) c − [32]1[23]0k (23,0) c (4.39) For ` = 0, the result is given by (4.31). For ` > 0, we use the recursion (4.37) in (4.39) which gives a recursion for the mixing matrix:

CSST(`, α1, α2, α3) = d 2+ ` − 2 d + ` − 3CSST(` − 1, α1+ 1 2, α2− 1 2, α3) +CSST(` − 1, α1−1₂, α2+1₂, α3) − CSST(` − 1, α1+1₂, α2−3₂, α3+ 1) −C_SST(` − 1, α1−3₂, α2+1₂, α3+ 1) + 2CSST(` − 1, α1−1₂, α2−1₂, α3+ 1) − ` − 1 d + ` − 3CSST(` − 2, α1, α2, α3). (4.40)

The structure of the mixing matrix for general ` becomes apparent by looking at the ` = 1, 2 cases: CSST(`, α1, α2, α3) = π d 2(d − α₃− 1)_` Γ d₂− α₁+₂` Γ d 2 − α2+ ` 2 Γ d 2 − α3 Γ α1+₂` Γ α2+₂` Γ(α3+ `) . (4.41) It is then easy to verify that this expression satisfies the recursion (4.40) with the right initial value.

2

These projector identities are very important for the calculation of conformal blocks. The proof is given in appendixA

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hSV T i

Now for a scalar, a vector, and a rank-` tensor, there are two independent tensor structures: Θ1_b¯_c(x1, x2, xj) = m(2j)_bc1 k (12,j) c2 · · · k (12,j) c` , Θ2_b¯_c(x1, x2, xj) = k_b(1j,2)k_c(12,j)1 · · · k (12,j) c` . (4.42)

Therefore T can be in either a symmetric traceless · · · or a mixed symmetry

irre-ducible representation · · ·.

Symmetric

Let us first look at the symmetric case. The relevant identity for contracting the 3-point structure with Ξ is xa2· · · x_a`Π[`]¯ a ¯ byb 2 · · · yb` = `!(x 2_y2₎`−2₂ 2`_`2 d 2 − 1 ` `2xa1yb1C d 2−1 ` (t) + (x2y2)12δa 1 b1 + t(1 − 2`)xa 1 yb1+ (x2y2) 1 2(` − 1) x a1 xb1 x2 + ya1yb1 y2 ! d dtC d 2−1 ` (t) + ya1xb1 + t2xa 1 yb1− t(x2y2) 1 2 x a1 xb1 x2 + ya1yb1 y2 ! d2 dt2C d 2−1 ` (t) , t = x · y (x2_y2₎12 . (4.43)

We can contract the external indices of V and T with either ka(13,2)2 k

(12,3) a1 3 · · · k(12,3) a` 3 or m(23)_a 2a13 k_a(12,3)2 3 · · · k(12,3) a` 3

. For the first case, we have

k(13,2)bΘ1_b¯_c(x1, x2, x0)Π[`]¯ c ¯ eΞ¯e¯a3(x0, x3)k(12,3)_a1 3 · · · k(12,3) a` 3 = [1 2]0[21]3k (12,0) c1 + [3₂]₀[2₃]₁k (23,0) c1 k(12,0)_c2 · · · k (12,0) c` Π [`]¯c ¯ e(ye 1 )ye2· · · ye`, (4.44) k(13,2)bΘ2_b¯_c(x1, x2, x0)Π[`]¯ c ¯ eΞ¯e¯a3(x0, x3)k(12,3)_a1 3 · · · k(12,3) a` 3 = 1 2 [ 2 1]0[12]3+ [12]0[21]3− [3 31 2]0[23]1[13]2 k (12,0) c1 · · · k (12,0) c` Π [`]¯c ¯ eye 1 · · · ye`, (4.45) k(13,2)bΘ1_b¯_c(x1, x2, x3)Π[`]¯ c ¯ a3k (12,3) a1 3 · · · k(12,3) a` 3 = k(13,2)bΘ2_b¯_c(x1, x2, x3)Π[`]¯ c ¯ a3k (12,3) a1 3 · · · k(12,3) a` 3 = (d − 2)` 2` d 2 − 1 ` , (4.46) While for the second case,

Θ1_b¯_c(x1, x2, x0)Π[`]¯ c ¯ eΞe¯¯a3(x0, x3)m(23)_ba1 3 k_a(12,3)2 3 · · · k(12,3) a` 3 = δe_c11 − 2k (23,0) c1 k (23,0)e1 k(12,0)_c2 · · · k (12,0) c` Π [`]¯c ¯ eye 2 · · · ye`, (4.47)

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Θ2_b¯_c(x1, x2, x0)Π[`]¯ c ¯ eΞe¯¯a3(x0, x3)m(23)_ba1 3 k_a(12,3)2 3 · · · k(12,3) a` 3 = ([2 3]0[32]1k (13,0) e1 + {[3₁]₀[1₃]₂− [_{1 3}2 2]₀[3₂]₁[1₂]₃}k (23,0) e1 )k (12,0) c1 × k(12,0)_c2 · · · k (12,0) c` Π [`]¯c ¯ eye 2 · · · ye`, (4.48) Θ1_b¯_c(x1, x2, x3)Π[`]¯ c ¯ a3m (23) ba1 3 k_a(12,3)2 3 · · · k(12,3) a` 3 = (d − 2)` d 2 + ` − 1 2`−1_` d 2 − 1 ` , (4.49) Θ2_b¯_c(x1, x2, x3)Π[`]¯ c ¯ a3m (23) ba1 3 k_a(12,3)2 3 · · · k(12,3) a` 3 = (d − 2)` 2` d 2 − 1 ` . (4.50)

Inserting these expressions into (4.24) results in four equations which are used to solve for each component of CSV T: Z ddx0 Kc e (x2₁₀)α1_(x2 20)α2(x230)α3 δ_ce− k(23,0) c k(23,0)e− x2 10x223 x2₂₀x2₁₃ 1₂ k(23,0)_c k(13,0)e + x 2 10x223 x2₃₀x2₁₂ 1₂ k(12,0)_c k(23,0)e− x 2 10x223 (x2₂₀x2₃₀x2₁₂x2₁₃)12 k_c(12,0)k(13,0)e ! = (d − 2)`+1 2`_` d 2 − 1 ` C_{SV T}1 1 (x2₁₂)d2−α3(x2 23) d 2−α1(x2 13) d 2−α2 , (4.51) Z ddx0 Kc e (x2₁₀)α1_(x2 20)α2(x230)α3 − `δ e c d + 2` − 2 − d − 2 d + 2` − 2k (23,0) c k(23,0)e + x 2 10x223 x2₂₀x2₁₃ 1₂ k_c(23,0)k(13,0)e− x 2 10x223 x2₃₀x2₁₂ 1₂ k_c(12,0)k(23,0)e+ x 2 10x223 (x2₂₀x2₃₀x2₁₂x2₁₃)12 k(12,0)_c k(13,0)e ! = (d − 2)`+1 2`_{(d + 2` − 2)} d 2 − 1 ` CSV T12 (x2₁₂)d2−α3(x2 23) d 2−α1(x2 13) d 2−α2 , (4.52) Z ddx0 Kc e (x2₁₀)α1_(x2 20)α2(x230)α3 x2₁₃x2₂₀− x2 10x223+ x212x230 2(x2 20x230x212x213) 1 2 k(12,0)_c k(13,0)e +x 2 10x223+ x212x230− x220x213 2(x2₁₀x2₃₀x2₁₂x2₂₃)12 k(12,0)_c k(23,0)e ! = (d − 2)`+1 2`_` d 2 − 1 ` CSV T21 (x2₁₂)d2−α3(x2 23) d 2−α1(x2 13) d 2−α2 , (4.53) Z ddx0 Kc e (x2₁₀)α1_(x2 20)α2(x230)α3 (d − 2)x2₁₃x2₂₀+ (d + 2` − 2)(x2₁₀x2₂₃− x2 12x230) 2(d + 2` − 2)(x2 20x230x212x213) 1 2 k_c(12,0)k(13,0)e +(2 − d − 2`)x 2 10x223+ (d − 2)(x212x230− x220x213) 2(d + 2` − 2)(x2₁₀x2₃₀x2₁₂x2₂₃)12 k_c(12,0)k(23,0)e ! = (d − 2)`+1 2`_{(d + 2` − 2)} d 2 − 1 ` CSV T22 (x2₁₂)d2−α3(x2 23) d 2−α1(x2 13) d 2−α2 , (4.54)

Bootstrapping CFTds

Institute for Theoretical Physics in Amsterdam

Master Thesis

Bootstrapping CFTds

by

Fernando G. Rej´

on-Barrera

10392165

Contents

Introduction

Conformal Symmetry

2.1

Conformal Group

2.2

Conformal Lie algebra

2.3

Field representations of the conformal group

2.4

Unitarity bounds

2.5

Operator product expansion

Correlation functions

3.1

Embedding space

3.2

Conformal correlators

3.3

Conserved operators

Conformal blocks

4.1

Conformal partial waves

4.2

Shadow formalism