Tree-level spectrum of supergravity on AdS3 × S3 from Lorentzian inversion formula

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Dipartimento di Fisica “Enrico Fermi”

Corso di Laurea Magistrale in Fisica

Tree-level spectrum of supergravity

on AdS

₃

× S

3

from Lorentzian

inversion formula

Author

Mattia Serrani

Supervisor

Prof. Alessandro Vichi

Master Thesis

Pisa, 16 November 2020

Anno accademico 2019/2020

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Chapter 1 Introduction

String theory is a beautiful mathematical and physical theory that aims to unify all forces of nature, including gravity, in a single quantum mechanical framework; in a sense is a theory of quantum gravity.

The theory describing the string worldsheet is a 2 dimensional conformal field theory (CFT), but the string coordinates live in a 10 dimensional target space. Moreover, internal consistency often requires the presence of supersymmetry.

The bosonic action of string theory is the Polyakov action: Sp = − 1 4πα0 Z d2σ√−ggαβ_∂ αXµ∂βXνηµν. (1.0.1)

This action describes the motion of a string of tension T = 1/(2πα0) in flat spacetime, and gαβ _{is the dynamical metric on the worldsheet.}

The bosonic action is not well defined, for example it contains tachyons in its spectrum. Furthermore the bosonic string theory does not admit fermionic particles in its spectrum. Therefore we have to introduce fermions and supersymmerty, in order to cure these two problems. Actually we have five different type of string theory: Type IIA, Type IIB, Type I, E8× E8 heterotic and SO(32) heterotic, and each of them with a different

super-symmetric CFT to described the worldsheet. For example type IIB superstring theory worldsheet is described by a 2d SCFT with N = (2, 0) supersymmetry.

All this theories are related among other by each dualities (S-duality and T-duality) and therefore we can pass from one to another using them. Moreover they are also conjec-tured to be all related to a more fundamental theory, called M-theory, that lives in 11 dimension.

The first observation of the AdS/CFT correspondence was made precisely in M-theory from Maldacena in 1997 [1]. In order to understand this duality it is useful to start from a surprising analogy between large N theory and string perturbation theory.

The former is a non-perturbative method introduced by ’t Hooft in 1973 [2], who pointed out that gauge theories based on the non-abelian gauge group G = SU (N ) simplify in the large N limit (N 1). In this limit there exists a new dimensionless parameter (1/N ) in which expand the theory. In particular it is possible to see that the expansion in order (1/N ) organize the Feynman diagrams according to their topology. For exam-ple, at leading order in 1/N we have only planar graphs; at the next order we have the first non-trivial topology and so on. This expansion is similar to the one found in string perturbation theory. In the string theory action, the term R d2σ√−gR, gives us the pa-rameter of the perturbation. This term in 2d is topological and is proportional to χ, the Euler-Poincar`e characteristic of the worldsheet surface swept by the string. Therefore

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also here we obtain an expansion in the topology of the worldsheet. If now we put the two expansions, the first non-perturbative and the second perturbative, one next to the other, we have: log ZQFT = ∞ X h=0 Nχf(h)(gc2N ), Zstring = ∞ X h=0 e−χφ0 Z Σh DXDge−Sp_. _(1.0.2)

Where in the first expansion we have that N is related to the gauge group SU (N ) of the theory and therefore also to the number of fields ∝ N2_{, g}

c is the gauge coupling constant

of the theory and λ = gcN2 is the ’t Hooft coupling constant and is kept fix in the large

N limit.

In the second expansion we have χ = 2 − 2h, where h is the genus of the surface, eφ0 _{= g}

s,

where φ0 is the VEV of the dilaton, and gs is defined as the coupling of string theory.

Therefore we see easily form eq.1.0.2 that the two expansion are of the same type if we identify gs∼ 1/N .

At this point we can introduce the AdS/CFT correspondence or duality. This is a beautiful way to make two apparently distinct fields of physics communicate. Indeed this correspondence basically tells us that the following two types of theories are dual:

Theories of quantum gravity on a d + 1 dimensional AdS space ←→

Non-gravitational conformal QFTs on the d-dimensional boundary.

Strictly speaking we can think the CFTdas living at the boundary of the higher-dimensional

AdSd+1 spacetime. In practice this duality tell us two important things: first, fields in

the bulk of AdSd+1 are dual to operators in the boundary CFTd, and we can exploit this

duality to compute things in one space and relating them to the other. Second, it implies that we have a relation between the partition functions of the two theories, in particular we have:

heR ddxϕ0(x)O(x)_i

CFT = Zbulk ˜ϕ(x, z)|z=0= ϕ0(x), (1.0.3)

where the LHS is the generating functional of correlators in the CFT side, functional of the sources ϕ0(x), and the RHS is the partition function of the bulk (which is typically

associated with a string theory), function of the boundary conditions ϕ0(x) at the

bound-ary of AdSd+1.

Now we are able to introduce the first example of AdS/CFT duality, that is based on the following correspondence:

U (N ) 4d N = 4 SYM theory ←→ Type IIB string theory on AdS5× S5. (1.0.4)

This correspondence becomes simpler and therefore easier to study when in the CFT side we have N → ∞ and large ’t Hooft coupling λ, and in the gravity side we have gs → 0

and α0 → 0, also called pure supergravity (SUGRA) limit.

This correspondence, or in general the AdS/CFT correspondences, are truly surprising, because no one would have previously thought of a comparison between a theory of gravity and a non-gravitational one. This correspondence opens the door to an interpretation of gravity ”without gravity”.

The work presented in this thesis takes place in this framework. In particular we will focus on two theories that are also conjectured to be dual:

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The supergravity theory on AdS3× S3 is derived form a type IIB string theory low-energy

limit. Indeed one starts with a type IIB string theory on AdS3 × S3 × K3, where K3

is a complex 4d manifold, and compactifies the theory using the low-energy limit in a background of AdS3× S3. At this point using the AdS/CFT correspondence one can find

the dual CFT, that turns out to be a SCFT in 2d with supersymmetry N = (4, 4). These theories should have dual states, and in particular one can see that certain Kaluza-Klein modes in the SUGRA theory (in the bulk) are dual to 1/2-BPS states in the SCFT. Contrary to Maldacena’s example, here there is one important difference, we do not know exactly the SCFT, so we cannot identify any coupling constant gc in the theory, and

neither any ’t Hooft coupling.

However we can always reduce the theories in their low-energy limit in this way: for the SUGRA theory, first we decouple the string states sending α0 → 0, second we take the limit gs → 0. This correspond respectively to consider in the dual SCFT only double-trace

operators, and take large central charge c → ∞.

Let us now present the work of this thesis, that is completely based on field theory computations.

Motivated by previous works [3, 4], in this thesis we study 4-pt correlation functions of particular 1/2-BPS supermultiplets of the SCFT in 1.0.5 [5]. In order to analyze these correlation functions in CFT we consider the conformal block decomposition:

hO1O2O3O4i ∼

X

O

λ12Oλ34OG∆∆12O,l,∆O34(u, v), (1.0.6)

where u, v are the usual cross-ratios, which correspond to two independent conformal invariants of the theory.

This tool allows us to divide the 4-pt functions in various contributions with a weight given by λ12Oλ34O which are the OPE coefficients. Each conformal block G∆∆12O,l,∆O34 encodes the

contribution of a single irreducible conformal multiplet with a primary of dimension ∆O

and spin lO.

At this point our job will be to derive the spectrum and OPE coefficients: in order to do that in a CFT exist various techniques both analytic and numerical. Our approach is purely analytic and uses a formula recently found by Caron-Huot [6]. This formula is called ”Lorentzian OPE inversion formula” and is an analytic formula that allows us to find OPE coefficients, but not only. The Lorentzian OPE inversion formula has the following form: c(l, ∆) = κl+∆ 4 Z 1 0 dzd¯zµ(z, ¯z)G∆+1−d,l+d−1(z, ¯z)dDiscG(z, ¯z), (1.0.7)

λ12Oλ34O = −Res∆0_=∆c(l, ∆0), (1.0.8)

where κl+∆it is a multiplying factor, µ defines the integration measure, G is the conformal

block, z, ¯z are the two only independent variable of the theory, and G is the 4-pt corre-lation function. The expression dDisc stands for ”double discontinuity” and represents a sum of two particular discontinuities in the complex plane of z, ¯z.

The quantity c(l, ∆) is like a generating function, and in general we can extract from its poles the various OPE coefficient λ∆,l of the theory. This formula also allows us to know

what operators are needed in one channel (for example s-channel) to reproduce the other one (for example t or u-channel), and in order to obtain the s-channel information we do not need the whole G(z, ¯z) but only the t-channel and the u-channel dDisc. In particular double-trace operators [O1O2]n,l emerge in this way and are fundamental in the theory.

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At the quantum level, operators that are not protected, unlike the 1/2-BPS we have in the 4-pt correlation functions, acquire anomalous dimensions, that we can extract using the procedure above. In particular we calculate anomalous dimensions γn,l of certain

double-trace operators.

We compared our results with the literature and we found complete agreement. Moreover we obtained new values of anomalous dimensions.

This thesis is organized as follows:

In chapter 2 we present the conformal field theories (CFTs): We focus on the construction of the conformal transformations and also in the description of new tools. In particular we concentrate our attention into the conformal block decomposition that is at the basis of the thesis work. Finally we quickly review the main difference that appear in 2d CFT. In chapter 3 we introduce the Lorentzian inversion formula: first we introduce some useful tools in order to understand the formula; second we describe it and all its properties and applications; finally we present the derivation of the 2d case which is simpler.

In chapter 4 we summarize some important concepts to better understand what will come next. In particular we quickly introduce basic concepts of: supersymmetric theories, su-perconformal field theories (SCFTs), string theory and large N expansion.

In chapter 5 we introduce the AdS/CFT correspondence. We present the Maldacena’s first example of this duality, and then we give a formal definition and interpretation of AdS/CFT.

In chapter 6 we introduce the theory we study in this thesis. Then we study the super-conformal algebra of this theory, whose spectrum we will calculate in the next chapter. In chapter 7 we present the original computations of this work. Here we calculate OPE coefficients and anomalous dimensions of double-trace operators using the Lorentzian in-version formula.

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Chapter 2 Conformal field theory (CFT)

Conformal Field Theories (CFTs) appear in many branches of theoretical physics: funda-mental interactions beyond the Standard Model, statistical physics, critical phenomena, string theory, AdS/CFT, quantum gravity and fluidodinamics. A general quantum field theory (QFT) can be thought of as a Renormalization Group (RG) flow starting from a CFT in the Ultraviolet (UV) and flowing to another CFT in the Infrared (IR); this is typical in an RG flow to go from a fixed point to another.

Conformal field theories are under better theoretical control than general QFTs, this can be expected, because we add a symmetry. However is amazing that with the addition of a simple (and reasonable for many physical system) symmetry the theories are much more constrained. Indeed in CFTs we have several tools which are not available or much less powerful without conformal symmetry. In the following we firstly introduce CFTs and the conformal transformations, then we pass to describe all the new tools that appear. In this chapter we follow [7, 8].

What is a CFT ?

A Conformal filed theory is a scale invariant QFT, however as we shall see, it will be much more. In general a theory without scales or with only dimensionless parameters are classically scale invariant. A simple example is the scalar field with only quartic interactions: S = Z d4x(∂φ)2− g 4!φ 4_, _(2.0.1)

where g is dimensionless, indeed the various dimensions are: [φ] = 1 and [∂x] = 1, so

[g] = 0. Remember that S is dimensionless in natural units, indeed has the dimension of ~ = 1.

The action is invariant if we simultaneously rescale the spacetime coordinates and the field with a specific weight: φ(x) = λ∆_{φ(λx). The parameter ∆ is called the scaling dimension}

of the the field and in this specific case it coincides with the canonical dimension we mention before [φ] = ∆ = 1. Is important to observe that the same theory with a mass term (in this case for example, if we add m2_φ2_{) is not invariant, because [m] = 1.}

In general a QFT that is classically invariant of scale is not usually invariant at quantum level. However, when this happens we have a CFT.

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Why CFTs should be useful ? (The RG flow)

One of the most powerful tool to study QFTs is the Renormalization Group (RG) flow, that pictorially is a trajectory in the space of theories, see fig.2.1. The end points of this trajectories are called ”fixed points”1 _{and play a fundamental role in our understanding}

of quantum physics. We can say that a QFT can be seen as the interpolation between different ”fixed points”, where some will be attractive, and others will be unstable. One of the aims of modern theoretical physicists is to classify these ”fixed points” and to enclose the theories in various classes of universality. In order to understand why com-pletely different theories should flow to the same fixed point (and therefore belong to the same universality class) it is useful to follow the approach proposed by Wilson [9]. If we are interested in the long distance behavior of a theory it makes sense to integrate out the high momentum modes of our fields2 and work with an effective action where only the important degrees of freedom appear.

The effect of integrating out shell of momenta is to redefine the coupling constant appear-ing in our Lagrangian, and in particular couplappear-ing constants are suppressed or enhanced according to the dimension of the interaction they parametrize. In a general d dimensional QFT we can have three different types of operators:

Relevant operators: [O] < d, They are important in the IR,

Marginal operators: [O] = d, They are equally important at all scales (from UV to IR), Irrelevant operators: [O] > d, They are suppressed in the IR.

This is very important because it tells us that if we have a theory with a fixed set of fundamental fields and a given global symmetry, only a finite set of interactions (those with dimension smaller than d can survive) in the IR limit.

Everything we have just said is the opposite for the UV case, in fact here the important operators are the one with [O] > d, then we cannot have a finite set of interactions, and this complicates the description.

1_{Fixed point because in that point of the trajectories the β-functions vanish (β =} ∂g

∂ log µ = 0).

There-fore here we have no flow but the coupling parameters are fixed.

2_{These modes are not produced as external states in our scattering experiments, otherwise we should}

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Figure 2.1: Pictorial representation of RG flow trajectories of CFTs from UV on the IR. In particular the left one end in a gapped theory, and the right one in another CFT fixed point. Note that the right second fixed point can be the starting point of another RG trajectory.

The IR theory induced by a relevant or marginal deformation can be of several kind: the IR physics can be gapped, i.e all the degrees of freedom decoupled, or it can consists of gappless excitations like photons, goldston bosons, or it can be a non-trivial CFT (see fig.2.1, for a pictorial representation of RG flows3_{). For instance we can interpret}

the ordinary φ4 theory as a free scalar CFT deformed by a relevant operator φ2 and a marginal one φ4_{. The QED can be seen as the sum of two CFTs (free photons and}

electrons) deformed by a relevant interaction ψ ¯ψ and a classically marginal one ¯ψγµ_ψA µ.

In these two theories, in the UV we have Landau poles, but we can flow in the IR. In particular both the φ4 _{theory and the QED are gapped or gapless, depending on whether}

we have or not mass deformations. Another important example is the QCD, that admit a UV Lagrangian description in terms of weakly coupled quarks and gluons. In the UV limit (µ → ∞) the gauge coupling vanish (asymptotic freedom). In the IR QCD becomes strongly coupled, the theory undergoes confinement and chiral symmetry breaking and we have bound states (mesons, baryons and glueballs), so the theory is gapped. Finally we have to say that in addition to fixed points we can also have a continuous family of CFTs (”conformal manifold”). This happens when one or more deformations on the theory are exactly (not only classically) marginal.

2.1 Conformal transformations

Here we describe how to obtain all conformal transformations, and also why the case d = 2 is special. Let us consider the metric tensor gµν of a d-dimensional spacetime.

The conformal group can be defined as the set of diffeomorphisms that leave the metric unchanged up to a overall scale factor, which in general can be coordinate dependent:

g0_µν(x0) = ∂x ρ ∂x0µ ∂xσ ∂x0νgρσ(x) = Ω 2_(x)g µν(x). (2.1.1)

3_{Notice that RG flow is irreversible. Indeed we cannot uniquely reconstruct the UV theory from where}

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As we already mentioned the group structure of the conformal group is the set of coordi-nate transformations:

x −→ x0, gµν(x) −→ g0µν(x 0

) = Ω2(x)gµν(x). (2.1.2)

However, we should point out that coordinate transformations (like the one above) do not do anything at all, in the sense that the metric is invariant, ds2 _{= ds}2_{, and all we did was}

relabel the points (this is an example of diffeomorphism). This tells us nothing more than the invariance under diffeomorphisms, typical of general relativity4_{.What we want in a}

CFT is to compare it at different distances, in this case we can say something useful about physics. For example, we would like to compare the correlators of physical observables separated at different distances. Conformal symmetry relates this kind of observables. Indeed to implement a conformal transformation, we need to change the metric via a Weyl rescaling:

x → x, gµν(x) → Ω2(x)gµν(x), (2.1.3)

which change physically the distance ds2 _{→ Ω}2_(x)ds2_{. Now acting with a coordinate}

transformation such as the one in eq.2.1.2, we obtain that: x → x0, Ω2(x)gµν(x) → g0µν(x 0

). (2.1.4)

This takes us to a coordinate system where the metric has the same form as the one we started with, but the points have all been moved around and pushed closer together or farther apart. Therefore, we can view the conformal group as those coordinate transfor-mations which can ”undo” a Weyl rescaling. Everything we have said can be rephrase by: the conformal group is generated by transformations that change the metric by: gµν → Ω2(x)gµν and leave the coordinate unchanged: x → x, so we can also associate

the conformal group with the Weyl rescaling that we usually see in general relativity, in particular more symmetric actions.

Let us go back to our conformal group, at the infinitesimal level we can write:

x0µ(x) = xµ+ µ(x), Ω(x) = 1 + ω(x)

2 , (2.1.5) using the condition of eq.2.1.1, we easily obtain the following result:

∂µν + ∂νµ= ω(x)gµν, (2.1.6)

taking the trace we have: ωd = 2∂µ

µ, where d is the dimension of the spacetime, if now

we substitute this equation into eq.2.1.6, we obtain the equation identifying conformal transformations at the infinitesimal level:

∂µν + ∂νµ=

2 d(∂

τ

τ)gµν. (2.1.7)

Now we have to play around with this equation and we finally obtain:

(2 − d)∂µ∂ν(∂ττ) = gµν∂ττ. (2.1.8)

4_{Notice that the invariance under diffeomorphisms is a reparametrization invariance of the theory, and}

tells us how the metric changes by changing coordinates, but the metric is not changing. The focal point of the invariance under diffeomorphisms is not how the metric changes, but the fact that the theory does not depend on the reference system and therefore the action must be invariant under reparametrizations that are precisely called diffeomorphisms.

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From here we can see that if we put d = 2 we have a different equation and in general we have infinite solutions and the conformal group is infinite dimensional. We will see the case d = 2 in more detail at the end of this chapter.

Taking the trace of eq.2.1.8 we obtain a second order differential equation for f (x) ≡ ∂τ

τ(x), and if we solve it we obtain all conformal transformations.

What we obtain in d > 2 is that the function f (x) must be at least linear in the coor-dinates, and counting the parameters contained in a general infinitesimal transformation we obtain the following:

cµ : d, aµν :

d(d − 1)

2 + 1, bµ : d, (2.1.9) for a total of (d + 1)(d + 2)/2 parameters. Now we can recognize the transformations associated to the above parameters:

x0µ = xµ+ cµ: Transaltions, (2.1.10) x0µ = (1 + λ)xµ: Dilatations, (2.1.11) x0µ = xµ+ ω_νµxν: Lorentz rotations, (2.1.12) x0µ = xµ+ 2(bρxρ)xµ− x2bµ: Conformal boosts. (2.1.13)

We can also derive the finite transformations associated with the infinitesimal transfor-mations: µ(x) = aµ, Ω2(x) = 1, (2.1.14) µ(x) = λxµ, Ω2(x) = λ2, (2.1.15) µ(x) = Λνµxν, Ω2(x) = 1, (2.1.16) µ(x) = xµ− bµx2 1 − 2(x · b) + b2_x2, Ω 2_{(x) = (1 − 2(x · b) + b}2_x2₎2_. _(2.1.17)

There is another method to derive all the conformal transformation, instead of writing down the most general transformation that respect the condition 2.1.1. We can derive the total algebra by introducing the scale invariant transformation xµ _{→ (1 + λ)x}µ _and

an extra discrete symmetry that acts as a conformal transformation:

xµ→

xµ

x2, (dx)

2 _{→ x}2_(dx)2_, _(2.1.18)

adding this inversion transformation we obtain the full conformal transformation group. We actually get something more, because now we have also the inversion x → −x, there-fore we have a double covering of the conformal group.

At this point we want to underline one important things about CFTs. At the beginning of the paragraph we said that a CFT is a scale invariant theory, this is not entirely true. Indeed a CFT, as we just seen, add to the theory not only the scale invariant transfor-mation, but also a new type: the ”special conformal transformations”.

However there is a very important, and in a sense extraordinary, fact: under mild condi-tions, a scale invariant theory is also conformal invariant.

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Conformal algebra and primary operators

Starting from the infinitesimal transformations in eq.2.1.10 we can define the differential form of generators:

Pµ = −i∂µ, Mµν = i(xµ∂ν − xν∂µ), (2.1.19)

D = −ixµ∂µ, Kµ= −i(2xµxρ∂ρ− x2∂µ). (2.1.20)

Now using the differential form we can easily compute the commutators and then construct the algebra:

[D, Pµ] = iPµ, [D, Kµ] = −iKµ, [Kµ, Pν] = 2i(ηµνD − Mµν), (2.1.21)

[Mµν, Pρ] = −i(ηµρPν − ηνρPµ), [Mµν, Kρ] = −i(ηµρKν − ηνρKµ), (2.1.22)

[Mµν, Mρσ] = −i(Mµρηνσ − Mµσηνρ− Mνρηµσ+ Mνσηµρ), (2.1.23)

[D, Mµν] = [Pµ, Pν] = [Kµ, Kν] = [D, D] = 0. (2.1.24)

The algebra above is isomorphic to SO(d, 2), and we can put it in standard form, with signature (−, +, ..., +, −) and generators JAB given by:

Jµν = Mµν, Jµd =

Kµ− Pµ

2 , Jµ,d+1 =

Kµ+ Pµ

2 , Jd,d+1= D. (2.1.25) In order to construct representations of the conformal group SO(d, 2) in Rd−1,1 _{we have}

to decompose it into representations of:

SO(d − 1, 1) × SO(1, 1), (2.1.26) where the first is the Lorentz group and the second are the dilatations. The dilaton D has eigenvalues −i∆, and fields that are eigenfunctions transform like:

φ(x) → λ∆φ(λx), (2.1.27) these operators are called quasi-primary fields. The operators Pµ raise the eigenvalue of

D, while Kµ lower it. Thus we can construct ”highest weight representations”, this is a

usual procedure when we have a semisimple group with raising and lowering operators. We start with operators with the lowest dimension, annihilated by Kµ (at x = 0), and in

some Lorentz representation. These are called primary operators and are fundamental in our work description:

Primary operators: [Kµ, O(0)] = 0. (2.1.28)

The representation is infinite dimensional5. All other operators are constructed with the action of Pµ and are called descendants.

Conformal frame

Given n points, we can make use of conformal transformation to rearrange them in a convenient way as we see in fig.2.2. In a general d-dimensional spacetime, a CFT have an SO(d, 2) symmetry due to conformal invariance. Now the question is: how can we use the whole symmetry to move points in convenient locations for calculations ?

Here we describe all the passage to be done in order to construct the conformal frame:

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• Using invariance under translations we put the center of our coordinate system in x1 then we have x1 = 0.

• Using rotation symmetries we choose the x axes along x3 and we use the scale

invariant symmetry to put x3 = 1.

• Finally we use special conformal transformations to put x4 → ∞. I can do this using

the finite transformation (look at eq.2.1.14) with bµsuch that: 1 − 2(x · b) + b2x2 = 0,

valid only for x4. Therefore we have µ→ ∞, then x4 → ∞.

In doing so, we have not exhausted the full symmetry, a residual SO(d − 1) remains. This symmetry can be used as the stabilizer group for the last point x2. If there are

more than four points, the stabilizer group of this configuration is instead SO(d + 2 − m), where m = min(n, d + 2). Therefore when n > d + 1, there is no residual symmetry and there are (n − d − 1) d-dimensional points that are completely unconstrained. The total unconstrained degrees of freedom, can be identified with the number of independent conformally invariant cross-ratios that can be built out of n points.

Figure 2.2: Usual conformal frame for the coordinates of 4 point. Then all the kinetic physics can be described in function of z that is a complex number.

Since we will be interested in 4-pt correlation functions, for us n = 4 is the relevant case. Here we have 2 degrees of freedom due to the fact that the last point x2 has only

SO(3) symmetry in a 4-dimensional spacetime.

These invariants are usually identified with the following two cross-ratios:

u = x 2 12x234 x2 13x224 , v = x 2 14x223 x2 13x224 . (2.1.29)

In Euclidean signature we can use the following parametrization:

x1 = (0, 0,#»0 ), x2 = 0,z − ¯z 2i , z + ¯z 2 , x3 = (0, 1,#»0 ), x4 = (0, ∞,#»0 ), (2.1.30)

where #»0 is a (d − 2)-vector. Here z, ¯z are complex conjugate variables. In Minkowski signature they become real and independent. Using this parametrization we can rewrite the cross-ratios:

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This coordinate frame can be thought of as a gauge fixing of most of the conformal sym-metry. Now in order to classify the independent tensor structures or to impose consistency conditions (such as crossing symmetry, that we see later on in this chapter), we can restrict to a conformal frame instead of working at generic coordinate configurations.

2.2 Radial quantization and states-operators duality

In QFTs we have to define what we mean by evolution operator. Indeed in general we want to compute correlation functions that corresponds to scattering amplitudes, and we need the concept of evolution of states in order to compute them. In ordinary QFTs we are used to consider time evolution as generated by the Hamiltonian H = P0 _{(the time}

evolution operator is e−iHt), namely the time component of the translation generator Pµ. Therefore we can evolve some initial state defined on a spacial slice at fixed time t1, to an

evolved state at time t2. This is very convenient because in QFTs we choose eigenstates

of Pµ as basis of the Hilbert space.

However in CFTs the situation is different, because the irreducible representations (irreps) of the theory are built from an highest state, the primary state, which is not an eigenstate of translations. This means that the usual evolution operator would evolve a primary state into a mixture of primary and descendants, which is inconvenient. In order to overcome this issue, it is amenable to choose the operator D as the new evolution operator of the theory (the time evolution operator become eiDα). By doing this choice, we are effectively choosing a different foliation of the spacetime: instead of fixed time slices, we are foliating by spheres with fixed radius.

To obtain this new type of foliation we have to change coordinates and use spherical ones; then we pass from Rd _{to S}d _{and obtain the following metric:}

ds2 = dxµdxµ= dr2+ r2dΩd−1. (2.2.1)

Is also possible, using conformal transformation, to give a cylindrical interpretation by the following change of coordinates:

τ = log r =⇒ r → λr become: τ → τ + log λ. (2.2.2) In this case we pass from Sd _{to R × S}d−1 _{and we obtain the following conformal}

equivalent metric:

ds2 = dr2+ r2dΩd−1−→ ds2 = r2(dτ2 + dΩd−1) = r2ds2cyl, (2.2.3)

where r2 _{is the conformal factor. The cylindrical case can be seen in fig.2.3. In particular}

r → ∞ correspond to τ → ∞ and r → 0 correspond to τ → −∞. The cylindrical interpretation is more familiar and a slice of constant τ is a (d − 1)-sphere. We are now able to verify the state operator correspondence in CFT. A state in a CFT is determined by the field configuration on a given slice of our spacetime foliation, hence a sphere. Then, as we seen above, we can use the dilaton operator D to evolve it back and forth, namely contract or expand the sphere. The sphere parameter is the ratio r/ri, where ri is the

starting radius and r is the actual one. Sending ri → 0 we can define the vacuum in radial

quantization on sphere of radius r:

|0ir =

Z

Br

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(a) (b)

Figure 2.3: In (a) we have a pictorial representation of radial quantization, in particular r indicate the radial coordinate and φ1, φ2 indicates two generic states. In (b) we have

the cylindrical quantization, where the τ → ∞ state is the out one hφ| and the τ → −∞ state is the in one hφ|. The variables n2 and n3 parametrize the Sd−1 sphere and specify

the particular states.

where the integral is performed inside a ball Br of radius r. Notice that if we use the

cylindrical prospective all become more familiar because we have: |0iτ =

Z

τ

D[OCF T]eiSCF T, (2.2.5)

where the integral is performed inside the cylinder from τ → −∞ to τ . Now we return to the usual radial quantization, and we continue our analysis in this coordinate system. If we insert an operator at the origin O(0) we obtain:

|Oir =

Z

Br

D[OCF T]O(0)eiSCF T. (2.2.6)

This state has the same quantum numbers of the operator, and in particular is a primary state. We can interpret it as the action of the operator on the vacuum O(0)|0i. We can also ask what happen if we insert an operator in a generic position O(x):

|O(x)ir = e−iP ·xO(0)eiP ·x|0ir= |O(0)ir+ ix · P |O(0)ir+ ..., (2.2.7)

which is a infinite superposition of primary and descendants. The description above permit us to identify a state starting from an operator in a CFT. Remember that an operator is completely defined by its matrix elements between other states of the theory, hence by its correlation functions. If we consider an n-pt correlation function:

hO1(x1)O2(x2) · · · On(xn)i. (2.2.8)

Suppose we want to define the operator Oi(xi), we can put xi = 0 due to translation

symmetry. We can replace the presence of the operator with the state defined on the sphere:

hOi(0)|other operatorsi. (2.2.9)

This operation can be done (is valid) for any operators. This fully define the CFT oper-ators in terms of states. Therefore we obtain a uniquely way to go from an operator to a state, and also from a state to an operator. Therefore we got a one to one map between states and operators in any CFT.

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2.3 Generalities of CFTs and Operator product

ex-pansion (OPE)

One of the most powerful tools in a CFT is the Operator Product Expansion (OPE). Let us briefly review this concept by following [10]. For simplicity here we will consider OPE between scalar operators. In any Lorentz-invariant QFT we can define an OPE, but in CFTs this tool is much more powerful. The OPE of φ(x)φ(0), in the short-distance limit (x → 0), can be approximated by a sum of local operators with c-number coefficient functions depending only on x and ∂:

φ(x)φ(0) ∼X

O

C(x, ∂)xµ1_{· · · x}µl_O

µ1···µl(0). (2.3.1)

The RHS will in general contain both scalars and operators of nonzero spin l, and in the latter case their indices have to be contracted as shown.

In a CFT we have three more specific properties. Firstly the x2 _{dependence of the}

coefficient functions is a power-law fixed in terms of the operator scaling dimensions6

C(x, ∂) ∼ (x2)−∆φ+∆O−l₂ _{(1 + O(∂)).} _(2.3.2)

Secondly conformal symmetry allow us to classify all local operators of the theory into primaries, which transform homogeneously under the conformal group, and their deriva-tives (descendants). We can now rewrite the OPE using a summation only over all the primaries O:

φ(x)φ(y) =X

O

λφφOC(x − y, ∂y)O(y). (2.3.3)

The index contractions are implicit if O has nonzero spin. The function C is a power series in ∂y and encodes the contribution of O and all of its descendants. The form of

this function is completely fixed by conformal symmetry in terms of the operator scaling dimensions up to the overall numerical coefficient λφφO.

The form of the 2-pt function is completely fixed and normalized by conformal symmetry and can be written as:

hO(y)O(z)i = |y − z|−2∆O_. _(2.3.4)

At this point one can argue if is possible to extract the function P . This is possible, but before we have to introduce the 3-pt function. Also this is fixed by conformal symmetry to be like this:

hφ(x)φ(y)O(z)i = λφφO|x − y|∆O−2∆φ|x − z|∆O|y − z|∆O. (2.3.5)

On the other hand we can also use the OPE to expand φφ and we obtain the following relation:

hφ(x)φ(y)O(z)i = λφφOC(x − y, ∂y)hO(y)O(z)i. (2.3.6)

6_{This is only guaranteed in a unitary theory, so that the dilaton operator can be diagonalized. There}

exist special class of non-unitary theories called logarithmic CFTs, were this expansion in not true, but we should have some logarithmic deviations.

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Now, using both eq.2.3.5 and eq.2.3.6, the function P can be fixed unambiguously. Here we write down the first few terms in the expansion:

C(x, ∂y) = |x|∆−2∆φ 1 + 1 2x µ_∂ µ+ αxµxν∂µ∂ν + βx2∂2+ ... , (2.3.7) α = ∆ + 2 8(∆ + 1), β = − ∆ 16 ∆ − d₂ + 1 (∆ + 1). (2.3.8) In general for non-identical operators we also have the dependence on the difference of the scaling dimension |∆φ1− ∆φ2|. Such an expansion show interesting structure, for example

the d dependence appears only in the terms multiplied by x2_{, and would therefore be}

subleading on the lightcone (that in this case corresponds to the limit x → 0). Notice also that the d-dependent term becomes singular when ∆O hits the scalar field unitarity

bound d/2 − 1. This is not a problem since such an O is necessarily free and so λφφO = 0.

When the OPE structure is determined we can use it to express any n-point function as a sum of (n − 1)-functions. Schematically:

hφ(x)φ(y) n Y i=1 ψi(zi)i = X O λφφOC(x − y, ∂y)hO(y) n Y i=1 ψi(zi)i. (2.3.9)

For n = 3 there is a single exchanged primary operator O = φ1, but for n > 3 the sum

will be infinite. Actually, it will be doubly infinite since C’s are infinite series in ∂y.

Here comes the third special property of CFTs, this infinite sum converges. By this we mean that the representations 2.3.9 are actually absolutely convergent at finite separation (x − y), rather than being just asymptotic expansions in the limit x → y. This property has two important applications:

• Correlation functions of arbitrarily high order can be computed by applying the OPE recursively. Of course, to do this we must know all primary operator dimensions ∆i

and all OPE coefficients λijk (collectively known as the CFT data).

• Eq.2.3.9 can also be used to constraint the CFT data, this is an old method which goes by the name of conformal bootstrap [11]. The key point is that in a 4-pt function we can apply the OPE in three different couples, we can do (12)-(34) or (13)-(24) or (23)-(14). The fact that the three result agree give us constraints on the CFT data.

Unitarity bounds (shortening conditions)

Unitarity constrains the scaling dimensions of our conformal field theory. This follows from the fact that, in a unitary theory all states must have non-negative norm. Therefore we have to impose that condition, to all states O of the theory:

|||Oi||2

= hO|Oi > 0. (2.3.10) This condition has strong consequences on the possible spectrum of operators in a generic CFT. Here we report the result for a generic dimension d and a operator Oµ1µ2···µl_{, in a}

generic symmetric and traceless irrep of SO(d):

Oµ1µ2···µl _=⇒

∆ > l + d − 2, for: l 6= 0. ∆ > d−22 , for: l = 0.

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We have a general derivation of these results in AppendixA.

Let us now try to understand the deeper meaning behind the unitarity bound, indeed we see below that this implies a conserved current in the theory.

When we have an operator that saturates the unitarity condition ∆ = ∆U, we have that

hPµ_Oµ1µ2···µl_{Xi = 0, where X is a general combination of operators; indeed if this quantity}

is not zero, violates the unitarity bound.

As consequence for the condition just found, we have that:

∂µhOµ1µ2···µlXi = 0, (2.3.12)

then Oµ1µ2···µl _{is a conserved current.}

For example if Tµν (l = 2) is conserved (this is needed for example in local field theories) we have:

h∂µTµνXi = 0 =⇒ [Tµν] = l + d − 2 = d, (2.3.13)

and for a generic conserved current Jµ_{, we have [J}µ_{] = d − 1}

Finally a very instructive example is the case of a massless scalar theory:

L = 1 2(∂µφ)

2

. (2.3.14)

We have ∆U = (d − 2)/2 = 1 in d = 4 and the corresponding unitarity condition is

|||P2_Oi||2 _{= 0. This condition is indeed verified by φ (has the correct dimension [φ] = 1)}

and correspond to the usual KG equation ∂µ_∂

µφ = φ = 0.

The unitarity bound is also called shortening condition, this because if we saturate the unitarity bound, as we have seen: hPµOµ1µ2···µl_{Xi = 0 and from now on if we act both}

with Pµ or Pµ we obtain another null operator. Therefore the multiplet becomes shorter,

We will see this concept better later on, when we talk about SCFT and BPS operators.

2.4 Conformal block decomposition

Let us now consider a 4-pt function. For simplicity we restrict to the case of 4 scalar fields, with respectively dimension ∆1, ∆2, ∆3, ∆4. Recall the OPE introduced in the previous

section, can be easily computed by for example pairing φ1(x1)φ2(x2) and φ3(x3)φ4(x4) [12].

This generally give us a decomposition on the 4-pt function of the following form: hφ1(x1)φ2(x2)φ3(x3)φ4(x4)i =

X

O

λ12Oλ34OWO, (2.4.1)

where WO ≡ WO(xi) are the conformal partial waves (CPWs) given by:

WO = C12O(x1, x2, y, ∂y)C34O(x3, x4, y0, ∂y0)hO(y)O(y 0

)i. (2.4.2) Since the 2-pt function is diagonal, the summation is over the same operator O in both OPEs. This decomposition above it will be useful to us, when we talk about the Lorentzian inversion formula in chapter 3.

At this point we can introduce the conformal block decomposition, in order to do this we have to extract from eq.2.4.2 the conformal invariant part. Indeed the CPWs transforms under conformal transformation in the same way as the 4-pt function itself, then we can extract a prefactor that transforms correctly under conformal transformations (like we do

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for 2-pt and 3-pt functions). The prefactor for a 4-pt function of scalars only is easy to obtain and we have:

hφ1(x1)φ2(x2)φ3(x3)φ4(x4)i = KG(u, v), (2.4.3) K ≡ K(∆i, xi) = 1 (x2 12) ∆1+∆2 2 (x2 34) ∆3+∆4 2 x2 24 x2 14 ∆12₂ _x2 14 x2 13 ∆34₂ , (2.4.4) where ∆12= ∆1− ∆2 and ∆34= ∆3 − ∆4.

We can now rewrite eq.2.4.2 using the prefactor and we obtain:

WO = KG_∆∆_O12_,l,∆_O34(u, v), (2.4.5)

where u, v are the cross-ratios we defined in the subsection 2.1, and G∆12,∆34

∆O,lO (u, v) is called

conformal block. In reality, in the end, conformal blocks and partial waves are the same thing and are often exchanged even in literature. Then we can write the conformal block decomposition:

G(u, v) =X

O

λ12Oλ34OG∆_∆12_O_,l,∆_O34(u, v). (2.4.6)

Note that each conformal block G∆12,∆34

∆O,lO (u, v) encodes the contribution of a single

irre-ducible conformal multiplet, with a primary of dimension ∆ and spin l (the conformal primary or the superconformal primary in a SCFT, and its descendants). The superscript ∆12 and ∆34 represents dependence on the dimensions of the four external operators φi.

Furthermore λ12O and λ34O are the OPE coefficients, and the sum runs over the particular

set of operators that we have in a given theory.

The above decomposition is not regular in the whole spacetime, but have a regularity domain that in general depends on the coordinate system we use to parametrize it. For example using (z, ¯z), the usual variable, we have that the region of regularity is given by X = C \ (1, +∞), as we can see in fig.2.4. Everywhere in this region the blocks will be analytic, except at z = 0 because of the |z|∆ _factor.

Figure 2.4: On the left we have the region of regularity of the conformal block decompo-sition for the z variable. On the right we have the region of regularity for the ρ variable.

As we have seen above in order to construct the conformal block decomposition we have to use the OPE in two pairs of the 4 points, this in general depends on the relative positions of the points.

Here is a bit tricky, for example we can couple (12) and (34) only if |z| < 1, if this is not true we can we can make different pairs. The point is that due to conformal symmetry we can always transform, for example, the following couple (23) and (34) to (12) and (34) and than have always |z| < 1. This is obvious if we use the conformal frame we defined

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above, in that case we always can have |z| < 1 and then using the couple (12) and (34). Assuming this last case, for every point in X we can find a sphere which separates x1 and

x2 from x3 and x4, in order to use the OPE. Choosing the center of this sphere as a radial

quantization origin, one can prove the regularity of the conformal block for such z. The blocks will be singular on the cut, because the separating sphere jumps when z crosses it. There exists other important parametrization, a one-to-one correspondence between the complex parameters (z, ¯z) and ρ, ¯ρ, fixed by demanding that the cross-ratios should agree. We will study this case better in chapter 3, for now we just need to know that the function ρ(z) maps the region X onto the unit disk. This suggests that these coordinates should be particularly suitable to analyze the blocks. Indeed conformal block representations as power series in ρ will converge for |ρ| < 1, as we can see in fig.2.4.

Now that we define well what is a conformal block decomposition, we have to find it, also because its usefulness is linked to the fact that there are techniques to obtain it. To determine the conformal block decomposition of a 4-pt function exist at least three methods:

1. We can use the definition (using directly the algebra), see for example [13], here was calculated for the first time the conformal block decomposition for d = 2 and d = 4, using a particular recursion relation.

2. The Zamolodchikov recursion relation, or pole recursion relation. 3. The conformal Casimr equation.

We analyze first the last method, which allows, as we will see, to obtain exact formulas in the case of even spacetime dimension.

In radial quantization we can write the 4-pt function as a a scalar product of two states hφ1(x1)φ2(x2)φ3(x3)φ4(x4)i = hφ1(x1)φ2(x2)|φ3(x3)φ4(x4)i, (2.4.7)

living on a sphere separating x1, x2 from x3, x4.

Then we can insert the identity operator between the two states:

hφ1(x1)φ2(x2)φ3(x3)φ4(x4)i = hφ1(x1)φ2(x2)|I|φ3(x3)φ4(x4)i, (2.4.8)

where we can decompose the identity operators in term of a complete set of projectors:

I = X ∆,l P∆,l = X ∆,l X

α,β=O∆,l,P O,P P O,...

|αiGαβ_hβ|, _(2.4.9)

here the dots in the summation indicates that we have to sums over primaries and all the descendants, and Gαβ = hα|βi is the Gram matrix of the multiplet, is like a metric in this

strange space of all the descendants of the primary operator O∆,l.

Then we obtain the following expression: X

∆,l

hφ1(x1)φ2(x2)|P∆,l|φ3(x3)φ4(x4)i = K

X

O∆,l

λ12Oλ34OG∆_∆,l12,∆34(u, v). (2.4.10)

Now we can notice that each projector P∆,l is in the irrep (∆, l) of SO(d + 1, 1), then if

we act with the quadratic Casimir operator C2 = 1₂JABJBA (where remember the JAB

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C4 = 1₂JABJBCJCDJDA, is proportional to the identity (using the second Shur’s lemma)

and we obtain the corresponding eigenvalues: CiP∆,l = C (i) ∆,lP∆,l, C (2) ∆,l = 1 2[∆(∆ − d) + l(l + d − 2)], (2.4.11) C_∆,l(4) = l(l + d − 2)(∆ − 1)(∆ − d + 1). (2.4.12) On the other hand the action of C2 or C4 on the RHS can be computed using the algebra

representation of the conformal generators on primaries as first-order differential opera-tors.

In conclusion the second order differential equation that the conformal blocks have to satisfy is the so called conformal Casimir equation:

C∆,lG∆,l(z, ¯z) = DG∆,l(z, ¯z), (2.4.13)

where as we already said D = D(z, ¯z) should be a second order differential operator. In particular we have: D = Dz+ Dz¯+ 2(d − 2) z ¯z z − ¯z[(1 − z)∂z− (1 − ¯z∂z)], (2.4.14) Dz = 2z2(1 − z)∂z2− (2 + ∆34− ∆12)z2∂z+ ∆12∆34 2 z, Dz¯= Dz(z ↔ ¯z). (2.4.15) Notice the symmetry z ↔ ¯z always present in the conformal blocks of a CFT, as we seen later this is not always true in a SCFT, due to supersymmetry.

Then as we can see from eq.2.4.13, we have to solve a second differential equation, and as we know in order to do this we must fix the boundary condition (the Dirichlet or Von Neumann), in this case Dirichlet boundary condition. In particular the leading z, ¯z → 0 behavior of the conformal block can be easily determined using the OPE. Considering in eq.2.4.5, the limit x12, x34→ 0 we can obtain the following boundary condition:

g∆12,∆34 ∆,l ∼z,¯z→∞Nd,l(z ¯z) ∆ 2Gegl z + ¯z 2√z ¯z , (2.4.16) where Geg_l(z) = C( d 2−1)

l , is a Gegenbauer polynomial, and the normalization Nd,l is given

by: Nd,l = l! (−2)l₍d 2 − 1)l , (2.4.17)

where d is the dimension of spacetime and l the spin of the exchanged operator.

At this point we can solve the Casimir equation 2.4.13, but in literature exist closed form only for even spin l. Indeed the general way to obtain all the even solutions is to find the d = 2 solution that is easier also due to the holomorphic property of the 2d CFT, and then exist a recursion relations relating blocks in d and d + 2 dimensions, that permits to calculate a closed form for all even dimension.

The first computation using Casimir equation was done by Dolan and Osborn in [14], in particular here they compute the exact form of conformal blocks for d = 6 for the first time and reproduce the d = 4 result they obtained jet using a recursion relation in [13]. Here we report the result for d = 2 and d = 4, in particular we will use the 2d conformal block decomposition in our work. They are expressed in terms of the basic function:

kβ(z) = z β 2₂F₁ β − ∆₁₂ 2 , β + ∆34 2 , β; z , (2.4.18)

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which satisfy: Dzkβ(z) = 1 2β(β − 2)kβ(z), kβ(z) ∼z→0 z β 2. (2.4.19)

In d = 2 the differential operator factorize D = Dz + Dz¯, as we expected due to the

holomorphic property of 2d CFT, then we obtain:

G∆12,∆34 ∆,l (z, ¯z) = 1 (−2)l_{(1 + δ} l,0) (k∆+lk∆−l(¯z) + z ↔ ¯z). (2.4.20) In d = 4 we have: G∆12,∆34 ∆,l (z, ¯z) = 1 (−2)l z ¯z z − ¯z(k∆+lk∆−l−2(¯z) + z ↔ ¯z). (2.4.21) In odd dimension, general closed-form solution of the Casimir equation are not available yet. Nevertheless there are very efficient series representations that are sufficient for practical purposes.

Now we quickly review the Zamolodchikov recursion relations, because it might be useful to better understand the derivation of the Lorentzian inversion formula that we will do in chapter 3 and in Appendix C.

This technique is an old idea and has been introduced by Zamolodchikov for Virasoro conformal blocks7 _{in [15], then there were several generalizations and applications also in}

d > 2 CFTs, for example in [16].

The basic idea of this recursion relation is to extract the conformal blocks using residue of poles due to null states in the CFT. In practice when we write the conformal block as in eq.2.4.10, if we diagonalize8 _G

αβ we obtain the following:

λ12Oλ34OG∆,l(x1, x2, x3, x4) =

X

α∈HO

hO1(x1)O2(x2)|αihα|O3(x3)O4(x4)i

hα|αi , (2.4.22) where HO is the irreducible representation of the conformal group associated with the

primary O (i.e., it is O with all its descendants). Then if we study the various states in HO, we can find that for some special values of ∆ = ∆∗A there exist a descendant state:

|OAi ∈ HO, with: ∆A= ∆∗A+ nA, (2.4.23)

that at level nA becomes a primary (Kµ|OAi = 0). Then we have a primary descendant:

easy to show must have zero norm (||OA||2 = 0). Therefore, it follows that the

denomi-nator of eq.2.4.22 becomes zero when ∆ = ∆∗_A. Than G∆,l has a pole for ∆ → ∆∗A and

its residue is proportional to a conformal block:

G∆,l(x1, x2, x3, x4) →

RA

∆ − ∆∗_AG∆A,lA(x1, x2, x3, x4), as: ∆ → ∆

∗

A. (2.4.24)

The residue RA will be given by three contributions:

RA= M (L) A QAM

(R)

A , (2.4.25)

where QAcomes from the inverse of the norm hα|αi, while M (L)

A and M (R)

A comes from the

three point functions hO1(x1)O2(x2)|αi and hα|O3(x3)O4(x4)i, and in general is possible

to compute all these values, using algebra and OPE.

7_{These are particular conformal blocks only for the case d = 2.} 8_{In general we can diagonalize G}

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2.5 Conformal field theory in d = 2

The content of this section was selected by [17]. We have already seen that in 2d the conformal transformation are special, indeed we have infinite generators, so in a sense infinite symmetry. Therefore in general the theory is much more constrained then in d > 2.

We work for simplicity in Euclidean signature, and we use the following coordinate system (z0, z1). We denote with φ the conformal map:

φ : gµν(z) −→ g0µν(w) = Λ(z)gµν(z). (2.5.1) Remember now the transformation of the metric under general change of coordinates:

gµν −→ ∂w µ ∂zα ∂wν ∂zβ gαβ = Λ(z)gµν(z), (2.5.2) where the last condition in the conformal symmetry one. The condition to be a conformal transformation than it became the following:

∂w0 ∂z0 2 + ∂w 0 ∂z1 2 = ∂w 1 ∂z0 2 + ∂w 1 ∂z1 2 , (2.5.3) ∂w0 ∂z0 ∂w1 ∂z0 + ∂w0 ∂z1 ∂w1 ∂z1 = 0. (2.5.4)

These equations are equivalent to the following condition:

∂z0w1 = ±∂_z1w0, ∂_z0w0 = ∓∂_z1w1, (2.5.5)

which is easy to see are the holomorphic (and anti-holomorphic) Cauchy-Riemann condi-tions. A complex function that satisfy this condition is an holomorphic function in some open set. We can exploit this property in 2d CFT performing a change of coordinates and passing to complex ones:

z = z0+ iz1, z = z¯ 0− iz1_, _∂ z = 1 2(∂0− i∂1), ∂z¯= 1 2(∂0+ i∂1). (2.5.6) Then the Cauchy-Riemann conditions become the usual: ∂z¯w(z, ¯z) = 0.

We have now that all the possible conformal tranformations are in one to one correspon-dence with the holomorphic ones, that are all the following:

z → w(z) → dw = dw dz

dz, (2.5.7)

where |dw/dz| is the dilaton factor and arg(dw/dz) is the phase responsible for rotation. Indeed remember that two important example of holomorphic transformations are trans-lations (z → z + a), rotations (z → eiθz) and dilatations (z → λz).

All what we said above is true only locally, if we want to define global conformal transfor-mations in 2d we have to require also that the transfortransfor-mations are defined everywhere and are invertible. The global conformal group in 2d (called the ”special conformal group”) is the set of all these maps:

f (z) = az + b

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where a, b, c, d ∈ C. These mappings are called projective transformations and are iso-morphic to SL(2, C) ' SO(3, 1), that is a group with 6 real parameters (3 complex ones). For a careful demonstration see [17].

Instead the local transformation generators give us the Virasoro algebra:

[Ln, Lm] = (n − m)Ln+m+ c 12n(n 2_{− 1)δ} n+m,0, (2.5.9) [ ¯Ln, ¯Lm] = (n − m) ¯L + c 12n(n 2_{− 1)δ} n+m,0, [Ln, ¯Lm] = 0, (2.5.10)

where Ln and ¯Ln are modes of the energy momentum tensor:

T (z) =X n∈Z z−n−2Ln, Ln= 1 2πi I dzzn+1T (z), (2.5.11) ¯ T (¯z) =X n∈Z ¯ z−n−2L¯n, L¯n= 1 2πi I d¯z ¯zn+1T (¯¯ z), (2.5.12) and c corresponds to the central charge of the 2d CFT algebra. Notice that this algebra contains a particular subalgebra formed by L0, L−1 and L1. This is associated to the

global algebra SL(2, C) defined by eq.2.5.8. For our purpose we need only the global part of the 2d CFT transformations. Moreover all the techniques we will use are the ones we have seen in the d > 2 case. Indeed the global SO(3, 1) group corresponds to SO(d + 1, 1), with d = 2, that is the Euclidean conformal group. Than we have the same global group structure as in the d 6= 2 cases.

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Chapter 3 The Lorentzian inversion formula

The Lorentzian inversion formula we derive here was first introduced by Caron-Huot [6], and it has already been used in several works. This formula is very powerful and permit us to extracts two fundamental data in the CFT side. An interesting thing, is that this formula is been derived using Lorentz frame. Typically in QFTs we use Wick rotation to go in an Euclidean frame, because is more simple and also because the partition functions converge better9 , but here we take the opposite path: we pass from Euclidean to Lorentzian metric, in doing so we are able to obtain a more powerful formula. We present both in the following.

Here we follow [6]: we review the most important concepts and emphasize certain aspects. For all technical calculation we refer to [6].

3.1 Preliminary concepts

How to extract information form dispersion relations

Consider an ”amplitude” which admits a low-energy Taylor series:

f (E) =

∞

X

l=0

flEl, (3.1.1)

and suppose to have the following information:

• f (E) is analytic except for a branch cuts at real energies |E| > 1. • f (E) E is bounded at infinity.

The function f (E) could for example represent the 4-pt correlator and its low-energy expansion will be provided by the OPE. At the thresholds E = ±1 some distances become timelike, and so we have branch cuts for |E| > 1, were distances become spacelike. With the above assumptions, an elementary contour deformation argument relates the

9_{Remember that the Wick rotation (t → −iτ ), in general simplify the QFT side. Indeed for example}

transform the partition function of a generic QFT to a new one, where convergence is easier to see and study:

Z = Z

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series coefficients to the discontinuity of the amplitude (this is what we typically call an integral dispersion relation10_{). Now, to determine the coefficients f}

l, firstly we use the

second Cauchy theorem of complex analysis and then in the second equality we use the assumed high-energy behavior to drop large arcs at infinity:

fl ≡ 1 2πi Z |E|<1 dE E E −l f (E) (3.1.2) = 1 2π Z ∞ 1 dE E E −l

(Disc f (E) + (−1)lDisc f (−E)), (l > 1), (3.1.3) where Disc f = −i[f (E(i + i)) − f (E(i − i))]. For example, one may take the function f (E) = − log(1 − E2_{), which has the following discontinuity Disc f = 2π, so the integral}

produces: fl= 1 2π Z ∞ 1 dE E E −l 2π(1 + (−1)l) = 1 + (−1) l l , (3.1.4) that is the result we expected using a simple Taylor expansion.

Now let us focus on on a single coefficient, say f2. It may seem paradoxical that it can

be recovered from the discontinuity of f (E), given that varying f2 alone in the above

equation clearly leaves Disc f (E) unchanged. The point is that given the constraint that f (E) E

is bounded at infinity, the coefficient f2 (or any finite number of coefficients) can-not be varied independently of all the others. Rather, the coefficients form a much more rigid structure, that is an analytic function of spin. This is explicit by the integral in the eq.3.1.2, which defines an analytic function provided that the real part of l is large enough. More precisely, there are two analytic functions, for even and odd spins, reflecting that there are two branch cuts.

These are the key features of the Euclidean Froissart-Gribov formula [19, 20] that we will see in Appendix C, which is conceptually the same but with Legendre functions instead of power-law ones.

We will show that OPE coefficients in unitary conformal field theories are of a similar type: they are not independent from each other, but rather organize into rigid analytic functions. Furthermore, they can be extracted from a “discontinuity” which would naively seem to annihilate each individual contribution.

An immediate quantitative implication can be illustrated in the example of large-N the-ories with a sparse spectrum, where the discontinuity is negligible below a gap ∆2

gap. The

formula 3.1.2 then gives a result which decays rapidly with spin:

fl ∼ Z ∞ ∆2 gap dE E E −l

Disc f (E) ∼ (∆gap)−2l, (3.1.5)

which in the context of gauge-gravity duality (AdS/CFT), that we will see in chapter 5, will be interpreted as the expected suppression of higher-derivative corrections, if the bulk theory is local to distances of order 1/∆gap times the AdS curvature radius.

10_{Dispersion relations are very old tools, indeed the first appearance in physics dates back to 1926−1927}

with the Kramers and Kronig relations [18].

Probably mathematicians had used them also before, for mathematical purposes, indeed these techniques are born in the theory of complex analysis.

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Generalities of inversion formulas

In order to get ready for the derivation of the Lorentzian Inversion formula, we have to review some important concepts:

• The conformal block decomposition. • The crossing equations.

• The lightcone coordinates (ρ, ¯ρ).

We have already seen the conformal block decomposition in chapter 2, so we do not review it again. Another fundamental tools in CFT are crossing symmteries, indeed when we consider a 4-pt correlation function, for example of scalar operators, and we expand it into conformal blocks, we have to decide how to expand it. Generally we can do (12) → (34) (usually called s-channel) or (32) → (14) (t-channel) or (13) → (24) (u-channel).

Figure 3.1: Crossing equation between s-channel (to the left) and t-channel (to the right) for the 4-pt function hO1O2O3O4i. Ok represent the exchanged operators.

Now since the choice of pairing the operators is completely arbitrary, the two expan-sions must give the same final result, therefore correlation functions must be crossing symmetric. It is easy to see that the first two channel s and t can be obtained each other by interchanging the point 1 and 3, see fig.3.1, this correspond to z → 1 − z. For a complete and general treatment see Appendix B.

Therefore taking into account the value of the prefactor in both channels, and equating the two conformal block decomposition, we can get the crossing relations. For exam-ple for the simexam-ple case in which we have four scalar operators with the same dimension ∆1 = ∆2 = ∆3 = ∆4 = ∆, we have the following:

X O λ12Oλ34O G∆O,lO(z, ¯z) (z ¯z)∆1+∆22 =X O0 λ32O0λ_14O0 G∆O0,lO0(1 − z, 1 − ¯z) ((1 − z)(1 − ¯z))∆3+∆22 =X O00 λ13O00λ_24O00 G∆O00,lO00( 1 z, 1 ¯ z) (1_z1_z_¯)∆1+∆32 , (3.1.6)

where the sums run over the operators O, O0 and O00 which appear in the OPE in the three channels. Focus now only on the crossing relation between s- and t-channel, that can be imposed at any point z, ¯z where both the channels converge. This corresponds to the plane of all complex z minus cuts along (1, +∞) where the s-channel diverges and (−∞, 0) where the t-channel diverges. Therefore we can do such an expansion in the

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space X = C \ [(1, +∞) ∪ (−∞, 0)]. Indeed the standard choice in numerical studies is to impose crossing in a Taylor expansion around the point z = ¯z = 1/2, which is well inside this region. In general we can also use the crossing equation for the u-channel in order to make other constraints. However in the case of 4 identical scalar operators is automati-cally satisfied. For non-identical external operators, the u-channel becomes important. The last fundamental thing to do is to change coordinates and use lightcone (also called Rindler) ones. Remember that four points can always be mapped to a plane via a con-formal transformation. Indeed z and ¯z can be viewed as coordinates on a 2-dimensional plane. Moreover since we will be interested in Lorentzian kinematics, we distribute the points symmetrically in two Rindler wedges. Now remember that when we use (z, ¯z) co-ordinates we are in the Euclidean frame, then in order to pass to the Lorentzian one, we have to use the Wick rotation and then we just make one coordinate imaginary, y → it. This has the following effect on our complex coordinates:

(z, ¯z) = (x + iy, x − iy) 7−→ (ρ, ¯ρ) = (x − t, x + t), (3.1.7) so the complex conjugate pair z and ¯z have become two independent real numbers. Now we rename the two variable (ρ, ¯ρ), but in general in literature they use (z, ¯z) inde-pendently in the Euclidean or Lorentzian frame.

Notice that in the Euclidean frame, we have that z and ¯z are complex conjugate vari-ables, usually we use them as general different complex coordinate, to make the most of analytical techniques, but in the end we must remember that ¯z = z∗. In the Lorentzian frame, z and ¯z become two independent real variables. They are now called lightcone coordinates since lines of constant ρ or ¯ρ are at 45◦ on a spacetime diagram. Now we can put11 _{our four points in the new plane, as in fig.3.2:}

x4 = (ρ, ¯ρ) = −x3, x1 = (1, 1) = −x2. (3.1.8)

We see that ρ → 0 is the s-channel OPE and in the double lightcone limit ¯ρ → 1 we approach the t-channel. Actually ¯ρ → −1 is a similar limit but now involving the u-channel. We can easily see from fig.3.2 that this way to write the coordinates is natural, in particular we have the following relations:

z = 4ρ (1 + ρ)2 =⇒ ρ = 1 −√1 − z 1 +√1 − z, ρ =¯ 1 −√1 − ¯z 1 +√1 − ¯z. (3.1.9) In ρ coordinates the conformal block change a bit and we can replace the OPE by a series in ρ and ¯ρ: G(ρ, ¯ρ) =X ∆,l ˜ c∆,lρ ∆−l 2 ρ¯ ∆+l 2 . (3.1.10) The product ρ∆−l2 ρ¯ ∆+l

2 is called ”scaling block”. A conformal block is a convergent sum

of scaling blocks with positive coefficients. Remember from chapter 2 that the conformal block in the Euclidean case are oscillatory functions, then are not so easy to control the asymptotic behavior. Besides in the Lorentzian frame the blocks appear very controlled, due to the presence of scaling blocks that are power-law functions. Since we have sum of positive terms which stays nonsingular for all ρ, ¯ρ < 1, it immediately follows that the ρ-series is absolutely convergent in the product of two discs |ρ| < 1 and | ¯ρ| < 1.

11_{We can always put the points in this way, acting with a global conformal tranformation to the}

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(a) (b)

Figure 3.2: In (a) we have the (ρ, ¯ρ) plane. We can always put 4 points x1, x2, x3 and x4 in

this way by using a global conformal tranfromation from the conformal frame. The pairs x4 − x1 and x2 − x3 are timelike separated for ¯ρ > 1. In (b) we have the corresponding

configuration of ρ and ¯ρ in the complex plane. Notice the singularities at ρ, ¯ρ = ±1.

Positivity and analyticity properties of Rindler wedge correlator

Here we try to extract information about the 4-pt correlation functions, and we want to use Lorentzian kinematics where the lightcone coordinates ρ and ¯ρ are independent real variables. As we can see in fig.3.2 when ρ and ¯ρ are both small (ρ, ¯ρ < 1), all points are spacelike separated and the physics is essentially Euclidean. Instead we are interested in the case 0 < ρ < 1 < ¯ρ, where the distance between x2 − x3 and x3 − x4 both became

timelike. According to the standard Feynman’s i-prescription (x0 _{−→ x}0_{(1 − i)), ¯}_ρ

should be slightly below the cut if we are computing the time-ordered correlator12, and above the cut for its complex conjugate.

These kinematics do not lie within the radius of convergence of the s-channel sum, but within the t−channel one (x2 → x3 corresponding to ρ, ¯ρ → 1). To obtain the conformal

blocks expansion in the t-channel we have to exchange z ↔ 1 − z, which using 3.1.9 gives a non trivial transformation for ρ:

G(ρ, ¯ρ) = (z ¯z) ∆1+∆2 2 ((1 − z)(1 − ¯z))∆2+∆32 X l0_,∆0 ˜ λ23O0λ˜_14O0 1 −√ρ 1 +√ρ ∆0+l0_{1 −}√ ¯ ρ 1 +√ρ¯ ∆0−l0 . (3.1.11)

To simplify the argument (and with no loss of generality) we will not use the full conformal blocks but rather just power-laws, that is we include both primaries and descendants independently.

Now if we pass to the Lorentzian region (0 < ρ < 1 < ¯ρ), using the i-prescription, the scaling blocks acquire a phase:

G(ρ, ¯ρ) = u ∆1+∆2 2 v∆2+∆32 X l0_,∆0 ˜ λ23O0λ˜_14O0 1 −√ρ 1 +√ρ ∆0+l0 1 − √1 ¯ ρ 1 + √1 ¯ ρ ∆0−l0 eiπ(∆0−l0−∆2−∆3)_, _(3.1.12)

where u, v are the usual cross-ratios. So we obtain the following:

|G(ρ, ¯_{ρ)| 6 G(ρ,}1 ¯

ρ) ≡ GEucl(ρ, ¯ρ), (0 < ρ < 1 < ¯ρ). (3.1.13)

12_{We have to use the standard Feynman’s i-prescription that indicate the path to compute the integral}

Tree-level spectrum of supergravity on AdS3 × S3 from Lorentzian inversion formula

Dipartimento di Fisica “Enrico Fermi”

Corso di Laurea Magistrale in Fisica