Conformal field theory at large N

(1)

Uppsala University

Master thesis in physics 30 HP

Department of physics and astronomy Division of theoretical physics

Conformal field theory at large N

Abstract

The conformal bootstrap method is a non-perturbative method that uses the symmetry in a conformal field theory to constrain and solve for the observables in the theory. The four-point correlation function of a quartic interaction of four general scalar fields in a conformal field theory can be written as a sum over primary operators. In order to study the four-point correlator a large-N expansion is made, where N comes from the symmetry group SU(N ). Using the conformal bootstrap method the anomalous dimension of the primary operators in the four-point correlator is calculated. Using the AdS/CFT correspondence the anomalous dimension of the primary operators is also calculated using Witten diagrams.

Author:

Nadia Flodgren

Supervisors:

Associate Prof. Agnese Bissi Giulia Fardelli Subject reader:

Assistant Prof. Marco Chiodaroli September 24, 2019

(2)

Sammanfattning

Konform fältteori är en kvantfältteori med konform symmetri. Kon- form symmetri är en symmetri som bevarar vinklar och lokalt ser ut som en kombination av en rotation och en förändring i skala. En metod för att beräkna de observerbara kvantiteterna i en konform fältteori är metoden "conformal boostrap". Denna metod går ut på att använda symmetrin i en konform fältteori för att begränsa och beräkna värdet på de observerbara kvantiteterna i teorin. En av de observerbar kvantiteterna i en fältteori är en korrelationsfunktion. Korrelationsfunk- tioner beskriver interaktionerna mellan partiklarna i en fältteori.

I detta arbete studerar vi en interaktion mellan fyra skalärfält genom att studera fyra-punkts korrelationsfunktionen för denna interaktion. Metoden vi använder är "conformal bootstrap" men vi testar också om AdS/CFT dualiteten håller för våra beräkningar. AdS/CFT dualiteten är en ekvivalens av två olika teorier, en strängteori i ett (d + 1)-dimensionellt anti-de Sitter (AdS) rum och en konform fält- teori (CFT) i den d-dimensionella gränsen av anti-de Sitter rummet.

Enligt denna dualitet kan en observerbar kvantitet beräknas från båda dessa två teorier och ge samma resultat. Teorin vi studerar har sym- metrigrupp SU(N ) och vi arbetar i dimension två. Vi arbetar också med att N , matrisrangen i teorin, är stort då detta är den gräns där AdS/CFT dualiteten gäller.

Enligt konform fältteori så kan en fyra-punkts korrelationsfunktion av fyra skalärer beskrivas som en summa över vad som kallas primära fält. Genom att använda "conformal bootstrap" metoden beräknas den anormala dimensionen, vilket är en korrektion av första icke-triviala ordning till dimensionen, av dessa primära fält. Samma kvantitet beräknas också från strängteorisidan av AdS/CFT dualiteten genom användandet av så kallade Witten diagram. Resultatet från båda sidor av dualiteten visas stämma överens.

(3)

1 Introduction

Quantum field theory (QFT) attempts to describe the physics of elementary particles by combining classical field theory with quantum mechanics and special relativity. An example of a quantum field theory is the standard model which describes three of the four fundamental forces, with the exception of gravity, and their elementary particles. In quantum field theory fields are the most fundamental objects and particles are described as excitations of these fields. An observable in QFT is the n-point correlation function. These functions describe interactions between n fields located at n points and correspond to the physical amplitude for the propagation of excitations. Interactions of fields in QFT can be calculated using a mathematical tool called Feynman diagrams. Using Feynman diagrams is a perturbative method and thus only works for weakly coupled systems.

Symmetry is a powerful property that is found in most physical systems.

It is the foundation of many physical theories, such as the standard model, and can be a useful tool when calculating observables since it provides constraints that need to be fulfilled. Quantum field theories are constrained by their symmetries. For example, the standard model is Poincaré invariant which means that it is invariant under translations and Lorentz transformations, which are the transformations of special relativity. However, the symmetry of interest in this thesis is conformal symmetry. In the large distance limit many quantum field theories are scale invariant but scale invariance is just one of the symmetries of conformal symmetry.

Conformal symmetry is a symmetry that preserves angles and it includes scale invariance. There are several theories that are conformally invariant for specific values of the parameters of the theory. For example, φ⁴-theory in three dimensions is a QFT that is not generally conformally invariant but for a certain value of its coupling constant to mass ratio the theory becomes conformally invariant. Similarly, the Ising model in three dimensions is also a CFT for a specific value of its coupling constant [1]. Furthermore, another system that becomes conformally invariant at the critical point in its phase diagram is water. Not only are all of these theories conformally invariant at these specific points but they are all described by the same CFT there. In fact, there are many theories that in their critical points can be described by the same conformal field theory. Note that this does not mean that the systems can be described by the same Lagrangian or Hamiltonian but rather that they behave similarly and that the microscopic details of the theories do not matter at the critical point. The fact that this equivalence of theories at critical points occurs for many different theories is called critical universality.

Critical universality implies that we can study the behaviour of one system

(5)

at its critical point and simultaneously learn about the other systems at their critical points.

The conformal bootstrap method is a method that takes advantage of conformal symmetry to solve a CFT. This method uses the conformal symmetry of the theory to calculate the observables of that theory. When we use the bootstrap method we study the CFT and not the full microscopic theory of whichever system we are interested in. When studying the CFT we look for all the ways conformal symmetry constrains the theory and then use these constraints to calculate the observables. This method is non-perturbative and thus might work for theories with strong couplings.

AdS/CFT correspondence is a duality between two theories, one is a string theory containing gravity which exists in anti-de Sitter (AdS) space and the other one is a conformal field theory that exists in the boundary of the AdS space [2]. The CFT thus lives in the d dimensional space which is the boundary of the d + 1 dimensional AdS space. The duality implies that there are two ways to calculate any observable. For example, we can calculate correlation functions from the boundary perspective using conformal field theory or from the bulk perspective using string theory and these calculations should yield the same observable. However, what makes this duality extremely useful is that it is a strong-weak duality. This enables us to perform calculation in a strongly coupled theory by doing the equivalent calculations from the weakly coupled side of the duality. For the purposes of this thesis, the bulk calculations we perform only require that we use Witten diagrams [3], which are similar to the Feynman diagrams of QFT.

In this thesis we will study the AdS/CFT correspondence, in particular we will study the four-point correlation function of a quartic interaction of four scalar fields from both sides of the correspondence. From the boundary perspective we will use the conformal bootstrap method to find the correlator.

In the bulk the same correlator will be studied using Witten diagrams. We will choose to work with the symmetry group SU(N ) where N is the rank of the matrices. Furthermore, we will work with large-N since the large-N region is where the AdS/CFT correspondence is valid and using a large N also simplifies our calculations. The object is to understand and be able to apply the conformal bootstrap method. We want to study the four-point correlation function of a quartic interaction of four general scalar fields. However, we will not be able to complete the full four-point function but will instead make an expansion in N . This is the large-N expansion and it will allow us to calculate the four-point function of this theory, up to order 1/N². When applying the bootstrap method we find that the four-point correlator we are looking to obtain can be written in terms of a sum over a specific type of operators. The main quantity we want to compute in this thesis is the

(6)

anomalous dimension of these operators, up to leading non-trivial order in the large-N expansion. These calculations will be performed for a conformal field theory in spacetime dimension two. However, it is also possible use the same exact method to perform these calculations in the case of spacetime dimension four. We will repeat calculations performed by Heemskerk et al.

in [4] and apply them to a more general case.

In the article [4] the authors calculate the four-point correlation function, up to first non-trivial order in the large-N expansion, for an interaction of four identical scalar fields from both sides of the AdS/CFT correspondence.

From the CFT perspective the authors use the bootstrap method and on the AdS side of the duality they use Witten diagrams. For this thesis we will study a more general interaction than the one in [4] where the quartic interaction is of four identical fields. We are interested in a quartic interaction of four general scalar fields that are not necessarily identical.

However, before calculating the observables we are interested in we will review what a conformal field theory is. In section 2 we discuss the basics and most important concepts of a conformal field theory and the conformal bootstrap method, this includes the operator product expansion and conformal block decomposition. In this section we follow the very useful article [5].

In section 3 we discuss the article [4] and describe the calculations of the four-point correlation function of a theory with a quartic interaction of four identical scalar fields. Then we move on calculating the four-point correlator of a quartic interaction four general scalar fields in section 4. The result is an expression for the anomalous dimension of the operators in the sum in the four-point function. The final step is to calculate the four-point correlation function from the bulk point of view using Witten diagrams, which is done in section 4.3. In section 5 we discuss the results and mention further applications of the calculations.

2 Background

2.1 Conformal symmetry

Conformal transformations are all transformations that change the metric with a rescaling

δg_µν = f (x)g_µν, (1)

here f (x) is a scalar function. There are several infinitesimal coordinate transformations x^0µ = x^µ+ ^µ(x) that can achieve this but they have to fulfill

∂_µ_ν+ ∂_µ_µ= c(x)g_µν (2)

(7)

where c(x) is a scalar function. The above equation (2) is called the conformal Killing equation. For spacetime dimensions larger than or equal to three there are four different infinitesimal transformations that fulfill eq. (2). They are translations, rotations, dilatations and special conformal transformations, listen in the same order below they are

p_µ = ∂_µ

m_µν = x_ν∂_µ− x_µ∂_ν d = x^µ∂_µ

k_µ= 2x_µ(x^ν∂_ν) − x^νx_ν∂_µ.

(3)

Translations and rotations are movement in a system while dilatations are changes in scale. Special conformal transformations consist of one inversion followed by a translation and then another inversion. In spacetime dimension two eq. (2) has infinitely many solutions and a two-dimensional CFT can be described by the Virasoro algebra. If we exponentiate the infinitesimal transformations we find the finite transformations

x^0µ = x^µ+ a^µ x^0µ = M_ν^µx^ν x^0µ = αx^µ

x^0µ = x^µ− b^µx² 1 − 2(b · x) + a²x²

(4)

where a^µ and b^µ are constants. The generators for the four transformations in (4) are, in the order stated above,

P_µ= i∂_µ

M_µν = i(x_µ∂_ν − x_ν∂_µ) D = ix^µ∂µ

K_µ = i(2x_µ(x^ν∂_ν) − x²∂_µ)

(5)

where we are using a Euclidean signature on a fixed background metric.

These are the generators of the conformal algebra. The generator D acting on an operator at the origin gives the dimension ∆ of that operator

[D, φ(0)] = i∆φ(0) (6)

and away from the origin it is

[D, φ(x)] = (ix^µ∂_µ+ i∆)φ(x). (7)

(8)

The momentum operators P_µ and K_µ act as raising and lowering operators for the dimensions, which we can see from the commutation relations

[D, Pµ] = iPµ

[D, K_µ] = −iK_µ. (8)

Let us apply P_µ to an operator at the origin with dimension ∆ and observe how it raises the dimension, using the commutation relations (8),

DP_µφ(0) = ([D, P_µ] + P_µD)φ(0)

= (iP_µ+ P_µD)φ(0) = (1 + ∆)iP_µφ(0). (9) Acting with Mµν on an operator gives the spin-tensor

[M_µν, φ^a(0)] = i(S_µν)_b^aφ^b(0). (10) Mµν generates the SO(d) algebra and a, b are spin indices for the SO(d) representation of φ. The conformal algebra is isomorphic to SO(d + 1, 1). We can use the generators to classify operators as what are called primary and descendant operators. A primary operator is an operator whose dimension cannot be lowered further, i.e. it fulfills

[K_µ, φ(0)] = 0, (11)

which all physical theories require, since dimension is bounded from below in these theories. Descendant operators are obtained by acting on the primary operator with the raising operator P_µ repeatedly

φ(0) → P_µ₁· · · P_µ_nφ(0)

∆ → ∆ + n. (12)

The primary operator and all of its descendant operators form a collection called a conformal multiplet.

Conformal symmetry fixes both two- and three-point functions of any field up to constants. For example, two-point functions of primary scalars are given by the expression

hφ₁(x₁)φ₂(x₂)i = Cδ_∆₁_,∆₂

|x₁− x₂|^∆¹^+∆² (13) where C is a normalization constant. We can see why this is the case by first noticing that since we want the correlator to be translation and rotation invariant it must be a function of the distance between the operators, i.e.

(9)

hφ₁(x₁)φ₂(x₂)i = f (|x₁ − x₂|). Next, if we apply the dilatation operator, given by (7), to φ₁(x₁)φ₂(x₂) we get

h0| [D, φ₁(x₁)φ₂(x₂)] |0i = h0| [D, φ₁(x₁)]φ₂(x₂) + φ₁(x₁)[D, φ₂(x₂)] |0i

= (ix^µ₁∂_µ+ i∆₁ + ix^µ₂∂_µ+ i∆₂) h0| φ₁(x₁)φ₂(x₂) |0i (15) where we used (7). Combining the two equations (14) and (15) we want to find a solution to

(x^µ₁∂_µ+ ∆₁+ x^µ₂∂_µ+ ∆₂)hφ₁(x₁)φ₂(x₂)i = 0. (16) Eq. (13) solves (16) and therefore two-point correlation functions of primary scalars have a fixed form¹. Similar calculations show that three-point functions of primary scalars are fixed up to a constant, f₁₂₃,

hφ₁(x₁)φ₂(x₂)φ₃(x₃)i = f₁₂₃

x^∆₁₂¹^+∆²^−∆³x^∆₂₃²^+∆³^−∆¹x^∆₃₁³^+∆¹^−∆². (17) In general correlation functions of primary scalars must also fulfill the Ward identity

hφ1(x1)φ2(x2) . . . φn(xn)i = Ω(x⁰₁)^∆¹. . . Ω(x⁰_n)^∆ⁿhφ1(x⁰₁)φ2(x⁰₂) . . . φn(x⁰_n)i (18) which contributes to determining the forms of the two-, three- and four-point correlation functions. Ω(xi) is a rescaling factor for the metric

g_µν∂x^0µ

∂x^α

∂x^0ν

∂x^β = Ω(x)²g_αβ, (19)

and all transformations that rescale the metic in this way are conformal, eq.

(19) is the same as (1).

1Note that the Kronecker delta in (13) does not comes from this equation. It comes from applying the Ward identity (18).

(10)

2.2 Operator product expansion

In quantum mechanics a unitary theory implies that the time evolution operator for a quantum state is a unitary operator. Let us make a physical example of what unitarity implies. For example, if we are calculating the probability of finding a particle in a volume and we choose that volume to be the entire space the particle is in, then the probability is always equal to one.

In a unitary CFT all local operators can be written as linear combinations of primary and descendant operators. For example, if we have two operators inside a sphere and perform a path integral over the interior we will get a state that can be decomposed as seen below

φ_i(x₁)φ_j(x₂) =X

k

C_ijk(x_12,∂₂)φ_k(x₂) (20)

where the φ_k’s are primary operators and the C_ijk(x₁₂, ∂₂)’s are differential operators that create descendant operators from the primary operator φk. Eq. (20) is called the operator product expansion (OPE). The operator product expansion is valid in correlation functions where the distances between the rest of the operators are greater than the distance between φi(x1) and φ_j(x₂). The slightly more general version of the OPE is

φ_i(x₁)φ_j(x₂) = X

k

C_ijka(x_12,∂₂)φ^a_k(x₂) (21)

where a is the spin index of the primary operator in the sum.

It can be shown that the operators C_ijk are determined only by the dimensions ∆_i, spins and OPE coefficients f_ijk, where the coefficient f_ijkcomes from the 3-point function. We can see why the the coefficient f_ijk appears in C_ijk by studying the three-point function. The three-point function is given by (17) but we can also take the OPE (20) with a third operator to obtain

hφ_i(x₁)φ_j(x₂)φ_k(x₃)i =X

l

C_ijl(x₁₂, ∂₂)hφ_l(x₂)φ_k(x₃)i. (22)

The expressions (17) and (22) are equal and if we input the two-point function (13) we obtain

f_ijk

x^∆₁₂ⁱ^+∆^j^−∆^kx^∆₂₃^j^+∆^k^−∆ⁱx^∆₃₁^k^+∆ⁱ^−∆^j = C_ijk(x₁₂, ∂₂)

(x₂− x₃)^2∆^k (23) which shows that C_ijk is proportional to f_jik. In eq. (20) we always have primary operators, both the φi, φ_j and φk are primary operators.

(11)

Using the OPE we can rewrite an n-point function as a sum over (n − 1)- point functions. Therefore it is possible to apply the OPE until the n-point function has been reduced to an expression in terms of two-point functions.

For example, applying the OPE to a four-point function by contracting the four-fields in pairs of two we obtain a two-point function. Thus any correlator can be determined in terms of only the dimensions and OPE coefficients, what is generally called the CFT data.

Although they are not completely fixed, four-point functions of scalars are constrained enough to have a certain form. A four-point function of four identical primary scalars φ with dimensions ∆ has the form [5]

hφ(x₁)φ(x₂)φ(x₃)φ(x₄)i = g(u, v)

x^2∆₁₂x^2∆₃₄ , (24) where u and v are the conformally invariant cross-ratios

u = x²₁₂x²₃₄

x²₁₃x²₂₄ v = x²₂₃x²₁₄

x²₁₃x²₂₄ (25) and x_ij = x_i − x_j. The coordinates x_i live in d-dimensional boundary of the AdS space, this space is locally flat Minkowski spacetime. The conformal symmetry constrains the function g so that it only depends on these variables.

Similarly, a four-point function of four primary scalars with dimensions

∆_i, i = 1, 2, 3, 4, is of this form [6], [7]

hφ₁(x₁)φ₂(x₂)φ₃(x₃)φ₄(x₄)i = 1 x^∆₁₂¹^+∆²x^∆₃₄³^+∆⁴

x₂₄ x₁₄

∆12 x₁₄ x₁₃

∆34

F₁₂₃₄(u, v) (26) where F₁₂₃₄(u, v) is a function only dependent on the conformally invariant cross-ratios and ∆_ij = ∆_i− ∆_j.

2.3 Crossing conditions

Crossing conditions are constraints on the n-point functions that we get by interchanging the positions of the operators in the correlation function. What we are doing is changing the pairs of operators that we apply the OPE to.

n-point correlation functions are invariant under these exchanges. All n- point functions should be equal regardless of which pairs of operators that are contracted with each other, see Figure 1. For example, the following two equations

hφ(x₁)φ(x₂)φ(x₃)φ(x₄)i = hφ(x₂)φ(x₁)φ(x₃)φ(x₄)i

hφ(x₁)φ(x₂)φ(x₃)φ(x₄)i = hφ(x₃)φ(x₂)φ(x₁)φ(x₄)i (27)

(12)

Figure 1:

are crossing conditions we obtain when interchanging x₁ ↔ x₂ and x₁ ↔ x₃ respectively for four identical fields. Making the change x1 ↔ x2is equivalent to x₃ ↔ x₄, and the change x₁ ↔ x₃ is equivalently x₂ ↔ x₄, which we can see from the form of (24). Inputting eq. (24) into the two crossing conditions above we find the crossing conditions on the four-point functions of four identical scalars. In terms of the function g(u, v) the two distinct conditions we obtain are

g(u, v) = g u v,1

v

(from interchanging x₁ ↔ x₂) (28)

g(u, v) = u v

∆

g(v, u) (from interchanging x₁ ↔ x₃). (29) To find the crossing conditions for the case of four general scalar fields we take eq. (26) and make the swaps; x₁ ↔ x₂, ∆₁ ↔ ∆₂ and x₁ ↔ x₃,

∆₁ ↔ ∆₂. It is necessary to change the dimensions like this ∆_i ↔ ∆_j since here we are considering four general fields and what we are interchanging are the fields and not simply the positions. The results are the following two crossing conditions on F₁₂₃₄(u, v)

F₁₂₃₄(u, v) = 1

v^∆³⁴^/2F₂₁₃₄(u/v, 1/v) = 1

v^∆³⁴^/2F₁₂₄₃(u/v, 1/v) F1234(u, v) = u¹²^(∆¹^+∆²⁾

v¹²^(∆²^+∆³⁾F3214(v, u) = u¹²^(∆¹^+∆²⁾

v¹²^(∆²^+∆³⁾F1432(v, u)

(30)

respectively. These types of conditions are very important and the main constraints we will be focused on solving.

(13)

Note, from eq. (25), that when we make the interchanges x₁ ↔ x₂ and x₁ ↔ x₃ they are equivalent to

(u, v) ↔ u v,1

v

and (u, v) ↔ (v, u) (31)

respectively. Furthermore, due to conformal invariance it is possible to place the operators in a plane at the positions x₁ = (0, . . . , 0), x₂ = (x, y, 0, . . . , 0), x₃ = (1, 0, . . . , 0), x4 → ∞ without loss of generality. Let us define z = x + iy and then the variables u, v become

u = z ¯z and v = (1 − z)(1 − ¯z). (32) Thus, in terms of z, ¯z the interchanges x₁ ↔ x₂ and x₁ ↔ x₃ become

(z, ¯z) ↔

z

z − 1, z¯

¯ z − 1

and (z, ¯z) ↔ (1 − z, 1 − ¯z) (33) respectively. Here z is a complex variable with conjugate ¯z but it is possible to through analytic continuation make z and ¯z into two independent variables.

For the rest of the calculations z, ¯z will be two independent variables.

2.4 Conformal block decomposition

The next important concept of conformal field theory is the so called conformal block decomposition. As we have seen we can describe a four-point function of identical scalars with the equation (24). However, we can also use the OPE to rewrite it in terms of a two-point function. Applying the OPE (21) twice we obtain

hφ(x₁)φ(x₂)φ(x₃)φ(x₄)i = X

φ_k,φ_l

f_φφφ_kf_φφφ_lC_a(x₁₂, ∂₂)C_b(x₃₄, ∂₄) hφ^a_k(x₂)φ^b_l(x₄)i.

(34) where we moved out the coefficients f_ijk from the C_ijk. The fields φ_k and φ_l have dimensions ∆_k, ∆_l respectively. The two-point functions for operators with spin is

hφ^a_k(x₂)φ^b_l(x₄)i = δ_∆_k_,∆_lI^ab(x₂₄)

x^∆₂₄^k^+∆^l (35)

where I^ab is an orthogonal tensor given by [8]

Iâb = δâb− 2xâx^b

x² . (36)

(14)

Eq. (34) is then

hφ(x₁)φ(x₂)φ(x₃)φ(x₄)i = X

φk

f_φφφ²

kC_a(x₁₂, ∂₂)C_b(x₃₄, ∂₄)I^ab(x₂₄) x^2∆₂₄^φk

. (37) The operators φ_k are the primary operators, with spin l_φ_k and dimensions

∆_φ_k, that we sum over in the OPE. By equating the two expressions (24) and (37) we get the following decomposition of g(u, v)

g(u, v) =X

φk

f_φφφ²

kg_∆_φk_,l_φk(u, v) (38) where the functions g_∆_φk_,l_φk(u, v) are called conformal blocks, they are given by

g∆_φk,l_φk(xi) = x^2∆₁₂x^2∆₃₄Ca(x12, ∂2)Cb(x34, ∂4)I^ab(x₂₄) x^2∆₂₄^φk

. (39)

Equation (38) is known as the conformal block decomposition, or the partial wave expansion. The constant f_φφφ_k is the OPE coefficient from the three- point correlation function which we have moved out from the differential operator C_φφφ_k.

Following the same procedure we find that for four general scalar fields the expression is the very similar but with two different OPE coefficients and new conformal blocks in the sum. The conformal block decomposition is in this case

F₁₂₃₄(u, v) = X

φ_k

f_φ₁_φ₂_φ_kf_φ₃_φ₄_φ_kg_∆_φk_,l_φk(u, v) (40)

with the conformal blocks

g∆_φk,l_φk(xi) = x^∆₁₂¹^+∆²x^∆₃₄³^+∆⁴ x₁₄ x₂₄

∆12 x₁₃ x₁₄

∆34

Ca(x12, ∂2)Cb(x34, ∂4)I^ab(x₂₄) x^2∆₂₄^φk

. (41) The expressions for the conformal blocks are known for spacetime dimensions two, four and six. The formulas for the conformal blocks in dimensions two and four are found in [9], [10] and [7]. For spacetime dimension d = 2 the conformal blocks, for four general scalar fields, are given by

g^∆_∆¹²^,∆³⁴

φk,l_φk(z, ¯z) = (−1)^l(z ¯z)¹²^(∆^φk^−l^φk⁾

z^l^φkF ∆_φ_k − ∆₁₂+ l_φ_k

2 ,∆_φ_k+ ∆₃₄+ l_φ_k

2 , ∆_φ_k+ l_φ_k, z

F ∆_φ_k − ∆₁₂− l_φ_k

2 ,∆_φ_k+ ∆₃₄− l_φ_k

2 , ∆_φ_k − l_φ_k, ¯z

+ (z ↔ ¯z)

.

(42)

(15)

The functions F (a, b, c, z) are the hypergeometric functions₂F₁(a, b, c, z) and z, ¯z are real variables that fulfill 0 ≤ z, ¯z ≤ 1.

Now we have all the necessary concepts and constraints that we will need when calculating the four-point function due to the quartic interaction of four scalar fields.

2.5 Conformal bootstrap method

As previously mentioned, in section 1, the conformal bootstrap method is about using the symmetry of a CFT to calculate the observables of the theory.

The first step in the bootstrap method is to choose to study a general CFT.

This is a CFT that describes several physical system at their critical points, for example the three dimensional Ising CFT. Therefore one does not need to consider one specific microscopic system, for example the three dimension Ising lattice.

The next step in the bootstrap method is to find out which constraints we get from conformal symmetry. As we have seen in the Background sec- tions conformal symmetry constrains the forms of both two- and three-point correlation functions of any operators and additional constraints come from the OPE and crossing conditions. After we have found all the constraints we apply them to the theory and finally try to solve the theory for its observables.

The bootstrap method does not need a Lagrangian and is not a perturbative method.

(16)

3 Problem and method

3.1 Presentation of the problem

The main object of this thesis is to calculate the anomalous dimension of the primary operators in the OPE of the the four-point function due to a quartic interaction of four general scalar fields φ_i with dimensions ∆_i, i = 1, 2, 3, 4.

We work in a CFT with the symmetry group SU(N ) and we assume Z² symmetry for the scalars. The Z2symmetry implies that φ_i → −φ_iwhich removes cubic interactions of the scalar fields. To calculate the anomalous dimension we need to calculate the four-point function up to the first non-trivial order in the large-N expansion, which we will do using the conformal bootstrap method. The corresponding four-point amplitude on the AdS side will also be calculated. The four-point functions on both sides of the AdS/CFT correspondence will then be compared and are expected to agree, i.e. give the same expressions for the anomalous dimensions. The anomalous dimension can tell us the number of solutions to the crossing condition we use in the calculations.

We will perform calculations similarly to the ones in [4] by Heemskerk et al. In [4] the CFT four-point function, for a quartic interaction of four identical scalar fields, is calculated up to order 1/N²in the large-N expansion using the conformal bootstrap method and then the the four-point amplitude from the bulk perspective is also calculated. They make sure that the results on both sides agree and then count the number of solutions on the CFT side.

However, we consider four general scalar fields instead of four identical ones. The problem is to apply the bootstrap method to the four-point function of our quartic interaction. In order to check if our results are correct we can set all the fields φ_i equal to each other and see if we recover the anomalous dimension calculated for four identical fields.

3.2 Method

In this section the calculation of the anomalous dimension of the primary operators in the four-point correlation functions due to a quartic interaction of four identical scalar fields in spacetime dimension two in [4] is reviewed.

The authors of [4] start with a restricted case that is not a full CFT and make the assumption that there is a restricted set of low-dimensional operators. Then one chooses to study a simple theory where the only low- dimensional single-trace operator is a scalar φ with dimension ∆ and Z2

symmetry. The reason one includes Z² symmetry is because this means the operator φ does not show up in the OPE of hφ(x₁)φ(x₂)φ(x₃)φ(x₄)i. Without

(17)

the Z2 symmetry the calculations would be more complicated.

In this simple theory, the operators with the lowest dimensions are the unit operator followed by φ² from which you can obtain all other operators

O_n,l = φ

↔

∂_µ₁. . .

↔

∂_µ_l(

↔

∂_ν

↔

∂^ν)ⁿφ − traces, (43) where n, l are integers and the defining properties of the operator. The operator O_n,l is primary and has spin l and dimension ∆(n, l) = 2∆ + 2n + l+O(1/N²). The reason only double-trace operators are considered is because when the large-N expansion is made it is up to order 1/N². To this order only double-trace operators will contribute to the OPE due to the Z2 symmetry.

If the expansion was made to a higher order one would have to include other kinds of operators than double-trace.

3.2.1 Conformal constraints on the four-point function

The next step is to use all of the constraints from conformal symmetry on the four-point function of the scalar field φ. First one places the operators in a plane at the positions x₁ = 0, x₂ = z, ¯z, x₃ = 1, x₄ = ∞, where z, ¯z are real independent coordinates fulfilling 0 ≤ z, ¯z ≤ 1. The four-point function is now given by combining equations (24) and (38)

hφ(0)φ(z, ¯z)φ(1)φ(∞)i ≡ A(z, ¯z)

= g(u, v) x^2∆₁₂x^2∆₃₄

= 1

x^2∆₁₂x^2∆₃₄ X

O_n,l

f_φφO²

n,lg_∆(n,l),l(u, v)

= 1

(z ¯z)^∆ +

∞

X

n=0

∞

X

l=0

p(n, l)g_∆(n,l),l(z, ¯z) (z ¯z)^∆

(44)

where we separated out the identity operator in the last row. Furthermore, p(n, l) = f_φφO²

n,l is the OPE coefficient squared and g_∆(n,l),l(z, ¯z) are the conformal blocks. In the final row, instead of summing over the primary operators O_n,l the sum is over their quantum numbers n, l. Since the four fields are identical l must be even due to Bose symmetry.

To find equation (44) one has so far used conformal symmetry in the form of the OPE and conformal block decomposition. The next step is to solve the conditions that remain and those are the crossing conditions. One can take one of the crossing conditions, here the condition (29) is used, and write

(18)

it in terms of the four-point function A(z, ¯z). First of all, we have from (44) A(z, ¯z) = g(u, v)

(z ¯z)^∆

⇒ g(u, v) = (z ¯z)^∆A(z, ¯z)

(45)

which when input into (29) gives (z ¯z)^∆A(z, ¯z) = (z ¯z)^∆

((1 − z)(1 − ¯z))^∆((1 − z)(1 − ¯z))^∆A(1 − z, 1 − ¯z). (46) where we used eq. (32) and (33). The crossing condition obtained from interchanging x1 and x3 is therefore

A(z, ¯z) = A(1 − z, 1 − ¯z). (47) The following calculations will show that it is possible to compute the anomalous dimension using only this crossing condition. That is, both conditions (28) and (29) are not needed to calculate the anomalous dimension of the primary operators. However, using the condition (28) should yield the same result for the anomalous dimension.

3.2.2 Solving the constraints

In order to solve the crossing condition one has to expand A(z, ¯z) in a large-N expansion up to order 1/N²

A(z, ¯z) = A₀(z, ¯z) + 1

N²A₁(z, ¯z) . . . p(n, l) = p₀(n, l) + 1

N²p₁(n, l) + . . .

∆(n, l) = 2∆ + 2n + l + 1

N²γ(n, l) + . . . .

(48)

N comes from the symmetry group SU(N ) and is the rank of the matrices.

The goal is to solve the condition up to the order 1/N² and the unknown quantities are the OPE coefficients squared p₀(n, l), p₁(n, l) and the anomalous dimension γ(n, l). However, we do calculate p0(n, l) explicitly and find γ(n, l) explicitly for (n, l) = (n, 0). How to find p₁(n, l) will be explained but p₁(n, l) will not be explicitly computed. The four-point function can now be written as two equations, one for each order in the expansion,

O(1) (z ¯z)^∆A₀(z, ¯z) = 1 +

∞

X

n=0

∞

X

evenl=0

p₀(n, l)g_2∆+2n+l,l(z, ¯z) (49)

(19)

O(1/N²) (z ¯z)^∆A₁(z, ¯z) =

∞

X

n=0

∞

X

l=0 even

p₁(n, l)g_2∆+2n+l,l(z, ¯z) + p₀(n, l)γ(n, l)1 2

∂

∂ng_2∆+2n+l,l(z, ¯z).

(50) Using the first of these two equations one is able to compute p₀(n, l) by first calculating A₀(z, ¯z) using Wick contractions, it is

A₀(z, ¯z) = 1 + 1

(z ¯z)^∆ + 1

[(1 − z)(1 − ¯z)]^∆, (51) and then Taylor-expanding both sides of (49). The result is

p₀(n, l) = [1 + (−1)^l]C_nC_n+l (52) where

Cn = Γ²(∆ + n)Γ(2∆ + n − 1)

n!Γ²(∆)Γ(2∆ + 2n − 1) . (53) When p₀(n, l) is known finding γ(n, l) and p₁(n, l) becomes possible. To compute γ(n, l) the one takes the expression for A₁(z, ¯z) from eq. (50) and inputs it into the crossing condition (47) as follows

∞

X

n=0

∞

X

evenl=0

p₁(n, l) + p₀(n, l)γ(n, l)1 2

∂

∂n

(z ¯z)ⁿ[z^lF (∆ + n + l, ∆ + n + l, 2∆ + 2n + 2l, z)

∗ F (∆ + n, ∆ + n, 2∆ + 2n, ¯z) + (z ↔ ¯z)]

=

∞

X

n=0

∞

X

evenl=0

p₁(n, l) + p₀(n, l)γ(n, l)1 2

∂

∂n

((1 − z)(1 − ¯z))ⁿ[(1 − z)^l

F (∆ + n + l, ∆ + n + l, 2∆ + 2n + 2l, 1 − z)F (∆ + n, ∆ + n, 2∆ + 2n, 1 − ¯z) + (z ↔ ¯z)], (54)

where the blocks were expanded. In this step it is necessary to introduce an upper bound to the spin. One way to see this is by observing the left hand side of (54), taking z to be small and noticing that this regulates both sums in the left hand side for the first term but not the second term, since it only has zⁿ and not z^l. To regulate the second term we set the upper bound of the spin to be L.

Next step in solving this version of the condition involves using properties of the conformal blocks. The conformal blocks in dimensions two and four contain hypergeometric functions 2F₁(a, b, c, z) and these functions have

(20)

many useful identities, see section A.1. For example, it is possible to write the hypergeometric function F (a, b, c, z) as two parts. One part is singular at z = 1 and the other one is holomorphic at z = 1. F (a, b, c, z) can be written as

F (a, b, a + b, z) = ˜F (a, b, 1, 1 − z) ln(1 − z)

+ holomorphic (at z = 1) part (55) where

F (a, b, 1, z) = −˜ Γ(a + b)

Γ(a)Γ(b)F (a, b, 1, z). (56) As we can see the first term in (55) has a logarithm ln(1 − z) and this will prove useful. It is possible to take the crossing condition (54) and expand the hypergeometric functions according to (55). This implies that the condition can be separated into several conditions because of the different factors in front of the terms in the condition. For example, all the terms with a factor ln(1 − ¯z) ln(z) will form one condition. By only taking the terms in the crossing condition that have factors of ln(1 − ¯z) ln(z) the condition can be simplified to the point where it only contains the unknown γ(n, l). Therefore, this is the condition one wants to find. In eq. (54) terms with ln(1 − ¯z) are found by expanding F (∆ + n, ∆ + n, 2∆ + 2n, ¯z) on the left hand side (LHS) and performing the differentiation

∂

∂n(1 − ¯z)ⁿ = (1 − ¯z)ⁿln(1 − ¯z) (57) on the right hand side (RHS). The condition from all terms with a factor of ln(1 − ¯z) is

∞

X

n=0 L

X

evenl=0

p₁(n, l) + p₀(n, l)γ(n, l)1 2

∂

∂n

[z^n+lz¯ⁿF (∆ + n + l, ∆ + n + l, 2∆ + 2n + 2l, z) F (∆ + n, ∆ + n, 1, 1 − ¯˜ z) + (n ↔ n + l)]

=

∞

X

n=0 L

X

evenl=0

p₀(n, l)γ(n, l)1

2[(1 − z)^n+l(1 − ¯z)ⁿ

F (∆ + n + l, ∆ + n + l, 2∆ + 2n + 2l, 1 − z)F (∆ + n, ∆ + n, 2∆ + 2n, 1 − ¯z) + (n ↔ n + l)].

(58) Similarly, keeping only terms with a factor ln(z) in (58) one obtains the

(21)

condition

∞

X

n=0 L

X

evenl=0

p₀(n, l)γ(n, l)[z^n+lz¯ⁿF (∆ + n + l, ∆ + n + l, 2∆ + 2n + 2l, z) F (∆ + n, ∆ + n, 1, 1 − ¯˜ z) + (n ↔ n + l)]

=

∞

X

n=0 L

X

l=0 even

p₀(n, l)γ(n, l)[(1 − z)^n+l(1 − ¯z)ⁿ

F (∆ + n + l, ∆ + n + l, 1, z)F (∆ + n, ∆ + n, 2∆ + 2n, 1 − ¯˜ z) + (n ↔ n + l)].

(59) As we can see, the only unknown quantity in (59) is the anomalous dimension.

However, it can still be simplified further and the next step is to get rid of one of the sums in the crossing condition by using the orthogonality of hypergeometric functions. The expression below shows when two hypergeometric functions are orthogonal

I

C

dz

2πiz^m−p−1F (a + m, b + m, c + 2m, z)F (1 − a − p, 1 − b − p, 2 − c − 2p, z) = δ_m,p (60)

It can be used to obtain a Kronecker delta which will remove the sum over n. As an example, let us take both sides of eq. (59) and multiply them with z^−p−1F (1 − ∆ − p, 1 − ∆ − p, 2 − 2∆ − 2p, z) (61) and then integrate both sides over z. On the LHS of (59) one will obtain a Kronecker delta for each of the two terms, for example the second term will give a δ_n,p,

I

C

dz 2πi

z^n−p−1F (∆ + n, ∆ + n, 2∆ + 2n, z)F (1 − ∆ − p, 1 − ∆ − p, 2 − 2∆ − 2p, z) p₀(n, l)γ(n, l)¯z^n+lF (∆ + n + l, ∆ + n + l, 1, 1 − ¯˜ z)

= δn,pp0(n, l)γ(n, l)¯z^n+lF (∆ + n + l, ∆ + n + l, 1, 1 − ¯˜ z)

(62) To remove the sum over n on the RHS of (59) one performs a similar calculation. The resulting expression is a crossing condition with integrals on both sides. The definition

J_∆(p, q) = C_p C_q

I

C

dz 2πi

(1 − z)^p z^q+1

F (∆ + m, ∆ + m, 1, z)F (1 − ∆ − q, 1 − ∆ − q, 2 − 2∆ − 2q, z)˜ (63)

(22)

makes the condition appear in a simpler form. The final form of the crossing condition is

L

X

l=0 even

γ(p, l)J_∆(p + l, q) +

L

X

l=2 even

γ(p − l, l)J_∆(p − l, q) = (p ↔ q) (64)

To compute the integrals J_∆(p, q) one can, for example, use the residue the- orem. Setting L = 0 the resulting γ(p, 0) is

γ(p, 0) = γ(0, 0) 2∆ − 1

2∆ + 2p − 1 (65)

which can be used to solve for other γ(n, l)’s. Eq. (64) is a recursion relation and by inputting a known γ(p, q) one can compute other γ’s. This also allows one to calculate the number of solutions to the crossing condition for a given L. The details will be explained later, in section 4.2.2. The number of solutions to (64) for L is

(L + 2)(L + 4)

8 . (66)

p₁(n, l) can now be found by inserting p₀(n, l) and γ(n, l) into a previous version of the crossing condition eq. (58). The authors of [4] find, although from examples and not an analytical derivation, however the analytical proof is found in [11], that

c(n, l) = 1 2

∂

∂nγ(n, l) (67)

where

c(n, l) = 2p₁(n, l) − γ(n, l)∂_np₀(n, l)

2p₀(n, l) . (68)

The ultimate step is to calculate the amplitude on the AdS side, which is done using Witten diagrams. The amplitude turns out to be given by the reduced-D function, the notation of which is ¯D [10] [12] [13]. This calculation will be performed in detail in section 4.3.1.

4 Main results

Next, we will perform similar calculations to the ones above which we used for the quartic interaction of four identical fields but apply them to our case

(23)

where we start with four scalar fields φ_i with dimensions ∆_i, i = 1, 2, 3, 4, where the fields are not necessarily identical. In order to be able to perform the calculations we take a simplified theory where the only low-dimensional single-trace operators that we have are the scalars φ_i. To the same purpose we assume that the total set of low-dimensional operators is limited. The lowest dimensional operators are then our fields φ₁, φ₂, φ₃, φ₄, from which we can obtain all the double-trace operators.

O_n,l = φ₁

↔

∂_µ₁. . .

↔

∂_µ_l(

↔

∂_ν

↔

∂^ν)ⁿφ₂ − traces O_n,l = φ₃

↔

∂_µ₁. . .

↔

∂_µ_l(

↔

∂_ν

↔

∂^ν)ⁿφ₄ − traces

(69) where the integers n, l are the quantum numbers of the operator, with l being the spin. These are the primary operators we sum over in the OPE’s and they have the dimensions

∆(n, l) = ∆₁+ ∆₂+ 2n + l + O(1/N²)

∆(n, l) = ∆₃+ ∆₄+ 2n + l + O(1/N²). (70) Again when we make the large-N expansion the only operators that will contribute up to order 1/N² are double-trace operators, which is why we only consider these types of operators here. As we can see from eq. (70) we get the condition

∆₁ + ∆₂ = ∆₃+ ∆₄ (71)

when we apply the OPE twice on the four-point function because from this we get a Kronecker delta which implies that the primary operators we sum over have to have the same dimension ∆(n, l).

Combining equations (26) and (40) the four-point function has the general form

A(z, ¯z) = hφ₁(0)φ₂(z, ¯z)φ₃(1)φ₄(∞)i

= 1

(z ¯z)¹²^(∆¹^+∆²⁾F₁₂₃₄(u, v)

= 1

(z ¯z)¹²^(∆¹^+∆²⁾

∞

X

n

∞

X

l

p(n, l)g_∆(n,l),l^∆¹²^,∆³⁴(z, ¯z).

(72)

where p(n, l) = f_φ₁_φ₂_O_n,lf_φ₃_φ₄_O_n,l. g^∆_∆(n,l),l¹²^,∆³⁴(z, ¯z) are the conformal blocks, which are given by equation (42) for dimension two. Applying the condition (71) and inputting the expression (70) the blocks in dimension two are given by

g^∆_∆¹²^,∆³⁴

1+∆2+2n+l,l(z, ¯z) = (−1)^l(z ¯z)¹²^(∆¹^+∆²⁾⁺ⁿ[z^lF (∆₂ + n + l, ∆₃+ n + l, ∆₁+ ∆₂+ 2n + 2l, z)

∗ F (∆₂+ n, ∆₃+ n, ∆₁+ ∆₂+ 2n, ¯z) + (z ↔ ¯z)].

(73)

(24)

4.1 Crossing conditions and large-N expansion

Now that we are considering four different scalar fields we need to be more careful when applying the crossing conditions. First of all, when we choose which operators to contract with each other we need to be more deliberate.

Let us consider the four fields to consist of two pairs of identical fields, because we will find this result in the next section, and contract the first two and the last two with each other as follows

A(z, ¯z) = hφ₁(x₁)φ₂(x₂)φ₁(x₃)φ₂(x₄)i. (74) Then we interchange x₁ ↔ x₃, which is equivalent to (z, ¯z) ↔ ((1−z), (1−¯z)), and obtain

A(1 − z, 1 − ¯z) = hφ₁(x₃)φ₂(x₂)φ₁(x₁)φ₂(x₄)i. (75) However, now equations (74) and (75) are no longer equal, which we can see if we consider the order of the positions of the fields. But we can also notice that the right hand side of eq. (75) is equivalent to

hφ₁(x₁)φ₂(x₂)φ₁(x₃)φ₂(x₄)i (76) which in turn is equivalent to eq. (74)²

hφ₁(x₁)φ₂(x₂)φ₁(x₃)φ₂(x₄)i = hφ₁(x₁)φ₂(x₂)φ₁(x₃)φ₂(x₄)i. (77) This is why it is possible to use the crossing condition

A(z, ¯z) = A(1 − z, 1 − ¯z) (78) in this case.

The first step towards solving the crossing condition (78) is making the large-N expansion of the four-point correlator

A(z, ¯z) = A0(z, ¯z) + 1

N²A1(z, ¯z) + . . . p(n, l) = p0(n, l) + 1

N²p1(n, l) + . . .

∆(n, l) = ∆1+ ∆2+ 2n + l + 1

N²γ(n, l) + . . . .

(79)

2Simply check that the fields and their positions are in the same order on both sides.

Conformal field theory at large N

Uppsala University

Master thesis in physics 30 HP

Conformal field theory at large N

Contents

1 Introduction

2 Background

2.1 Conformal symmetry

2.2 Operator product expansion

2.3 Crossing conditions

2.4 Conformal block decomposition

2.5 Conformal bootstrap method

3 Problem and method

3.1 Presentation of the problem

3.2 Method

4 Main results

4.1 Crossing conditions and large-N expansion