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## Abstract

Conformal field theory finds applications across diverse fields, from
statistical systems at criticality to quantum gravity through the AdS/CFT
correspondence. These theories are subject to strong constraints, enabling a
systematic non-perturbative analysis. Conformal defects provide a controlled
means of breaking the symmetry, introducing new physical phenomena while
preserving crucial benefits of the underlying conformal symmetry. This thesis
investigates conformal line defects in both the weak- and strong-coupling
regimes. Two distinct classes of models are studied. First, we focus on the
supersymmetric Wilson line in ${\mathcal{N}}=4$ Super Yang–Mills, which serves
as an ideal testing ground for the development of innovative techniques such
as the analytic conformal bootstrap. The second class consists of magnetic
lines in Yukawa models, which have fascinating applications in $3d$ condensed-
matter systems. These systems have the potential to emulate phenomena observed
in the Standard Model in a low-energy setting.

$\star$

Die konforme Feldtheorie findet in verschiedenen Bereichen Anwendungen, von
statistischen Systemen in der Nähe kritischer Punkte bis hin zur
Quantengravitation durch die AdS/CFT-Korrespondenz. Diese Theorien unterliegen
starken Einschränkungen, die eine systematische nicht-perturbative Analyse
ermöglichen. Konforme Defekte bieten eine kontrollierte Möglichkeit, die
Symmetrie zu brechen und neue physikalische Phänomene einzuführen, während
wichtige Vorteile der zugrunde liegenden konformen Symmetrie erhalten bleiben.
Diese Dissertation untersucht konforme Liniendefekte sowohl im schwachen als
auch im starken Kopplungsregimes. Es werden zwei verschiedene Klassen von
Modellen untersucht. Wir konzentrieren uns zuerst auf die supersymmetrische
Wilson-Linie in ${\mathcal{N}}=4$ Super Yang–Mills, die als ideales Testfeld
für die Entwicklung innovativer Techniken wie dem analytischen konformen
Bootstrap dient. Die zweite Klasse besteht aus magnetische Linien in Yukawa-
Modellen, die faszinierende Anwendungen in $3d$ kondensierten Materiesystemen
haben. Diese Systeme haben das Potenzial, Phänomene des Standardmodells in
einem Niedrigenergieszenario nachzubilden.

## Acknowledgements

First and foremost, I want to express my deepest gratitude to my two
supervisors, Jan Plefka and Valentina Forini, for their continuous support
throughout these three beautiful years. I am especially thankful to Jan, whose
availability, kindness, and trust have been invaluable. His guidance and
expertise have consistently provided me with the necessary support and
knowledge to progress in my research.

I am greatly indebted to Pedro Liendo for introducing me to the world of
conformal defects and analytic bootstrap. Without his guidance and mentorship,
none of the works presented here would have been possible. Our countless
discussions about physics, artsy movies, and vintage video games have been
both enjoyable and enlightening.

I would like to thank the additional members of my PhD committee, Benjamin
Lindner, Nikolay Gromov, and Agostino Patella, for showing dedication in
reading my thesis as well as for making insightful comments.

I wish to extend my sincere gratitude to the remarkable individuals with whom
I have had the privilege of collaborating: Daniele Artico, Gabriel Bliard,
Ilija Burić, Pietro Ferrero, Aleix Gimenez-Grau, Carlo Meneghelli, Giulia
Peveri, Volker Schomerus, and Philine van Vliet. I have learned something
special from each of you, and I am eagerly looking forward to continuing these
fruitful collaborations in the future. I wish you all the best in your future
endeavors, both professionally and personally.

Being in the research training group Rethinking Quantum Field Theory has been
a truly enjoyable experience, and I would like to thank everyone in this
program for that. The friendships that have formed there go beyond physics,
and I am especially grateful for the camaraderie and support from Ilaria
Costa, Tomás Codina, Felipe Diaz-Jaramillo, Moritz Kade, Allison Pinto, and
Jasper Roosmale Nepveu. I deeply enjoyed the environment at HU, and I have
also greatly benefited from the interactions with Roberto Bonezzi, Luke
Corcoran, Michele Galli, Olaf Hohm, Emanuel Malek, Agostino Patella, and all
the other members of the string theory and mathematical physics groups. Our
limited lunch options have brought us all closer!

I wish to express my gratitude to the team of Non-Standard Models, our
precious outreach project: Daniele Artico, Ilaria Costa, Michele Galli,
Claudio Iuliano, Giulia Peveri, and Allison Pinto. Working with all of you on
this project has brought me a great sense of fulfillment, and I will always be
proud of what we achieved. I would like to also address special thanks to Jan
and Agostino for always supporting us, even when our ideas seemed unrealistic.
I want to express my appreciation to Alessandro Cotellucci, Tim Meier, Davide
Scazzuso, and others mentioned earlier for their occasional contributions to
this project.

I would also like to thank the individuals across the world who have welcomed
me to their institutions: T. Bargheer (DESY), D. Chicherin (LAPTh), N. Drukker
(King’s College), P. Ferrero (Stony Brook), F. Galvagno (ETH), S. Giombi
(Princeton), V. Rosenhaus (CUNY), A. Tseytlin (Imperial College), and K.
Zarembo (Nordita).

I have also enjoyed insightful discussions with A. Cavagliá (King’s College),
N. Gromov (King’s College), J. Henn (MPI), J. Julius (King’s College), E.
Pomoni (DESY), and M. Preti (King’s College).

I want to thank my friends scattered all over the world and my family for
their unwavering support. There are too many names to mention here, but each
of you holds a special place in my heart. I should not forget to express my
gratitude to Leonie, my cat, for waking me up at 5 am when I was already
overworked.

Finally, I wish to express my deepest gratitude and heartfelt appreciation to
my sunshine, Anne Reitz, for bringing immense joy and giving meaning to
everything I do. I cherish all of our moments together and the infinite
happiness that you bring into my world.

## Publications

This thesis is based on the following publications.

[1] |  J. Barrat, A. Gimenez-Grau, and P. Liendo, Bootstrapping holographic defect correlators in $\mathcal{N}$ = 4 super Yang-Mills, JHEP 04 (2022) 093;
---|---
[2] |  J. Barrat, P. Liendo, G. Peveri, and J. Plefka, Multipoint correlators on the supersymmetric Wilson line defect CFT, JHEP 08 (2022) 067;
[3] |  J. Barrat, A. Gimenez-Grau, and P. Liendo, A dispersion relation for defect CFT, JHEP 02 (2023) 255;
[4] |  J. Barrat, P. Liendo, and G. Peveri, Multipoint correlators on the supersymmetric Wilson line defect CFT II: Unprotected operators, 2210.14916;
[5] |  J. Barrat, P. Liendo, and P. van Vliet, Line defect correlators in fermionic CFTs, 2304.13588.

Some of the content to appear in the following works is also presented:

[6] |  D. Artico, J. Barrat, and G. Peveri, Four-point functions at two-loop on the Wilson line defect CFT;
---|---
[7] |  J. Barrat, G. Bliard, P. Ferrero, C. Meneghelli, and G. Peveri, Bootstrapping multipoint correlators on the Wilson line defect CFT;
[8] |  J. Barrat, I. Burić, P. Liendo, V. Schomerus, and P. van Vliet, Scalar-fermion correlators in Yukawa CFT across dimensions.

Finally, the unpublished notes [9] are presented in Section 3.4. Parts of the
author’s master’s thesis are also included:

[10] |  J. Barrat, P. Liendo, and J. Plefka, Two-point correlator of chiral primary operators with a Wilson line defect in $\mathcal{N}$ = 4 SYM, JHEP 05 (2021) 195.
---|---

Each paper has received equal contributions from all the authors.

## Declaration of authorship

I hereby certify that the thesis that I am submitting is entirely my own
original work, except where otherwise stated. I am aware of the University’s
regulations concerning plagiarism, including those regulations concerning
disciplinary actions that may result from plagiarism. Any use of the works of
any other author, in any form, is properly acknowledged at their point of use.

Julien Barrat

###### Contents

  1. 1 Introduction
    1. 1.1 An invitation to conformal field theory
      1. 1.1.1 Critical phenomena
        1. Phase transitions
        2. Universality
      2. 1.1.2 Signposts in the space of quantum field theories
        1. The RG flow
        2. Universality again
      3. 1.1.3 From CFT to gravity
        1. The holographic principle
        2. The AdS/CFT conjecture
    2. 1.2 Conformal defects
      1. 1.2.1 Breaking the symmetry
        1. That’s what defects do
        2. The bootstrap program
      2. 1.2.2 Line defects
        1. As a point in space
        2. As a brane
    3. 1.3 Structure of the thesis
      1. 1.3.1 Outline
      2. 1.3.2 Conventions
  2. 2 Foundations
    1. 2.1 A brief introduction to conformal defects
      1. 2.1.1 Conformal field theory
        1. Conformal symmetry
        2. Representations
        3. Correlation functions
      2. 2.1.2 Conformal defects
        1. Symmetry breaking
        2. Defect correlators
        3. Defect  representations
      3. 2.1.3 From weak to strong coupling
        1. Perturbation theory
        2. The conformal bootstrap
    2. 2.2 The Wilson-line defect CFT
      1. 2.2.1 ${\mathcal{N}}=4$ Super Yang–Mills
        1. The action
        2. The AdS/CFT correspondence
        3. Feynman rules
        4. Half-BPS operators
      2. 2.2.2 The Wilson-line defect CFT
        1. Supersymmetric Wilson loops
        2. Defect Feynman  rules
        3. Correlation functions
    3. 2.3 Line defects in fermionic CFTs across dimensions
      1. 2.3.1 Yukawa CFTs in $4-\varepsilon$ dimensions
        1. Yukawa CFTs
        2. Feynman rules
      2. 2.3.2 The Wilson–Fisher–Yukawa fixed point
        1. The critical point
        2. Bulk operators
      3. 2.3.3 The magnetic line
        1. Defect Feynman rules
        2. The critical point
        3. Defect operators
  3. 3 Bootstrapping holographic  defect correlators
    1. 3.1 An invitation
      1. 3.1.1 Exact results
        1. Three-point functions
        2. One-point functions
        3. Bulk-defect two-point functions
      2. 3.1.2 Two-point functions of half-BPS operators
        1. Superconformal Ward identities
        2. Crossing equation
        3. The topological subsector
      3. 3.1.3 A first glimpse at weak coupling
        1. Leading order
        2. Next-to-leading order
      4. 3.1.4 The supergravity perspective
        1. Leading order
        2. Next-to-leading order
    2. 3.2 A dispersion relation for defect CFT
      1. 3.2.1 Main result and derivation
        1. Analytic structure
        2. Main result
      2. 3.2.2 Convergence and subtractions
        1. Convergence
        2. Subtractions
    3. 3.3 Two-point functions at strong coupling
      1. 3.3.1 An example: $\langle\\!\langle\,\mathcal{O}_{2}\mathcal{O}_{2}\,\rangle\\!\rangle$
        1. The method
        2. The discontinuity
        3. Applying the  dispersion relation
        4. Supersymmetrization
        5. Final result and consistency checks
      2. 3.3.2 The general case $\langle\\!\langle\,\mathcal{O}_{\Delta_{1}}\mathcal{O}_{\Delta_{2}}\,\rangle\\!\rangle$
        1. Generalization
        2. A closed form for the highest  $R$-symmetry channel
    4. 3.4 The microbootstrap
      1. 3.4.1 Microbootstrapping two-point functions
        1. Another look at the topological subsector
        2. Multiplicity
      2. 3.4.2 Solving the microboostrap equations
        1. A warm-up: $SU(2)$
        2. Large $N$
    5. 3.5 Summary and outlook
      1. 3.5.1 ’t Hooft loops
        1. $S$-duality
        2. Strong coupling
      2. 3.5.2 Bulk-defect-defect correlators
        1. Kinematics
        2. Ward identities
        3. A perturbative result
      3. 3.5.3 The Witten diagram bootstrap
        1. Mellin space
        2. Closed form
  4. 4 Multipoint correlators on the  Wilson-line defect CFT
    1. 4.1 An invitation
      1. 4.1.1 Four-point functions of half-BPS operators
        1. The four-point Ward identity
        2. Weak coupling
        3. Strong coupling
      2. 4.1.2 Correlators of half-BPS operators
        1. General considerations
        2. Combs and snowflakes
        3. The topological sector
      3. 4.1.3 Correlators of unprotected operators
        1. The curious case of $\phi^{6}$
        2. More unprotected operators
    2. 4.2 Correlation functions of scalars at weak coupling
      1. 4.2.1 Correlators of fundamental scalars
        1. Leading order
        2. Next-to-leading order
      2. 4.2.2 Applications
        1. Kinematically-fixed correlators
        2. Four-point  functions
        3. Higher-point functions: selected  results
        4. Conformal block expansion
      3. 4.2.3 The four-point function at NNLO
        1. Solution of the Ward identity
        2. Bulk diagrams
        3. Boundary diagrams
        4. Numerical integration
        5. Final result
    3. 4.3 Superconformal Ward identities
      1. 4.3.1 Multipoint Ward identities
        1. The conjecture and checks
        2. Comments and a  word of caution
      2. 4.3.2 Solving the Ward identities
        1. An example: $\langle\,\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{2}\,\rangle$
        2. Revisiting the  weak coupling
    4. 4.4 Multipoint correlators at strong coupling
      1. 4.4.1 Superconformal blocks
        1. The symmetric channel
        2. The asymmetric channel
      2. 4.4.2 Bootstrapping the correlator
        1. Generalized free-field theory
        2. First correction
    5. 4.5 Summary and outlook
      1. 4.5.1 Multipoint correlators of fermions
        1. Sketching a fermionic recursion relation
        2. Correlators  of superfields
      2. 4.5.2 More Ward identities
        1. ${\mathcal{N}}=4$ Super Yang–Mills
        2. A network of Ward  identities?
  5. 5 Line defect correlators in fermionic CFT
    1. 5.1 An invitation
      1. 5.1.1 The Gross–Neveu–Yukawa model
        1. The fermionic model
        2. UV completion
        3. The critical point
      2. 5.1.2 The NJLY and chiral Heisenberg models
        1. The NJLY model
        2. The chiral Heisenberg model
      3. 5.1.3 The magnetic-line defect CFT
        1. Codimension across dimensions
        2. Dimension across  dimensions
    2. 5.2 Correlators of defect operators
      1. 5.2.1 Kinematically-fixed correlators
        1. Two scalars
        2. Two displacements
        3. Two  fermions
        4. Three scalars
        5. One scalar and  two fermions
      2. 5.2.2 Four-point functions
        1. Four scalars
        2. Four displacements
        3. Four fermions
        4. Two scalars and two fermions
    3. 5.3 Correlators of bulk operators with a defect
      1. 5.3.1 Kinematically-fixed correlators
        1. Squared-scalar one-point function
        2. Bilinear one-point  function
        3. Bulk-defect two-point functions
      2. 5.3.2 Two-point functions of scalars
        1. The perturbative result
        2. Defect channel
        3. Bulk channel
    4. 5.4 Summary and outlook
      1. 5.4.1 Two-point functions of fermions
        1. The leading order
        2. A challenging integral
      2. 5.4.2 Take a walk on the bulk side
        1. A perturbative result
        2. Tensor structures
  6. 6 Conclusions
    1. 6.1 Summary of results
    2. 6.2 Future directions
  7. A Spinors
    1. A.1 Dirac and Weyl spinors
    2. A.2 Dirac and Pauli matrices
  8. B Conformal blocks
    1. B.1 Two-point functions of half-BPS operators
      1. B.1.1 Bosonic blocks
        1. Bulk channel
        2. Defect channel
      2. B.1.2 Half-BPS superblocks
    2. B.2 Multipoint conformal blocks
      1. B.2.1 Comb channel
        1. Bosonic blocks
        2. Five-point superblocks
      2. B.2.2 Snowflake channel
  9. C Integrals
    1. C.1 Preliminaries
      1. C.1.1 Regularization schemes
        1. Point splitting
        2. Dimensional regularization
      2. C.1.2 Harmonic polylogarithms
    2. C.2 Feynman integrals
      1. C.2.1 Finite integrals
        1. $4d$ integrals
        2. $1d$ limits
      2. C.2.2 Divergent integrals
    3. C.3 Defect integrals
      1. C.3.1 Elementary integrals
      2. C.3.2 One-loop integrals
        1. $T$-integrals
        2. $U$-integrals

## CHAPTER 1 Introduction

Conformal field theories (CFTs) are of central importance in modern
theoretical physics, finding applications across various fields ranging from
high-energy to condensed-matter physics. The investigation of conformal
defects has recently emerged as a focal point, yielding a wealth of new
insights and discoveries. This thesis is devoted to the study of line defects
within the framework of conformal field theories. In this chapter, we present
an introduction to the themes that motivate this work.

###  1.1 An invitation to conformal field theory

Conformal field theories are quantum field theories (QFTs) possessing scale
invariance, meaning that physical quantities remain unchanged under the
rescaling of space and time. More generally, they provide a framework to
describe the behavior of fields and their interactions in a way that is
invariant under conformal transformations, an extension of the Lorentz
transformations used to describe relativistic physics [11]. In the following,
we motivate the study of conformal field theory through three applications:
understanding phase transitions in statistical systems, characterizing the
space of quantum field theories, and exploring the AdS/CFT correspondence as a
realization of the holographic principle.

####  1.1.1 Critical phenomena

Quantum field theories in Euclidean spacetime provide a framework to describe
and understand the collective behavior and statistical properties of large
ensembles of particles in classical statistical physics.111See for instance
[12] and the textbooks [13, 14, 15]. Critical phenomena, including phase
transitions, often exhibit scale-invariant physics and can be analyzed using
conformal field theories [16].

##### Phase transitions

Phase transitions are fundamental phenomena found in various physical systems
when a sudden change occurs in their macroscopic properties, such as the
emergence of long-range order or the appearance of new collective behavior.
Consider water in its liquid and gas states as an example. Despite having the
same elementary components, these two phases exhibit distinct behaviors. The
liquid phase is typically simulated using molecular dynamics, while the gas
phase is described by the ideal-gas model, where the water molecules are
considered as non-interacting point particles.

Conformal field theory plays a crucial role in understanding phase transitions
and critical phenomena. Second-order phase transitions exhibit scale
invariance, meaning that the physics of the system remains the same at all
length scales. Physical quantities, such as correlation functions and
thermodynamic properties, exhibit scaling behavior with respect to specific
variables, such as temperature or distance from the critical point. These
scaling relations allow for the determination of critical exponents that
characterize the critical behavior of the system. In most cases, the scale
invariance is intimately connected to the presence of conformal symmetry,
making the critical point correspond to a Euclidean conformal field theory.

##### Universality

Table 1.1: Measurements and theoretical estimations of the critical exponents $\eta$ and $\nu$ in fluid and magnet phase transitions (on the experimental side) and the $3d$ Ising model (on the theoretical side), illustrating universality. This table reproduces excerpts of Table 1 in [17], with the results based on [18, 19, 20, 21, 22, 23]. | Experiment | Theory
---|---|---
| Liquid-gas | Ferro-para | Bootstrap | Monte Carlo |  $\varepsilon$-exp.
$\eta$ | 0.042(6) |  | 0.0362978(20) | 0.036284(40) | 0.03653(65)
$\nu$ | 0.62(3) | 0.64(1) | 0.629971(4) | 0.62998(5) | 0.62977(22)

At critical points, different physical systems can exhibit the same behavior,
implying that a single conformal field theory can describe the observables of
very different systems [24, 25]. Despite the apparent fundamental differences
between water and magnets, it turns out that the liquid-gas phase transition
in water shares the same critical exponents as the ferromagnetic-paramagnetic
phase transition observed in uniaxial magnets (see Table 1.1 and [26]). This
intriguing connection categorizes water and magnets into the same $3d$ Ising
universality class, which is also known to describe phase transitions in
binary mixtures in polymers [27, 28, 29] and alloys [30, 31]. This is
illustrated in Figure 1.1.

The concept of universality extends well beyond the $3d$ Ising universality
class and encompasses various categories of critical behavior. Universality
implies that microscopic details become irrelevant at critical points, and
only a set of robust properties is necessary to describe the physics. The
study and understanding of conformal field theory thus provide essential tools
for the description and classification of critical phenomena.

Figure 1.1: The $3d$ Ising universality class is known to describe the phase
transitions of various systems, such as the change from liquid to gas of water
(left) and the ferromagnetic-paramagnetic transition of uniaxial magnets
(right).

####  1.1.2 Signposts in the space of quantum field theories

The space of quantum field theories is vast and diverse, and finding effective
methods for classifying them is crucial. Due to the correspondence between
Euclidean quantum field theories and statistical systems, we can reformulate
the principles of criticality described above in the language of QFT. There
exists a remarkable common feature among all members of this family. In the
context of the renormalization group (RG), conformal field theories serve as
distinctive signposts that facilitate the classification of quantum field
theories flowing into CFTs.

##### The RG flow

Quantum field theories are characterized by a finite set of parameters, such
as masses and coupling constants, which can be modified to study the behavior
of the theory. This gives rise to the concept of RG flow, illustrated in
Figure 1.2, which describes how theories evolve as these parameters are varied
[32, 33, 34, 35, 36]. The trajectories traced out in the space of theories
converge towards fixed points, representing special models which, remarkably,
exhibit scale invariance. These fixed points correspond to conformal field
theories and serve as infrared (IR) attractors towards which specific quantum
field theories flow from their ultraviolet (UV) descriptions. The RG flow
provides a powerful tool for understanding the underlying symmetries of
quantum field theories across different energy scales.

##### Universality again

From this perspective, the space of quantum field theories can be viewed as a
landscape of RG flows, with the IR fixed points serving as the destinations
where the flows converge. This statement is fully equivalent to the discussion
of the previous section about the $3d$ Ising universality class. We encounter
again the principle of universality since distinct quantum field theories can
share common critical behavior, therefore belonging to the same class. In the
IR phase, the microscopic details of individual models become irrelevant, and
what dictates the physics are the universal properties and symmetries encoded
by the fixed point. This remarkable structure connects seemingly disparate
models in their behavior at long distances, giving conformal symmetry a
special place in the realm of quantum field theories.

####  1.1.3 From CFT to gravity

Figure 1.2: The image on the left represents the flow of models in the space
of quantum field theories. Conformal field theories serve as fixed points in
the RG flow. The holographic principle is depicted on the right. The
information contained in the $d$-dimensional gravitational theory in AdS space
is encoded on its boundary, where a conformal field theory resides.

The Standard Model of particle physics stands as a remarkable achievement in
science, yielding unprecedented agreement between experimental observations
and theoretical predictions [37]. It is widely recognized however that the
Standard Model is not complete. It necessitates the ad hoc fixing of $19$ free
parameters, does not provide a unified description of the electroweak and
strong forces, and fails to account for dark matter and dark energy, which
constitute about $95{\%}$ of the Universe (see for instance the reviews [38,
39]). Furthermore, the theory lacks a quantum version of gravity. We now
explore how conformal field theory plays a fundamental role in the modern
approach to resolving this open problem.

##### The holographic principle

The origins of this story lie in the theoretical study of black holes, where
an intriguing connection between gravity, spacetime, and information was
discovered in the early 1970s [40, 41, 42]. It was observed that the entropy
of a black hole, which measures its internal disorder, is proportional to its
surface area rather than its volume. In other words, the information contained
within a black hole is encoded on its boundary, much like a hologram captures
the essence of a three-dimensional object on a two-dimensional surface. More
generally, the holographic principle postulates that a higher-dimensional
theory with gravity can be fully described by a lower-dimensional theory
without gravity [43].

##### The AdS/CFT conjecture

The first concrete realization of the holographic principle emerged with the
development of the AdS/CFT correspondence. It was conjectured in [44] that
${\mathcal{N}}=4$ Super Yang–Mills (SYM) theory in four dimensions is dual to
a (super)string theory in a five-dimensional anti-de Sitter (AdS) space. In
other words, the gravitational physics of the bulk theory is fully encoded by
the quantum field theory residing on the boundary of the AdS space.
${\mathcal{N}}=4$ SYM stands out among quantum field theories due to its
remarkable features, including exact quantum conformal symmetry, hidden
integrability in the large $N$ limit, and profound dualities such as AdS/CFT
and $S$-duality [45, 46, 47, 48, 49, 50]. Since then, numerous other instances
of AdS/CFT dualities have been discovered [51, 52, 53, 54, 55, 56], providing
an exciting application of conformal symmetry for the study of quantum gravity
and black holes.

The AdS/CFT correspondence can be used to address another important problem
apart from the nature of quantum gravity [57, 58, 59, 60, 61, 62, 63]. This
duality establishes a connection between the gravitational theory at weak
coupling to the quantum field theory at strong coupling. The strong coupling
regime is notoriously difficult to analyze, as it cannot be accessed through
small perturbations of a free theory. This regime is important for instance in
the study of quantum chromodynamics (QCD) to understand features such as
confinement [64, 65, 66, 67]. The AdS/CFT correspondence unlocks access to
this regime, by perturbing the gravitational theory starting from a free
model. This has led to numerous advancements in the fields of particle physics
[68, 69, 70, 71, 72, 73, 74] and condensed matter [75, 76, 77, 78, 79, 80, 81,
82, 83, 84, 85].

###  1.2 Conformal defects

Extended operators, or defects, are important objects in quantum field theory
for probing new dynamics that would otherwise be inaccessible using local
operators alone. Defects serve as an ideal bridge between low- and high-energy
physics, facilitating the connection between experimental observations and
theoretical considerations. They find wide-range applications in both
condensed-matter systems and high-energy physics, offering opportunities for
shedding light on phenomena such as quark confinement in the strong force [64,
86, 87] and the AdS/CFT correspondence in supersymmetric theories [88, 89, 90,
91, 92, 93, 94, 54, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106,
107, 108, 109, 110, 111].

####  1.2.1 Breaking the symmetry

The presence of defects breaks the symmetry of the original system. There
exists however in conformal field theory a special class of defects, called
conformal, which retain crucial aspects of the underlying symmetry. This class
of defects is interesting, as the concepts presented above for defect-free
conformal field theory find analogous counterparts in defect conformal field
theory (dCFT).

##### That’s what defects do

In a (Euclidean) $d$-dimensional conformal field theory, the presence of a
conformal defect introduces a controlled breaking of the conformal group
$SO(d+1,1)$ [112, 113, 114, 115, 116, 117, 118, 119, 120]. These defects are
called conformal because they preserve a subset of the conformal symmetry:

$\underbrace{SO(d+1,1)}_{\begin{subarray}{c}\text{conformal group}\\\ \text{in
$d$
dimensions}\end{subarray}}\longrightarrow\underbrace{SO(p+1,1)}_{\begin{subarray}{c}\text{conformal
group}\\\ \text{in $p$
dimensions}\end{subarray}}\times\underbrace{SO(q)}_{\begin{subarray}{c}\text{orthogonal}\\\
\text{rotations}\end{subarray}}\,,$ (1.1)

where $p$ denotes the dimension of the defect and $q:=d-p$ its codimension.

Defects break translation invariance in the transverse directions, leading to
non-vanishing one-point functions of bulk operators. This feature proves
valuable for instance in numerical simulations of critical theories [121,
122]. While the study of spontaneous symmetry breaking in statistical systems
typically relies on the behavior of two-point functions, the quadratic
suppression of these functions can pose challenges. The introduction of a
defect offers a solution to increase the accuracy of the simulations. Despite
the symmetry breaking induced by the defect, the parameters of the bulk theory
remain unchanged far from the defect, enabling an efficient investigation of
the critical behavior through one-point functions.

##### The bootstrap program

The defect conformal bootstrap program investigates the implications of the
residual symmetry in quantum systems. In bulk theories, the bootstrap concept
was initially proposed in $S$-matrix theory to explain hadronic structures
[123, 124]. It was eventually shadowed by the advent of quantum
chromodynamics. However, it has experienced since then a revival in the
context of conformal field theory [125], achieving remarkable success in the
analysis of quantum field theories beyond the weak-coupling limit [126, 127,
128, 129, 130, 131, 132, 133, 21, 134, 135, 136, 137, 138, 139, 140, 141, 142,
143, 144, 145, 146, 147]. The bootstrap program has notably provided the most
precise determination of critical exponents for the $3d$ Ising model to date
(see [148, 149, 21] and Table 1.1).

In recent years, significant strides have been made in the analytic conformal
bootstrap, which relies on the behavior of operators at large spin and has
culminated into the development of the Lorentzian inversion formula [150,
151]. This formula, applicable to any conformal field theory in any dimension,
is a non-perturbative result that facilitates the derivation of strong-
coupling correlation functions under suitable conditions (see [152, 153, 154,
155, 156, 157, 158, 159, 160, 161, 162, 163] and references therein).

The application of bootstrap ideas to defects has been actively pursued [164,
165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179,
180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191]. Since the
presence of a defect does not significantly alter local physics, the
fundamental properties that enable the effectiveness of the bootstrap at
strong coupling in the parent theory remain intact. This program offers a
promising avenue for the study of correlation functions in the presence of
defects across various regimes.

####  1.2.2 Line defects

In this thesis, our primary focus is on line defects, characterized by $p=1$
in (1.1). This type of defect can be found in various condensed-matter
systems, including thin films [192, 193, 194, 195, 196, 197] and layered
materials [198, 199, 200, 201]. Another example of line defects is a magnetic
field localized along an infinite line [202, 203, 204, 205, 206, 207, 182,
184, 185, 208]. In the context of gauge theories, the supersymmetric Wilson
line has played a central role in the study of the AdS/CFT correspondence (see
references above).

##### As a point in space

Let us gain some intuitive understanding of line defects. In quantum field
theory, line defects that extend along the time direction can be visualized as
physical phenomena that manifest as points within the spatial domain. These
points serve as sources of disturbance for the surrounding system and exert an
influence on the nearby operators and observables.

To provide an analogy for this concept, we can imagine observing a serene lake
from an aerial perspective. Suppose now that we drop a small pebble into the
water, creating a localized disturbance. This disturbance affects the
propagation of nearby ripples, altering the overall pattern of motion in the
lake. From our vantage point above, we perceive this event as a point
dynamically influencing its environment. A line defect in a quantum system
operates similarly, influencing the dynamics of its parent theory. The pebble
can be considered as a probe, allowing measurements of specific properties of
the lake through its response to the disturbance. Similarly, line defects can
serve as probes to investigate the properties of the bulk theory.

##### As a brane

The AdS/CFT correspondence provides a completely different perspective on the
line defects described above. One notable example of a line defect is the
supersymmetric Wilson line operator in the ${\mathcal{N}}=4$ SYM theory in
four dimensions. This particular defect captures the interaction between a
heavy probe particle and the gauge fields of the theory. In the holographic
dual description, the Wilson line corresponds to a fundamental string or a
$D3$-brane that stretches into the extra dimension of the AdS space.

Certain correlation functions involving the Wilson line can be computed
exactly, which has enabled explicit verification of the AdS/CFT conjecture.
Conversely, the AdS interpretation of the Wilson line offers valuable handles
to constrain the strong-coupling behavior of correlation functions involving
local operators associated with the defect. This setup therefore not only
facilitates the examination of the AdS/CFT correspondence but also presents an
appealing opportunity for the defect bootstrap program.

###  1.3 Structure of the thesis

The topics explored in this thesis are intimately connected to the themes
discussed above. With these foundations in mind, we now present the structure
of the thesis.

####  1.3.1 Outline

The chapters are organized as follows. Chapter 2: Foundations provides a
compact review of the necessary concepts for the main chapters of the thesis.
After a general introduction to conformal defects, the specific examples of
the Wilson-line defect CFT in ${\mathcal{N}}=4$ SYM and of the magnetic line
in Yukawa CFTs are discussed.

In Chapter 3: Bootstrapping holographic defect correlators, we present the
results of [1, 3] as well as the unpublished notes [9]. In ${\mathcal{N}}=4$
SYM, correlation functions of operators in the presence of a Wilson-line
defect are computed at strong coupling using analytic bootstrap methods and
input from the AdS/CFT correspondence.

Chapter 4: Multipoint correlators in the Wilson-line defect CFT is dedicated
to the study of excitations along the defect, and summarizes the findings of
[2, 4, 6, 7]. Correlation functions of multiple operators are considered, both
in the weak- and strong-coupling regimes.

The case of a magnetic line in Yukawa CFTs is considered in Chapter 5: Line
defect correlators in fermionic CFT, which is based on [5] and presents
preliminary results to appear in [8]. Perturbative techniques are used to
calculate the correlation functions of $3d$ fermionic systems, which are
potentially relevant for condensed-matter systems and the study of certain
aspects of the Standard Model.

Chapters 3-5 begin with an invitation to the topic, and are concluded with an
outlook sketching potential follow-ups.

Chapter 6: Conclusions provides a concise summary of the results and discusses
future research directions related to the topics encountered throughout the
thesis.

####  1.3.2 Conventions

We consider spacetime to be Euclidean, i.e., the metric is given by

$g_{\mu\nu}(x):=\delta_{\mu\nu}\,,$ (1.2)

with $\mu\,,\nu=0\,,1\,,\ldots\,,d-1$. The direction $x^{0}:=\tau$ is referred
to as the Euclidean time. All the line defects studied here are chosen to
extend in the $\tau$-direction. Quantities are given in natural units, i.e.,
$c=\hbar=G=1$.

Note that the conventions used here might differ from [1, 2, 3, 4, 5, 6, 7, 8,
9, 10]. Changes have been made to ensure consistency throughout the thesis.

## CHAPTER 2 Foundations

The presence of conformal symmetry imposes significant constraints on the
observables of a conformal field theory, making them more tractable compared
to general quantum field theories while retaining key features. Conformal
defects provide a way to break conformal symmetry in a controlled manner. They
introduce non-trivial boundary conditions on the fields that give rise to new
interesting physical phenomena. Despite this symmetry breaking, correlation
functions associated with defects remain highly constrained, occasionally
allowing exact results to be derived. Defects play a crucial role in
holography, where a plethora of results have been obtained in the past two
decades [91, 92, 209, 210, 211, 212, 213, 100, 214, 215, 216, 217, 218, 219,
220, 221, 222, 223, 224, 225], but also in condensed-matter physics, where
they characterize critical phenomena in systems with impurities [226, 227,
228, 229, 230, 231].

This chapter serves as an introduction to line defects in conformal field
theories. We review how the correlation functions of a bulk theory are
impacted by the presence of a defect, and how the residual symmetry can be
used to derive results in both the weak- and strong-coupling regimes. Our
focus lies on two distinct classes of models: ${\mathcal{N}}=4$ Super
Yang–Mills with a supersymmetric Wilson-line defect, and Yukawa CFTs in the
presence of a magnetic line. The first system serves as an ideal laboratory
for the development of analytic conformal bootstrap techniques, due to its
abundance of symmetries. On top of conformal symmetry, ${\mathcal{N}}=4$ SYM
possesses supersymmetry and is believed to be integrable, while its
holographic dual has been identified and studied extensively. On the other
hand, Yukawa CFTs are interesting because they are closer to real-world
physical phenomena, such as phase transitions in graphene sheets in three
dimensions [232, 233, 234, 235, 236, 237, 238], which exhibit chiral symmetry
breaking akin to the Standard Model. Here we take these models across
dimensions to study strongly-coupled systems as a perturbation from free
theory.

This chapter draws on various resources, which are acknowledged at the
beginning of each section and in the text. We aim to present the material in
an accessible and engaging way, carefully selecting topics and methods that
are particularly relevant to the focus of this thesis and complement the
existing literature.

###  2.1 A brief introduction to conformal defects

We begin our journey in the world of conformal defects with an overview of how
the behavior of conformal field theories is impacted by their presence. In
this section, we provide a review of the key concepts of conformal symmetry in
$d$ dimensions, with a special focus on the correlation functions of local
operators. We then analyze how the constraints of conformal symmetry are
modified in the presence of a defect. Although the concepts discussed here
apply to more general defects, we specialize some of our formulas to the case
of line defects. By the end of this section, we hope to have established a
solid foundation for the topics presented in this thesis.

####  2.1.1 Conformal field theory

Before considering defects, we revisit the general principles of conformal
field theory and familiarize ourselves with the structure of correlation
functions. The content of this section is mostly based on [239, 49].111See
also the lectures [240, 241].

##### Conformal symmetry

Figure 2.1: An example of a conformal transformation: the Peirce quincuncial
projection [242] of the surface of the Earth shows the world in a square while
preserving the original angles.

Conformal symmetry is a rich mathematical concept that can be intuitively
understood as the group of transformations that preserve angles. The example
of a conformal transformation is presented in Figure 2.1. Conformal symmetry
extends the fundamental Poincaré group, which underlies our understanding of
spacetime symmetries.

For a transformation to be angle-preserving, the transformed metric must be a
scaling of the original metric by a conformal factor:

$g^{\prime}_{\mu\nu}(x^{\prime})=\kappa(x)g_{\mu\nu}(x).$ (2.1)

The Poincaré group can be restored by setting $\kappa(x)=1$. The conformal
group in $d$ dimensions consists of $(d+1)(d+2)/2$ generators, which can be
classified into four categories:

$\displaystyle P_{\mu}:=-i\partial_{\mu}$ $\displaystyle\qquad\text{($d$
translations)}\,,$ (2.2)
$\displaystyle(J_{\rho\sigma})_{\mu\nu}:=-i(\delta_{\rho\nu}\delta_{\mu\sigma}-\delta_{\sigma\nu}\delta_{\mu\rho})$
$\displaystyle\qquad\text{($d(d-1)/2$ LTs)}\,,$ $\displaystyle
D:=-ix_{\mu}\partial_{\mu}$ $\displaystyle\qquad\text{($1$ scalings)}\,,$
$\displaystyle K_{\mu}:=i(x^{2}\partial_{\mu}-2x_{\mu}x_{\nu}\partial_{\nu})$
$\displaystyle\qquad\text{($d$ SCTs)}\,,$

where LTs refers to Lorentz transformations and SCTs to special conformal
transformations. SCTs can be seen as the compositions of an inversion, a
translation, and another inversion, while $D$ rescales distances. The question
of whether scaling invariance automatically implies invariance under SCTs has
been the subject of much debate. While there have been several counterexamples
found in which scaling symmetry was preserved alone, these examples are
limited and have so far been constrained to the case $d=2$ (see [243, 244,
245, 246] and references therein).

The conformal algebra is formed by promoting the generators of the conformal
group to operators satisfying specific commutation relations:

$\displaystyle\begin{split}[D,K_{\mu}]&=-iK_{\mu}\,,\\\
[D,P_{\mu}]&=iP_{\mu}\,,\\\
[K_{\mu},P_{\nu}]&=2i(\delta_{\mu\nu}D-J_{\mu\nu})\,,\\\
[K_{\mu},J_{\nu\rho}]&=i(\delta_{\mu\nu}K_{\rho}-\delta_{\mu\rho}K_{\nu})\,,\end{split}$
(2.3)

which have to be complemented by the commutators of the Poincaré algebra (see
for instance [247, 248]). All other possible commutators vanish, leading to a
simple and elegant algebraic structure.222In fact, the conformal algebra in
$d$ dimensions can be expressed as a Poincaré algebra in $d+2$ dimensions,
known as the embedding formalism (see for instance [249]). This approach
offers an interesting perspective on conformal field theory, but will not be
treated in this thesis.

##### Representations

In this section, we direct our attention towards the representations of the
conformal algebra. These are also representations of the Lorentz algebra, and
we refer the reader to [247] for a review of the scalar, spinor, and vector
representations in this context. We denote the corresponding (local) fields by
$\phi(x)$ (scalar), $\psi(x)$, ${\bar{\psi}}(x)$ (spinors) and $A_{\mu}(x)$
(vector). In the following, we primarily focus on scalar fields, while the
case of spinors is treated in Chapter 5 to some extent.

In a CFT, scalar fields undergo new transformation laws in addition to the
Poincaré transformations. These are given by

$\displaystyle\begin{split}D\phi(x)&=i(x_{\mu}\partial_{\mu}+\Delta)\phi(x)\,,\\\
K_{\mu}\phi(x)&=i(x^{2}\partial_{\mu}-2x_{\mu}x_{\nu}\partial_{\nu}-2x_{\mu}\Delta)\phi(x)\,.\end{split}$
(2.4)

$\Delta$ is known as the scaling dimension of the field $\phi(x)$, and it is
defined by the action of a scaling transformation on the spacetime
coordinates:

$\phi(\lambda x)=\lambda^{-\Delta}\phi(x)\,,$ (2.5)

with $\lambda$ a real number.

Vector and spinor representations also carry scaling dimensions. These
quantities are important for the classification of representations in a
conformal field theory. At the classical level, they are typically integer-
valued. However, at the quantum level, they can be subject to corrections.
Small deviations from classical behavior are commonly referred to as anomalous
dimensions, which can arise in both the weak- and strong-coupling regimes,
where they can be expanded as

$\Delta=\Delta^{(0)}+\sum_{\ell=0}^{\infty}g^{\ell}\,\gamma^{(\ell)}\,,$ (2.6)

with $g$ a coupling constant. Anomalous dimensions play a central role
throughout this thesis.

We specialize our analysis to the case of unitary CFTs. For these theories,
there exists a fundamental lower bound for the scaling dimension $\Delta$,
known as the unitarity bound. The generators $P_{\mu}$ increase $\Delta$ while
the $K_{\mu}$ decrease it, and as a consequence the fields that saturate the
lower bound are the ones satisfying

$[K_{\mu},\phi(0)]=0\,.$ (2.7)

Such fields are called conformal primaries. A conformal multiplet consists of
a conformal primary and all its descendants, which are the fields with higher
dimension $\Delta$ that can be constructed by applying $P_{\mu}$ an arbitrary
number of times on the primary.

##### Correlation functions

Conformal symmetry imposes strong constraints on correlation functions of the
local operators presented in the previous section. The symmetry uniquely
determines the kinematics of two- and three-point functions, while higher-
order correlators are constrained to depend on a limited number of cross-
ratios. One-point functions of any operator vanish since that would break the
translation invariance of the vacuum.333As mentioned in Section 1.2.1, this
will not be true any longer once we add defects to the theory. In the
following, we review the correlation functions of scalar fields.

Let us begin with the two-point functions. It is straightforward to show that
the correlators of two operators $\mathcal{O}_{\Delta_{1}}$ and
$\mathcal{O}_{\Delta_{2}}$ take the form

$\langle\,\mathcal{O}_{\Delta_{1}}(x_{1})\mathcal{O}_{\Delta_{2}}(x_{2})\,\rangle=n_{\Delta}\frac{\delta^{\Delta_{1}\Delta_{2}}}{x_{12}^{2\Delta}}\,,$
(2.8)

with $x_{ij}:=x_{i}-x_{j}$, and where we have defined
$\Delta:=\Delta_{1}=\Delta_{2}$, since all other two-point functions are
orthogonal and thus vanish. The constant $n_{\Delta}$ is known as the
normalization constant. In this thesis, we choose the operators to be unit-
normalized, i.e., they satisfy

$\langle\,\mathcal{O}_{\Delta_{1}}(x_{1})\mathcal{O}_{\Delta_{2}}(x_{2})\,\rangle=\frac{\delta^{\Delta_{1}\Delta_{2}}}{x_{12}^{2\Delta}}\,.$
(2.9)

This can always be achieved by rescaling the operators:

$\mathcal{O}_{\Delta}\longrightarrow\frac{1}{\sqrt{n_{\Delta}}}\mathcal{O}_{\Delta}\,.$
(2.10)

Although $n_{\Delta}$ is a convention-dependent quantity (and, as such,
unphysical), determining it is still crucial for writing the operators
explicitly. In a perturbative setting, the two-point functions are also
essential for obtaining the anomalous dimensions of a given operator.

Conformal symmetry also places constraints on three-point functions, which
take the form

$\langle\,\mathcal{O}_{\Delta_{1}}(x_{1})\mathcal{O}_{\Delta_{2}}(x_{2})\mathcal{O}_{\Delta_{3}}(x_{3})\,\rangle=\frac{\lambda_{\Delta_{1}\Delta_{2}\Delta_{3}}}{x_{12}^{2\Delta_{123}}x_{23}^{2\Delta_{231}}x_{31}^{2\Delta_{312}}}\,,$
(2.11)

where $\Delta_{ijk}:=\Delta_{i}+\Delta_{j}-\Delta_{k}$. The three-point
functions (2.11) are therefore completely determined by the scaling dimensions
$\Delta_{i}$ and the constants $\lambda_{\Delta_{1}\Delta_{2}\Delta_{3}}$,
which we refer to as OPE coefficients for reasons that will soon become clear.

The kinematic dependence of higher-point functions is no longer fully
determined by conformal symmetry. In $d>1$ dimensions, the four-point
functions of scalars depend on two cross-ratios $u$ and $v$:

$\langle\,\mathcal{O}_{\Delta_{1}}(x_{1})\mathcal{O}_{\Delta_{2}}(x_{2})\mathcal{O}_{\Delta_{3}}(x_{3})\mathcal{O}_{\Delta_{4}}(x_{4})\,\rangle={\mathcal{K}}\,F(u,v)\,,$
(2.12)

where ${\mathcal{K}}$ represents a trivial conformal prefactor, which will be
defined precisely later. The cross-ratios are defined as

$u:=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}\,,\qquad
v:=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}\,.$ (2.13)

This choice of variables for expressing the reduced correlator $F(u,v)$ is not
unique and depends on the application. Another set of cross-ratios is often
used:

$\chi\bar{\chi}:=u\,,\qquad(1-\chi)(1-\bar{\chi}):=v\,.$ (2.14)

In line defect CFTs, the case $d=1$ is particularly important. In this case,
the cross-ratios are not independent of each other. Instead, the reduced
correlators depend on a single cross-ratio $\chi=\bar{\chi}$, leading to the
mappings

$u\overset{1d}{\longrightarrow}\chi^{2}\,,\qquad
v\overset{1d}{\longrightarrow}(1-\chi)^{2}\,.$ (2.15)

This fact is important for the study of the correlation functions of operators
inserted along a line defect.

The observations made above about the four-point functions can be extended
straightforwardly to the higher-point functions. Generally, a correlator
consisting of $n$ operators depends on $n(n-3)/2$ cross-ratios444This number
holds for small $n$. For $n\geq d+1$, it would be $nd-\frac{1}{2}(d+1)(d+2)$,
where the first term corresponds to the naive number of degrees of freedom,
while the second term is the number of generators. and can be written as

$\langle\,\mathcal{O}_{\Delta_{1}}(x_{1})\ldots\mathcal{O}_{\Delta_{n}}(x_{n})\,\rangle={\mathcal{K}}\,F(\\{u\,,v\,,w\\})\,,$
(2.16)

where $\\{u\,,v\,,w\\}$ refers to the following set of cross-ratios:

$\displaystyle u_{i}$
$\displaystyle:=\frac{x^{2}_{1(i+1)}x^{2}_{(n-1)n}}{x^{2}_{1(n-1)}x^{2}_{(i+1)n}}$
$\displaystyle\qquad\left(n-3\text{ cross-ratios}\right)\,,$ (2.17)
$\displaystyle v_{i}$
$\displaystyle:=\frac{x^{2}_{1n}x^{2}_{(i+1)(n-1)}}{x^{2}_{1(n-1)}x^{2}_{(i+1)n}}$
$\displaystyle\qquad\left(n-3\text{ cross-ratios}\right)\,,$ $\displaystyle
w_{ij}$
$\displaystyle:=\frac{x^{2}_{1(n-1)}x^{2}_{(i+1)n}x^{2}_{(j+1)n}}{x^{2}_{1n}x^{2}_{(n-1)n}x^{2}_{(i+1)(j+1)}}$
$\displaystyle\qquad\left(\frac{(n-3)(n-4)}{2}\text{ cross-ratios}\right)\,.$

Conformal field theory possesses a remarkable feature known as the operator
product expansion (OPE), which is a powerful tool for understanding the
behavior of local operators and the correlation functions introduced above.
The OPE allows to expand a product of two local operators in terms of all the
possible local operators in the theory, and it takes the form555See for
instance [239] for a detailed derivation of the OPE.

$\mathcal{O}_{\Delta_{1}}(x_{1})\mathcal{O}_{\Delta_{2}}(x_{2})\sim\sum_{\mathcal{O}_{\Delta}\text{
prim.}}\lambda_{\Delta_{1}\Delta_{2}\Delta}C_{\Delta_{1}\Delta_{2}\Delta}(x_{12}\,,\partial_{2})\,\mathcal{O}_{\Delta}(x_{2})\,,$
(2.18)

where the sum runs over all the primaries of the theory.
$C_{\Delta_{1}\Delta_{2}\Delta}$ is a differential operator encoding the
construction of descendants, while $\lambda_{\Delta_{1}\Delta_{2}\Delta}$ is
the three-point function coefficient introduced in (2.11).

While the idea of an OPE is not unique to CFT, conformal symmetry guarantees
that the expansion converges with a non-zero radius [250, 251], making it an
especially valuable tool. This expansion was first introduced in [252] several
decades ago, and since then it has become a fundamental concept in the study
of CFTs.

To understand the power of the operator product expansion, let us look at the
four-point functions defined in (2.12). By considering the pairs of operators
$\mathcal{O}_{\Delta_{1}}\mathcal{O}_{\Delta_{2}}$ and
$\mathcal{O}_{\Delta_{3}}\mathcal{O}_{\Delta_{4}}$, it can be expressed as a
sum over conformal primaries, the exchanged operators $\mathcal{O}_{\Delta}$.
This gives rise to conformal blocks $f_{\mathcal{O}}^{12,34}(u,v)$, which are
fully characterized by the symmetry group. The reduced correlator $F(u,v)$ can
then be written as

$F(u,v)=\sum_{\mathcal{O}}\lambda_{\Delta_{1}\Delta_{2}\Delta}\lambda_{\Delta_{3}\Delta_{4}\Delta}\,f^{12,34}_{\mathcal{O}}(u,v)\,.$
(2.19)

The conformal blocks offer an expansion of correlation functions in terms of
basis functions, which are determined by a corresponding Casimir equation
fixed by the symmetry group. The explicit form of the blocks in four
dimensions can be found for instance with the Mathematica notebook attached to
[253]. Furthermore, the OPE coefficients and the scaling dimensions provide
the necessary information to fully determine the remaining parts of the
correlator. Together, this information is commonly referred to as CFT data,
and it provides a powerful tool for analyzing and characterizing CFTs.

Before moving our attention to conformal defects, let us notice an interesting
consequence of (2.18). There is another OPE channel that can be considered to
express the correlator (2.12), which is to pair the operators as
$\mathcal{O}_{\Delta_{1}}\mathcal{O}_{\Delta_{4}}$ and
$\mathcal{O}_{\Delta_{2}}\mathcal{O}_{\Delta_{3}}$. This yields an identical
correlator, but expressed as a sum over different coefficients, giving a
consistency relation known as crossing symmetry:

$\sum_{\mathcal{O}}\lambda_{\Delta_{1}\Delta_{2}\Delta}\lambda_{\Delta_{3}\Delta_{4}\Delta}\,f_{\mathcal{O}}^{12,34}(u,v)=\sum_{\mathcal{O}^{\prime}}\lambda_{\Delta_{1}\Delta_{4}\Delta^{\prime}}\lambda_{\Delta_{2}\Delta_{3}\Delta^{\prime}}\,f_{\mathcal{O}^{\prime}}^{14,23}(u,v)\,.$
(2.20)

This relation is represented graphically in Figure 2.3. It plays a crucial
role in the numerical conformal bootstrap, and a defect counterpart will be
presented shortly.

####  2.1.2 Conformal defects

Conformal symmetry imposes strong constraints on the observables of a
conformal field theory. One way to introduce new physical phenomena while
preserving the benefits of conformal symmetry is to incorporate conformal
defects into the theory. In this section, we explore the concept of conformal
defect and examine the implications for the underlying bulk CFT.

##### Symmetry breaking

Conformal defects are non-local extensions of a bulk theory that deliberately
break conformal symmetry in a controlled manner. These defects preserve a
specific subgroup of the original symmetry of the parent theory. For instance,
in a $d\geq 2$ CFT with a line defect extending in the time direction,
translations are broken in the orthogonal directions while remaining preserved
in the time direction (see Figure 2.2).

$x$$y$$\tau$$x$$y$$\tau$ Figure 2.2: An example of a line defect extending in
the time direction in a $(2+1)$-dimensional spacetime. The grid in both
figures represents the two-dimensional space. The left figure shows the bulk
theory without any defects, while the right figure depicts the introduction of
a line defect. As explained in Section 1.2.2, line defects can be understood
as point-like impurities residing in the two-dimensional space.

As we have already seen in (1.1), a conformal defect can be viewed as a
deformation that breaks conformal symmetry in the following way:

$SO(d+1,1)\longrightarrow SO(p+1,1)\times SO(q)\,,$ (2.21)

where $p$ is the dimension of the defect and $q$ its codimension, such that
$p+q=d$. This symmetry breaking leads to constrained correlation functions in
the defect CFT compared to a homogeneous CFT. Although conformal symmetry is
preserved in the $p$-dimensional subspace $SO(p+1,1)$, it is broken in the
full $d$-dimensional space, with the second subgroup $SO(q)$ representing
$q$-dimensional rotations around the defect.

Not all quantum field theories can be described by a Lagrangian, and this is
also true for defect CFTs. However, in fortunate instances, it is possible to
express the bulk theory and the defect in terms of an action, given by666This
expression should be understood as schematic. In the case of non-Abelian gauge
theories, it has to be adapted to accommodate the gauge group.

$S_{\text{defect}}:=S_{\text{bulk}}+h\int d^{p}x\,\mathcal{O}(x^{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu})\,,$ (2.22)

where $h$ is the defect coupling constant and $\mathcal{O}(x^{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu})$ depends solely on the $p$-dimensional coordinates $x^{{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}:=(x^{0},\ldots,x^{p})$. To ensure consistency, $\mathcal{O}(x)$ must
have mass dimension $[\mathcal{O}]=p$ if $h$ is dimensionless. The directions
orthogonal to the defect are denoted by $x^{\perp}$. This thesis primarily
focuses on defects with dimension $p=1$,777The main exception is the
dispersion relation in Section 3.2, which is valid for all defects of
codimension $q\geq 2$. which have codimension $q=d-1$. Following the
conventions given in 1.3.2, we choose these line defects to extend along the
(Euclidean) time direction $x^{0}=\tau$.

We conclude this section with a brief discussion on boundary defects, which
have codimension $q=1$ and preserve the largest subgroup of conformal
symmetry. They can be seen as a separation in the $d$-dimensional space.
Although we do not consider boundaries specifically here, they are a
fascinating setup with numerous physical applications, particularly in
condensed-matter physics, where they arise naturally in the study of finite-
temperature systems [254, 255, 256, 257, 258, 259, 260]. Boundary defects have
also been used to describe impurities in systems at criticality, interfaces
between different phases, and have applications in the AdS/CFT correspondence.
Boundary defects present several conceptual and technical challenges, such as
the emergence of edge modes [261, 262] and the need to properly define the
boundary conditions. Constraints on supersymmetric boundary defects are
briefly discussed in the outlook 4.5.2 of Chapter 4.

##### Defect correlators

In the presence of a defect, it is necessary to revisit some of the results
that we established when considering the correlation functions of a CFT. We
previously determined that a quantum field theory with conformal symmetry was
completely fixed by the CFT data
$\\{\Delta_{i},\lambda_{\Delta_{i}\Delta_{j}\Delta_{k}}\\}$. The defect breaks
this symmetry and introduces new physics, and as a result a new set of
numbers, known as defect CFT data, is needed for characterizing the defect
theory, in complete analogy with our previous analysis.

As in Section 2.1.1, we focus on scalar operators. The correlators of $n$
local bulk operators $\mathcal{O}_{\Delta_{k}}(x_{k})$ in the presence of a
defect ${\mathcal{D}}$ are defined as

$\langle\\!\langle\,\mathcal{O}_{\Delta_{1}}(x_{1})\ldots\mathcal{O}_{\Delta_{n}}(x_{n})\,\rangle\\!\rangle:=\langle\,\mathcal{O}_{\Delta_{1}}(x_{1})\ldots\mathcal{O}_{\Delta_{n}}(x_{n}){\mathcal{D}}\,\rangle\,.$
(2.23)

The first observation is that one-point functions are non-vanishing in a
defect CFT, meaning that operators acquire expectation values due to their
interactions with the defect. These one-point functions are given by

$\langle\\!\langle\,\mathcal{O}_{\Delta}(x)\,\rangle\\!\rangle=\frac{a_{\Delta}}{|x^{\perp}|^{\Delta}}\,,$
(2.24)

where the coefficients $a_{\Delta}$ are functions of the coupling independent
of the kinematic variables, similarly to
$\lambda_{\Delta_{1}\Delta_{2}\Delta_{3}}$. Since the normalization of
$\mathcal{O}_{\Delta}$ is determined by the bulk two-point functions,
$a_{\Delta}$ contains dynamic information about the defect CFT that cannot be
absorbed into the operator.888One may question whether the parameter
$a_{\Delta}$ can be absorbed into the defect itself. It is standard to
normalize the defect such that the vacuum of the theory is
$\langle\,{\mathcal{D}}\,\rangle=1$. This makes $a_{\Delta}$ a physical
observable in its own right.

Note that (2.24) also contains information about the bulk theory. One-point
functions can be used to extract the scaling dimensions of the bulk fields. In
fact, in certain experiments and simulations, it is sometimes easier to
measure defect one-point functions than bulk two-point functions.

In a defect CFT, two-point functions are the first correlation functions to
exhibit a non-trivial dependence on kinematic variables. For a defect of
codimension $q\geq 2$, the correlators of two scalar operators take the form

$\langle\\!\langle\,\mathcal{O}_{\Delta_{1}}(x_{1})\mathcal{O}_{\Delta_{2}}(x_{2})\,\rangle\\!\rangle=\frac{F(r,w)}{|x_{1}^{\perp}|^{\Delta_{1}}|x_{2}^{\perp}|^{\Delta_{2}}}\,,$
(2.25)

where the variables $r$ and $w$ are defect cross-ratios defined via

$r+\frac{1}{r}=\frac{|x_{12}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}|^{2}+|x_{1}^{\perp}|^{2}+|x_{2}^{\perp}|^{2}}{|x_{1}^{\perp}|^{2}|x_{2}^{\perp}|^{2}}\,,\quad
w+\frac{1}{w}=\frac{2x_{1}^{\perp}\cdot
x_{2}^{\perp}}{|x_{1}^{\perp}|^{2}|x_{2}^{\perp}|^{2}}\,.$ (2.26)

Another useful set of cross-ratios $z$, ${\bar{z}}$ can be defined through

$z+{\bar{z}}:=2(x_{1}^{\perp}\cdot
x_{2}^{\perp})\,,\quad(1-z)(1-{\bar{z}}):=x_{12}^{2}\,.$ (2.27)

Geometrically, $z$ and ${\bar{z}}$ are coordinates in a plane orthogonal to
the defect, while $r$ corresponds to a radius and $w$ to an angle. Without
loss of generality, we can place the operator in a conformal frame by setting
$x_{1}=(0,1,0,0)$ and $x_{2}=(0,x,y,0)$ (see Figure 2.3). We can then identify

$z=rw=x+iy\,,\quad{\bar{z}}=\frac{r}{w}=x-iy\,.$ (2.28)

Note that in the Lorentzian signature, one would find instead that $z$ and
${\bar{z}}$ are real and independent.

In a sense, two-point functions in the presence of a defect can be viewed as
the counterpart of four-point functions without a defect. Both involve two
cross-ratios and an expansion in conformal blocks of (2.25) can be performed
using the bulk OPE given in (2.18):

$\langle\\!\langle\,\mathcal{O}_{1}(x_{1})\mathcal{O}_{2}(x_{2})\,\rangle\\!\rangle=\sum_{\mathcal{O}_{\Delta,\ell}\text{
prim.}}\lambda_{\Delta_{1}\Delta_{2}\Delta}C_{\Delta_{1}\Delta_{2}\Delta}(x_{12},\partial_{2})\langle\\!\langle\,\mathcal{O}_{\Delta,\ell}(x_{2})\,\rangle\\!\rangle\,.$
(2.29)

Similarly to the analysis in (2.19), this can be written as

$F(r,w)=\left(\frac{rw}{(r-w)(rw-1)}\right)^{\frac{\Delta_{1}+\Delta_{2}}{2}}\sum_{\Delta,\ell}\lambda_{\Delta_{1}\Delta_{2}\Delta}a_{\Delta}\,f_{\Delta,\ell}(r,w)\,.$
(2.30)

Here, the functions $f_{\Delta,\ell}(r,w)$ are conformal blocks fixed by the
symmetry and the quantum numbers $\Delta$ and $\ell$ (spin) of the exchanged
operators $\mathcal{O}_{\Delta,\ell}$. Their precise form can be found in
(B.6). Note that the three- and one-point function coefficients depend on
$\ell$ as well. However, we suppress this dependence here to avoid cluttering.
We reinstate it whenever confusion may arise.

Bulk crossing equation
$=$
Defect crossing equation
$=$

$x$$y$$\mathcal{O}(x_{1})$$\mathcal{O}(x_{2})$

Figure 2.3:  The bulk and defect crossing equations, given respectively in
(2.20) and (2.30)-(2.31), are represented in the left figure, where the red
lines correspond to the exchanged operators. The right figure shows the two-
point function in the presence of a line defect (the bold line) in the
conformal frame $x_{1}=(0,1,0,0)$ and $x_{2}=(0,x,y,0)$.

Another similarity between (2.25) and (2.12) is that bulk four-point functions
depend on a single cross-ratio in $d=1$. This is also true for defect two-
point functions with codimension $q=1$, i.e., for boundaries. In a more
general way, there exist interesting group-theoretical relations between bulk
and defect theories, which are briefly discussed in Section 4.5.2 for a
supersymmetric case.

The higher-point functions of a defect CFT can be derived from lower-point
functions through iterative application of the OPE (2.18). As a result, all
these correlation functions can be written in terms of the CFT data, which now
includes the scaling dimensions of the local operators $\Delta_{i}$, their
one-point function coefficients $a_{\Delta_{i}}$, and the bulk OPE
coefficients $\lambda_{\Delta_{i}\Delta_{j}\Delta_{k}}$.

##### Defect representations

There exists another OPE that one can take to expand the defect two-point
functions, which involves bringing one bulk operator close to the defect and
expanding around that limit. This is called the defect OPE, leading to a
second block expansion of (2.25)

$F(r,w)=\sum_{\hat{\Delta},s}b_{\Delta_{1}\hat{\Delta}}b_{\Delta_{1}\hat{\Delta}}\hat{f}_{\hat{\Delta},s}(r,w)\,.$
(2.31)

The defect blocks $\hat{f}_{\hat{\Delta},s}(r,w)$ (with $s$ the $SO(q)$
quantum number) can be found in (B.7). Similarly to (2.30), we suppress the
dependence on $s$ in $b_{\Delta\hat{\Delta}}$. Note that, throughout this
thesis, hats always refer to the defect channel.

In (2.31), the exchanged operators, denoted by
$\hat{\mathcal{O}}_{\hat{\Delta},s}$, are new defect operators that only exist
in the defect CFT. They correspond to local excitations on the defect, and as
such they reside in the $p$-dimensional CFT preserved by the defect. It is
important to notice that the scaling dimensions $\hat{\Delta}$ of these
operators are independent of the bulk operators. The OPE coefficients
$b_{\Delta\hat{\Delta}}$ correspond to bulk-defect two-point functions,
defined for the case of scalars ($s=0$) as

$\langle\\!\langle\,\mathcal{O}_{\smash{\Delta_{1}}}(x^{\perp}_{1})\hat{\mathcal{O}}_{\smash{\hat{\Delta}_{2}}}(x^{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}_{2})\,\rangle\\!\rangle=\frac{b_{\Delta_{1}\hat{\Delta}_{2}}}{(|x_{1}^{\perp}|^{2}+|x_{12}^{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}|^{2})^{\hat{\Delta}_{2}}|x_{1}^{\perp}|^{\Delta_{1}-\hat{\Delta}_{2}}}\,.$
(2.32)

As a consequence, the CFT data listed at the end of the previous section is
not sufficient for solving the defect CFT as a whole. The correlation
functions that involve defect operators $\hat{\mathcal{O}}_{\hat{\Delta},s}$
form a $p$-dimensional CFT, and consequently they follow the constraints
derived in (2.8)-(2.16). For a line defect ($p=1$), these correlators are one-
dimensional and depend on $n-3$ cross ratios, as explained in (2.15). All the
additional bulk-defect correlators can be obtained from the knowledge of
$b_{\Delta\hat{\Delta}}$ and the $1d$ CFT data.

We can now provide a complete list of the CFT data necessary for fully solving
a defect CFT:

$\\{\Delta_{i},\hat{\Delta}_{i},a_{\Delta_{i}},b_{\Delta_{i}\hat{\Delta}_{j}},\lambda_{\Delta_{i}\Delta_{j}\Delta_{k}},\lambda_{\hat{\Delta}_{i}\hat{\Delta}_{j}\hat{\Delta}_{k}}\\}\,,$
(2.33)

where the coefficients
$\lambda_{\hat{\Delta}_{i}\hat{\Delta}_{j}\hat{\Delta}_{k}}$ refer to the
three-point functions of defect operators. This thesis focuses on the study of
correlation functions in specific defect CFTs, both in the weak- and strong-
coupling regimes, from which an infinite amount of CFT data can be extracted
by expanding them in blocks.

To conclude this section, note that the expansion (2.31) provides another
parallel to the bulk four-point functions (2.12): a defect crossing equation.
This relation is obtained by equating (2.30) with (2.31), and is represented
graphically in Figure 2.3. Notice however that there is a crucial difference
with (2.20): while the two sides of the equation in a bulk CFT involve the
same type of OPE coefficients $\lambda_{\Delta_{i}\Delta_{j}\Delta_{k}}$, in
the defect CFT case we have $a_{\Delta}\lambda_{\Delta_{1}\Delta_{2}\Delta}$
on one side and $b_{\Delta_{1}\hat{\Delta}}b_{\Delta_{2}\hat{\Delta}}$ on the
other. Therefore, there is no real crossing, and the name is an abuse of
language that comes from the study of CFTs in the bulk. Nevertheless, the
defect crossing equation is a powerful identity that strongly constrains the
defect CFT data.

####  2.1.3 From weak to strong coupling

The residual conformal symmetry in defect CFTs provides strong handles on the
associated correlation functions, making them potentially tractable even in
regimes that would otherwise be challenging to explore using traditional
techniques. In this thesis, we study these correlators perturbatively in both
the weak- and strong-coupling regimes.

This section offers an overview of the methodologies used in each context. At
weak coupling, Feynman diagrams are computed using the machinery of
perturbation theory. In the strong-coupling regime, the analytic conformal
bootstrap is our preferred approach.

##### Perturbation theory

The calculation of correlation functions in weakly-coupled quantum field
theories is a topic that has been extensively studied and refined in the past
seventy years. There exist many introductions of great quality to this topic
(see for instance [247]), and here we only discuss the important notions
related to the content of this thesis.

One efficient way of studying weakly-coupled systems is to consider the
interacting theory as a small deviation from a free theory, which can be
solved exactly. Such computations are represented by Feynman diagrams, which
are assembled using a set of basic ingredients known as Feynman rules. Here we
focus on ${\mathcal{N}}=4$ SYM and on Yukuwa CFTs, for which Feynman rules are
given respectively in Sections 2.2.1 and 2.3.1. This concept extends
straightforwardly to defect CFTs, in which the coupling to the defect can be
represented by Feynman rules as well. For the models mentioned above, they are
given respectively in Sections 2.2.2 and 2.3.3.

Feynman diagrams give rise to Feynman integrals, for which the difficulty
increases as the order of the coupling (and hence the precision of the
calculation) increases. Such integrals have been studied in great depth, in
particular for deriving precise estimations of the processes observed in high-
energy colliders. Appendix C.2 presents the results for various one- and two-
loop integrals. In the case of defect CFT, we encounter new integrals due to
the coupling with the defect, about which much less is known. There, the
symmetry is often of great help for obtaining a full solution (see for
instance [10]).

Feynman integrals typically contain divergences that need to be accounted for.
As a consequence, operators with anomalous dimensions of the form given in
(2.6) require a renormalization procedure. Expanding the two-point function at
small coupling $g\sim 0$, we have

$\langle\,\mathcal{O}_{\Delta}(x_{1})\mathcal{O}_{\Delta}(x_{2})\,\rangle=\frac{1}{x_{12}^{2\Delta^{(0)}}}\biggl{\\{}1+g\,\gamma_{\Delta}^{(1)}\,\log\frac{\varepsilon^{2}}{x_{12}^{2}}+\ldots\biggr{\\}}\,,$
(2.34)

with $\varepsilon\sim 0$ encoding the logarithmic divergence. Higher orders of
$g$ produce new divergences of the form $\log^{\ell}\varepsilon^{2}$. Note
that we are using in this case the point-splitting regularization scheme,
which consists of taking the limit of two operators approaching each other to
regulate the divergences. This method is explained in more detail in Appendix
C.1.1. To cancel the divergence, we promote $\mathcal{O}_{\Delta}$ to its
renormalized version:

$\mathcal{O}_{\Delta}^{R}(x):=\mathcal{O}_{\Delta}(x)\left\\{1-g\,\gamma_{\Delta}^{(1)}\,\log\frac{\varepsilon^{2}}{\mu^{2}}+\ldots\right\\}\,,$
(2.35)

where $\mu$ corresponds to some choice of scale. The function between the
brackets is referred to as the renormalization constant $Z^{-1}_{\Delta}$.
This results in a finite two-point function:

$\langle\,\mathcal{O}^{R}_{\Delta}(x_{1})\mathcal{O}^{R}_{\Delta}(x_{2})\,\rangle=\frac{1}{x_{12}^{2\Delta^{(0)}}}\biggl{\\{}1+g\,\gamma_{\Delta}^{(1)}\,\log\frac{\mu^{2}}{x_{12}^{2}}+\ldots\biggr{\\}}\,.$
(2.36)

Note that, despite having introduced a scale, this correlation function is
still conformal upon renormalization of the dilatation operator. In the rest
of this work, we drop the superscript as we always refer to renormalized
operators.

In Chapter 5, we study CFTs across dimensions by using dimensional
regularization (see Appendix C.1.1). The renormalization procedure is the same
as for point-splitting, with the difference that the logarithmic divergence is
encoded in inverse powers of $\varepsilon$:

$\langle\,\mathcal{O}_{\Delta}(x_{1})\mathcal{O}_{\Delta}(x_{2})\,\rangle=\frac{1}{x_{12}^{2\Delta^{(0)}}}\biggl{\\{}1-\frac{g}{\varepsilon}\gamma_{\Delta}^{(1)}\,\log\frac{x_{12}^{2}}{\mu^{2}}+\ldots\biggr{\\}}\,,$
(2.37)

where we have absorbed a scale in $g$ to keep it dimensionless.

##### The conformal bootstrap

Solving a conformal field theory exactly remains a challenging task due to the
infinite number of unknowns involved in (2.19) (for the bulk case) or (2.30)
and (2.31) (for the defect case). In recent years, a lot of progress has been
achieved in the computation of correlation functions in the strong-coupling
regime with the analytic bootstrap. Perhaps the biggest accomplishment was the
derivation of the Lorentzian inversion formula in [150], which can be used to
reconstruct the CFT data from the double discontinuity of a four-point
correlator in the following way:

$C_{\Delta,\ell}\sim\int
dz\,d{\bar{z}}\,\mu_{\Delta,\ell}(z,{\bar{z}})\,\operatorname{dDisc}F(z,{\bar{z}})\,,$
(2.38)

where the left-hand side corresponds to the product of OPE coefficients
$\lambda_{\Delta_{1}\Delta_{2}\Delta}\lambda_{\Delta_{3}\Delta_{4}\Delta}$ for
an exchanged operator of scaling dimension $\Delta$ and spin $\ell$, while on
the right-hand side $\mu_{\Delta,\ell}$ is a known kernel related to conformal
blocks. The double discontinuity can be regarded as the absorptive part of the
correlator. This formula is useful because the absorptive part of the
correlator is often simpler than the full correlator. In particular, the
spectrum of exchanged operators is sparse at strong coupling, and using
insights from holography in combination with (2.38) can lead to new strong-
coupling results.

We can go one step further and use (2.38) as an input in (2.19). This leads to
the following dispersion relation for conformal four-point functions:

$F(z,{\bar{z}})=\int
dz^{\prime}\,d{\bar{z}}^{\prime}\,K(z,{\bar{z}},z^{\prime},{\bar{z}}^{\prime})\,\operatorname{dDisc}F(z^{\prime},{\bar{z}}^{\prime})\,,$
(2.39)

where the integration kernel $K(z,{\bar{z}},z^{\prime},{\bar{z}}^{\prime})$
was derived in [263].999See [264] for an alternative formulation with a single
discontinuity. This is analogous to the dispersion relation for the four-
particle scattering amplitude ${\mathcal{M}}(s,t)$, which involves the
discontinuity of ${\mathcal{M}}(s,t)$ and takes the form

${\mathcal{M}}(s,t)\sim\int\frac{dt^{\prime}}{t-t^{\prime}}\operatorname{Disc}{\mathcal{M}}(s,t^{\prime})+(t\leftrightarrow
u)\,,$ (2.40)

where $s$ and $t$ are the Mandelstam variables. The relation (2.39) is
interesting because it gives a direct way to derive the correlator from its
absorptive part, without having to go through the CFT data.

Similar developments have followed for defect CFTs. A defect Lorentzian
inversion formula was derived in [168, 173]. The principle of this relation is
to extract the bulk-defect coefficients presented in (2.31) from the two-point
functions (2.25) in the presence of a defect. This inversion formula takes the
form

$B(\hat{\Delta},s)\sim\int
dr^{\prime}\,dw^{\prime}\,\hat{\mu}(r^{\prime},w^{\prime})\,\operatorname{Disc}F(r^{\prime},w^{\prime})\,,$
(2.41)

where $B(\hat{\Delta},s)$ represents the product
$b_{\Delta_{1}\hat{\Delta}}b_{\Delta_{2}\hat{\Delta}}$ for an exchanged
operator of scaling dimension $\hat{\Delta}$ and spin $s$. Note that a single
discontinuity suffices in this context.

It is therefore natural to expect a dispersion relation for defect
CFT101010The interpretation of such a formula as a dispersion relation should
be taken lightly, as the discontinuity of defect correlators does not have
well-defined positivity properties, and thus it cannot be interpreted as the
absorptive part of a physical process. with a form similar to (2.39) and
(2.40). Such a formula is derived in Chapter 3.2, where it is then used to
obtain novel strong-coupling results in ${\mathcal{N}}=4$ SYM with a Wilson-
line defect.

Analytic bootstrap methods have also been developed for the defect correlators
discussed in Section 2.1.2. In Chapter 4, we use the techniques developed in
[265, 266] for computing a five-point function at strong coupling.

The method can be summarized as follows. The crossing symmetry (2.20)
naturally generalizes to higher-point functions and leads to a network of
relations highly constraining for the correlator. In holographic systems, the
leading order can be readily computed using Witten diagrams and serves as an
input for the bootstrap routine. The next-to-leading order is determined by
assuming a Regge behavior for the anomalous dimensions and using the leading
order as an input. In fortunate cases, the correlator is then fixed by the
crossing symmetry relations. One can then proceed iteratively to extract
higher orders, by reinserting the CFT data of the lower orders. This procedure
becomes however increasingly difficult, due to the fact that operators are
mixed. Indeed, different operators can have the same scaling dimensions at low
orders in the coupling. This mixing must be resolved to progress order by
order. The authors of [266] were able to reach the next-to-next-to-next-to-
leading order (NNNLO) at strong coupling.

###  2.2 The Wilson-line defect CFT

Amidst the vast landscape of conformal field theories, $4d$ ${\mathcal{N}}=4$
Super Yang–Mills is a particularly intriguing model. It is renowned for its
abundance of symmetries, which make it one of the simplest interacting
theories in four dimensions. Its importance in the study of the AdS/CFT
correspondence, integrability, and the (super)conformal bootstrap has earned
it the nickname of hydrogen atom of quantum field theories. A canonical defect
to consider in ${\mathcal{N}}=4$ SYM is the supersymmetric Wilson loop. This
extended operator plays a central role in holography, and as such it has been
the object of intense research. The associated defect CFT preserves many of
the properties of the bulk theory, making it a prime candidate for the study
of defect physics.

In the following, we introduce the bulk theory ${\mathcal{N}}=4$ SYM and
Wilson loops, as well as the resulting Wilson-line defect CFT. Most of the
content presented here relies on [49].

####  2.2.1 ${\mathcal{N}}=4$ Super Yang–Mills

We begin with a short review of the bulk theory ${\mathcal{N}}=4$ SYM. On top
of the properties mentioned above, this model presents the advantage of being
a rare example of a CFT with a coupling constant, and thus it can be studied
both in the weak- and strong-coupling regimes. We are interested in particular
in the large $N$ limit, in which the observables of ${\mathcal{N}}=4$ SYM
correspond to a free $4d$ quantum field theory at weak coupling and to an
integrable $2d$ string worldsheet at strong coupling.111111The notion of weak
and strong coupling is somewhat subtle at large $N$. The appropriate coupling
constant will be discussed in (2.44).

A comprehensive review of supersymmetry can be found for instance in [267,
268, 269]. For an introduction to string theory, we refer the reader to [270,
271, 272, 273].

##### The action

${\mathcal{N}}=4$ SYM corresponds to the maximally extended supersymmetric
theory in four dimensions. In other words, it is the realization of
supersymmetry with the largest number of supercharges $Q^{A}$
($A=1,\ldots,{\mathcal{N}}$) with a multiplet representation of spin $\leq 1$,
i.e., without a graviton. The field content of ${\mathcal{N}}=4$ SYM is the
following:

$\displaystyle\mathbf{1}\text{ gluon }\oplus\,\mathbf{4}\text{ Weyl fermions
}\oplus\,\mathbf{6}\text{ scalars}\,.$

As required by supersymmetry, the bosonic and fermionic degrees of freedom are
equal: eight each. The transformations between bosons and fermions are
generated by the supercharges mentioned above (see for instance [49]).

The action takes the following form:

$\begin{split}S_{{\mathcal{N}}=4}=\ &\frac{1}{g^{2}}\operatorname{tr}\int
d^{4}x\,\biggl{(}\frac{1}{2}F_{\mu\nu}F_{\mu\nu}+D_{\mu}\phi^{I}D_{\mu}\phi^{I}-\frac{1}{2}[\phi^{I},\phi^{J}][\phi^{I},\phi^{J}]\\\
&+i\,{\bar{\psi}}\not{D}\psi+{\bar{\psi}}\Gamma^{I}[\phi^{I}\,,\psi]+\partial_{\mu}{\bar{c}}D_{\mu}c+\xi(\partial_{\mu}A_{\mu})^{2}\biggr{)}\,,\end{split}$
(2.42)

where $\mu=0,\ldots,3$ are the spacetime directions and $I=1,\ldots,6$ are the
$R$-symmetry indices. This notation introduces a single $16$-component
Majorana fermion composed of the $4$ Weyl fermions mentioned above. All the
fields are in the adjoint representation of $SU(N)$ and carry a generator in
their definitions:

$\phi:=T^{a}\phi^{a}\,,\quad\psi:=T^{a}\psi^{a}\,,\quad
A_{\mu}:=T^{a}A_{\mu}^{a}\,,$ (2.43)

where $a=1,\ldots,N$ corresponds to the color index of the $SU(N)$ gauge
group.

The action (2.42) exhibits conformal symmetry at the quantum level, meaning
that the $\beta$-function of the coupling is zero non-
perturbatively.121212There exists another class of vacua for ${\mathcal{N}}=4$
SYM. Here we choose the superconformal phase, where all the vacuum expectation
values of the scalars vanish. The other possibility is the Coulomb phase
[274], where at least one scalar has a non-zero expectation value. This breaks
both conformal symmetry and the gauge group. We will not consider the Coulomb
phase in this thesis. In addition, ${\mathcal{N}}=4$ SYM is UV-complete in
perturbation theory and is invariant under the $S$-duality group
$SL(2,{\mathds{Z}})$, leading to an interesting weak/strong-coupling duality
of the electric/magnetic type and known as Montonen–Olive duality [45, 46,
47]. Although not the main topic of this thesis, $S$-duality will be briefly
discussed in Section 3.5.1.

##### The AdS/CFT correspondence

We are now turning our attention to one of the most fascinating discoveries of
modern theoretical physics: the anti-de Sitter/conformal field theory
(AdS/CFT) correspondence. This duality was discovered by Juan Maldacena in
1997 [44] and relates a quantum field theory on flat spacetime (the CFT) to a
string theory in anti-de Sitter space. This is a remarkable relation since
string theory is a promising candidate framework to describe quantum gravity,
while quantum field theory on flat space does not appear to contain gravity.
The AdS/CFT correspondence implies that these two theories describe the same
physics.

${\mathcal{N}}=4$ SYM was the first realization of the holographic principle
to be unraveled. The holographic principle [43] states that the number of
degrees of freedom of a gravitational theory in a volume $V$ scales as the
surface area $\partial V$ of that volume. In other words, it postulates that
gravity in $d+1$ dimensions can be described by a theory living on the
boundary of that space, i.e., in $d$ dimensions. This is a generalization of
the black hole example provided in Section 1.1.3. In the case of
${\mathcal{N}}=4$ SYM, the CFT is four-dimensional and is dual to a five-
dimensional string theory in AdS space. ${\mathcal{N}}=4$ SYM can therefore be
thought of as being defined on the (conformal) boundary of the AdS space.

The weakest form of the conjecture, which is also the most studied one, is
that the large $N$ limit of ${\mathcal{N}}=4$ SYM at strong coupling is dual
to a type IIB superstring theory in AdS${}_{5}\times S^{5}$ at weak coupling.
In this context, it is convenient to define the ’t Hooft coupling [275]

$\lambda:=g^{2}N\,.$ (2.44)

On the AdS side, the two free parameters are the string coupling $g_{s}$ and
the ratio $L^{2}/\alpha^{\prime}$, where $L$ is the radius of curvature of the
AdS space and $\alpha^{\prime}$ is the square of the string length. The
correspondence establishes the following relations between the parameters of
the field and string theories:

$g^{2}=2\pi g_{s}\,,\quad\lambda=\frac{L^{4}}{2\alpha^{\prime 2}}\,.$ (2.45)

As mentioned above, one important observation is that the AdS/CFT
correspondence is an example of weak/strong-coupling duality:131313Note that,
with respect to $g$, the theory is weakly coupled, even when $\lambda\gg 1$.
when the dual gravitational theory is considered at the classical level, the
quantum field theory is strongly coupled. For this reason, holographic models
are prime examples for studying field theories in the strong-coupling regime.

The supergravity (SUGRA) limit of the string theory is studied perturbatively
through Witten diagrams. In Chapter 3, we use insights from the dual SUGRA
theory to obtain valuable input for our bootstrap computations. More
generally, holographic theories are interesting as they provide additional
handles to study correlation functions in the strong-coupling regime.

##### Feynman rules

The action given in (2.42) generates correlation functions, which can be
computed at weak coupling using the laws of quantum field theory and the
Feynman rules of the action. The (free) propagators take the form

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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.0pt}{23.02953pt}\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\footnotesize\makebox[0.0pt][c]{$\smash{b}$}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.45276pt}{23.02953pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=g^{2}\delta^{ab}I_{12}\,,\end{split}$
(2.46)

where we have introduced the $4d$ scalar propagator

$I_{12}=\frac{1}{4\pi^{2}x_{12}^{2}}\,.$ (2.47)

Note that, in our conventions, free propagators carry the (dimensionless)
coupling $g^{2}$.

The dynamics of a quantum field theory are controlled by its vertices. In
${\mathcal{N}}=4$ SYM, we have eight vertices, which relate to the coupling
constant by a prefactor $g^{-2}$. The cubic vertices are

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