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Imagine that we lived in a world much like our own, with many cultures and languages, but lacking any means of translation. Different communities would be unable to exchange any information when they encounter each other. We would never learn to be humble from the story of Daedalus and Icarus, we would never get inspired to be as heroic as Hercules and we would never grasp the profound life lessons of Dostoevsky. In some sense, this artificial world portrays the landscape of theoretical physics established at the end of the 20th century. Physics of the 20th century is marked by two intellectual revolutions. On the one hand, the famous theory of general relativity, formulated by Einstein, provided a new description of space, time and gravity. Einstein’s theory allows us to understand phenomena on the scale of large and heavy objects, such as planets, stars, and galaxies. On the other hand, during the previous century, it was observed that a quantum nature of reality emerges once we probe Nature at the smallest possible length scale, even smaller than the atom. The subsequent development of particle physics required a new theoretical framework, known to physicists as quantum field theory, in order to explain this behaviour. These two theories, general relativity and quantum field theory, are both formulated in their own language and, like the hypothetical world we fantasized above, lack any means to talk to each other. This is a conceptual problem in theoretical physics, as some situations require us to deal with large matter densities at small scales. For example, at the dawn of our universe, an event known as the Big Bang, the entire observable universe was compressed into a size comparable to that of an atom. Describing this ‘primeval atom’ is impossible with general relativity or quantum field theory alone. Hence, we need a new theory, called quantum gravity, to provide us with an accurate description of such phenomena. Currently, the most viable candidate for quantum gravity is string theory. Out of studies of string theory came a remarkable discovery, which plays the role of the deus ex machina in our story. That is, string theory brought forward a dictionary that allows us to translate expressions from the language of gravity theories to the language of quantum field theory and vice versa. This link, known as the AdS/CFT correspondence, gives physicists new computational tools that provide a window into the phenomena of socalled strongly coupled quantum field theories. The term ‘strong coupling’ signifies that it is hard or even impossible to do computations in the theory. An exciting feature of the AdS/CFT dictionary is that it translates these ‘hard problems’ of a quantum field theory into ‘easier problems’ formulated in a gravity theory, enabling us to compute and solve these problems. In this thesis, we explore applications of this interesting aspect of the AdS/CFT dictionary. In particular, we gain insight on the dependence of strongly coupled quantum field theories on the energy scale with the help of calculations performed in a gravity theory.

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It’s a well-known fact: at a certain temperature ice melts, water boils,… In other words, at certain temperatures, matter goes from one form into another. We can ask general questions as what does this temperature depend on? Can we describe this type of phenomenon in a general framework? The general framework is given by phase transitions. Boiling water is normally a first-order phase transition, meaning we have to put in extra energy to make the water boil. This extra energy is called latent heat. For boiling water, the temperature it starts to boil depends on the pressure. At a certain pressure, the latent heat is zero. The temperature it boils at these pressure is called the critical point. Near this critical point the average density and the correlation of the density at two different points in space scale very specifically with temperature or pressure. The exponents of these scaling are called the critical exponents. It turns out these critical exponents are universal; they are the same for a wide class of physical systems. For example, the critical exponents for our boiling water example and a ferromagnetic material are the same. These critical exponents can be measured, but how to calculate them theoretically? To solve this question, we have to look at the underlying theory. The underlying physics for a system near criticality is that the system is described in the same manner no matter how closely we look. This behaviour is called self-similarity. A physical theory showing self-similarity is called a conformal field theory. A central object of study in conformal field theory is the 4-point correlation function. This object is like the two-point function that describes the correlation of observables at two different points in space, but now for four different points in space. In a general quantum field theory this object is an immensely complicated object, but the self-similarity behaviour of a conformal field theory simplifies a great deal, because a theory showing self-similarity must obey a lot of symmetries, and the existence of a symmetry restricts how a correlation function looks like. Additionally, it turns out that the form of a four-point function in conformal field theory puts constraints on the critical exponents of before. The method to do this is called the conformal bootstrap: it combines the techniques of quantum field theory with the use of symmetries to derive bounds on the critical exponents. In this thesis, I considered the building blocks for the conformal bootstrap, the so-called conformal blocks. They are entirely fixed by the symmetries of a conformal field theory and thus do not depend on a specific theory. In two dimensions, the conformal symmetry is larger and the conformal blocks are therefore harder to derive. It is however possible to write approximate expressions for these blocks. We can also imagine making the theory even more symmetric than merely conformal symmetry, combining the symmetries of a conformal field theory with supersymmetry. Supersymmetry has been observed in condensed matter systems. In this case it is also harder to derive blocks, but it is possible to derive exact results. The reward is that bounds on the critical exponents are stronger.

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In our everyday life, there are two theories which describe the physics around us. General relativity describes how gravity affects our surroundings. The Standard Model, a quantum field theory, describes the other three fundamental forces: the weak and strong nuclear interactions and the electromagnetic interaction. These two theories act in different regimes of physics. General relativity is important for objects with a very big mass, such as stars and planets, and in quantum field theory one considers physics at very small scales, the scale of electrons and quarks. Here on Earth these two regimes are indeed separate, but there are certain cases in which they overlap. One of them is the Big Bang - back when our Universe was contained in a tiny space. The other is a black hole - an extremely massive but very small object. In these special cases, a physical theory is needed that simultaneously describes the four fundamental forces, a theory of quantum gravity. At the moment, string theory is the leading candidate for such a quantum gravity theory. There is not just one string theory, but there are multiple versions of it. These different string theories are all captured by a single overarching theory: M-theory. Surprisingly, M-theory can only be constructed in eleven dimensions. This may sound absurd - we only observe four dimensions (three space dimensions and one time), certainly not eleven - but it is possible that these extra dimensions exist, only we cannot see them. The extra dimensions are just too small to be observed. Although black holes fit in the formalism of string theory, there are still some unsolved questions. One very important question is related to the entropy of a black hole. Entropy measures how many possible states a system has, and it turns out that black holes have a very high entropy. However, it was also shown that (classically) black holes have a very limited amount of possibilities, so there are almost no possible states. This is clearly a paradox, and string theory will hopefully resolve this problem. A very useful tool offered to us by string theory is the AdS/CFT correspondence, which gives a relation between two theories. On one side is a gravitational theory, and on the other a quantum field theory in one dimension less. This is very convenient when studying black hole entropy, because a black hole in such a gravity theory corresponds to something without gravity. And there, the states of a solution are clear. Although the gravity theories in question are not realistic in the sense that they could describe our Universe, we can still learn a lot from them. In this thesis, I consider one of these AdS/CFT correspondences, derived from M-theory, and I “prepare” black holes so that they are ready to be analyzed using the correspondence. The black holes are originally five-dimensional, but are uplifted first to seven and then to eleven dimensions. They are then expressed in terms of objects living in M-theory called M5-branes. These M5-branes are similar to magnetic monopoles in electromagnetism, only they are not point particles but five-dimensional things. The black hole solution is made from these objects, and its entropy can be expressed in terms of quantities like the number of M5-branes and how these branes are “wrapped around themselves”. The black holes are then ready to be analyzed with the AdS/CFT correspondence, but this is left for future research.

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Quantum field theory is a very powerful theoretical framework used in physics to describe interactions between particles. Physicist have a whole set of mathematical tools to make calculations in quantum field theory. However, there is a class of quantum field theories, called strongly interacting field theories, where the interaction between particles is so intense that these mathematical tools brake down, making it rather difficult to study them. Famous topics related to strongly interacting systems include the strong nuclear interaction, high-energy superconductivity, and black holes. In this thesis we are interested in a strongly coupled field theory in three dimensions. In addition, we want this theory to have supersymmetry. Supersymmetry is a symmetry of spacetime between two types of particles, bosons and fermions. Concretely, this means that each boson in our theory has a fermion partner and vice versa. These types of strongly interacting supersymmetric theories are relevant for the description of some condensed matter systems. The goal of this thesis is to see what happens to our theory in the low-energy limit. This is achieved by a renormalization procedure in 4-varepsilon dimensions, which gets rid of the high-energy part of our theory. After this procedure is done the low-energy quantities will be expressed as a function of varepsilon. In the end we put varepsilon = 1 to obtain results for three dimensions.

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In the second half of the previous century, there were two important developments in theoretical physics. The first was the Standard Model of theoretical physics, describing three of the fundamental four forces of nature. One of these three is the strong force described by quantum chromodynamics. Because this theory is strongly coupled, it is hard to do calculations in this framework, since most techniques of quantum field theory work only for weak interactions. The other is the development of possible theories of quantum gravity describing possible unifications of quantum field theory and general relativity. Perhaps the most popular candidate is string theory. However, this theory also presents its own problems as it is hard to treat string theory on curved spaces quantum mechanically. A promising new discovery in this respect that could provide insights into the problems present in both these developments is the AdS/CFT conjecture. This conjecture relates conformal field theories to string theories on a space of which a part resembles an Anti de Sitter space.( a well-known space in theoretical physics) Conformal field theories are strongly coupled quantum field theories which are invariant under a change of scale. While these are not exactly related to quantum chromodynamics, they are very important in physics, and more information on quantum field theory can be gathered by a deeper understanding of these theories. Thus, this conjecture can be used to relate calculations which might be hard on one side of the correspondence to some on the other side which could be much less cumbersome. A possible way to learn more about both sides of this correspondence is through deformations of a known solution on one side. For example, if we know a deformation of a conformal field theory which gives us a family of other conformal field theories, we may translate it to the other side of the conjecture, and get a deformation of the original AdS solution of string theory. This is analysed in this project for two possible deformations. The first is a deformation known on the conformal field theory side, called the β-deformation. On the gravity side, we get a transformation known as the TsT transformation. Another is a deformation relating conformal field theories in different dimensions. These are combined in this project.