ETD Collection
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Item Supersymmetric quantum mechanics and path integrals(2017) Ayad Mohamed Ali, AhmedSupersymmetry plays a main role in all current thinking about superstring theory. Indeed, many remarkable properties of string theory have been explained using supersymmetry as a tool. In this dissertation, we review the basics formulation of supersymmetric quantum mechanics starting with introducing the concepts of supercharges and superalgebra. We show that, if there is a supersymmetric state, it is the zero-energy ground state. If such a state exists, the supersymmetry is unbroken otherwise it is broken. So far, there has been no unbroken supersymmetry observed in nature, and if nature is described by supersymmetry, it must be broken. In fact, supersymmetry may be broken spontaneously at any order of perturbation theory, or dynamically due to non-perturbative e ects. The goal of this dissertation is to study the methods of supersymmetry breaking. For this purpose, a special attention is given to discuss the normalization of the ground state of the supersymmetric harmonic oscillator. Then we explain that perturbation theory gives us incorrect results for both the ground state wave function as well as the energy spectrum and it fails to give an explanation to the supersymmetry breaking. Later in the dissertation, a review of the uses of instantons in quantum mechanics is given. In particular, instantons are used to compute the tunneling e ects within the path integral approach to quantum mechanics. As a result, we give evidence that the instantons, which are a non-perturbative e ect in quantum mechanics and can not be seen in perturbation theory, leads to calculate the corrections to the ground state energy and provides a possible explanation for the supersymmetry breaking.Item Computational methods in string and field theory(2018) Pontiggia, Luca TerzioLike any field or topic of research, significant advancements can be made with increasing computational power - string theory is no exception. In this thesis, an analysis is performed within three areas: Calabi–Yau manifolds, cosmological inflation and application of conformal field theory. Critical superstring theory is a ten dimensional theory. Four of the dimensions refer to the spacetime dimensions we see in nature. To account for the remaining six, Calabi-Yau manifolds are used. Knowing how the space of Calabi-Yau manifolds is distributed gives valuable insight into the compactification process. Using computational modeling and statistical analysis, previously unseen patterns of the distribution of the Hodge numbers are found. In particular, patterns in frequencies exhibit striking new patterns - pseudo-Voigt and Planckian distributions with high confidence and exact fits for many substructures. The patterns indicate typicality within the landscape of Calabi–Yau manifolds of various dimensions. Inflation describes the exponential expansion of the universe after the Big Bang. Finding a successful theory of inflation centres around building a potential of the inflationary field, such that it satisfies the slow-roll conditions. The numerous ways this can be done, coupled with the fact that each model is highly sensitive to initial conditions, means an analytic approach is often not feasible. To bypass this, a statistical analysis of a landscape of thousands of random single and multifield polynomial potentials is performed. Investigation of the single field case illustrates a window in which the potentials satisfy the slow-roll conditions. When there are two scalar fields, it is found that the probability depends on the choice of distribution for the coefficients. A uniform distribution yields a 0.05% probability of finding a suitable minimum in the random potential whereas a maximum entropy distribution yields a 0.1% probability. The benefit of developing computational tools extends into the interdisciplinary study between conformal field theory and the theory of how wildfires propagate. Using the two dimensional Ising model as a basis of inspiration, computational methods of analyzing how fires propagate provide a new tool set which aids in the process of both modeling large scale wildfires as well as describing the emergent scale invariant structure of these fires. By computing the two point and three point correlations of fire occurrences in particular regions within Botswana and Kazakhstan, it is shown that this proposed model gives excellent fits, with the model amplitude being directly proportional to the total burn area of a particular year.Item Gravitational description of the conformally invariant quantum mechanics of large matrices(2017) Hanmer, Jeffrey ThomasWe study the collective field theory of a free multi-matrix model in the radial sector, which has an emergent 1/r2 term, and take the large N limit. We show that it is possible to generate 2−d metrics with generic dependence on the collective field Lagrange multiplier (μ) and potential and which are distinguished by the choice of the potential. The Lagrange multiplier is shown to depend on an induced scale parameter after an I.R. regularization and breaks scale invariance. The collective field sl(2, R) algebras of the free Hamiltonian and a related alternative compact operator only close in the absence of μ. We point out that the broken conformal symmetry is contained in the associated metrics which suggests that they are related to a Near-AdS2 geometry. We also comment on the resemblance of these metrics to black hole solutions.Item Holographic descriptions of CFT scattering(2017) Shrif, Esra Mohammed Shrif Mohammed Salih MohammedThe holographic computation of extremal correlators is often frustrated by divergences. The interpretation of these divergences is incomplete. The primary goal of this study is to develop a better understanding of these divergences. Towards this end, working within the AdS/CFT correspondence we review the computation of correlators. In the field theory we review well known matrix model techniques useful to study the planar limit, as well as methods exploiting group representation theory that are useful for the computation of correlators in large N but non-planar limits. On the gravity side of the correspondence, we describe in detail the computation of two- and three point functions of a scalar field coupled to gravity on the Euclidian AdSd+1 space, three-point functions of two giant gravitons and one pointlike graviton as well as correlators of Kaluza-Klein gravitons. A key observation of this study is that extremal correlators are mapped to scattering amplitudes of particles with parallel momenta. These are naturally accompanied by involve collinear divergences. Therefore, we suggest that the divergences in the computation of extremal correlators are linked to collinear divergences. A lot more work is needed to establish this connection.Item Investigation of higher spin theory through holography and mathematical methods in free conformal field theory(2017) Rabe, RandleIn this dissertation, we will develop novel methods for determining the spectrum of primary operators in a free conformal eld theory (vector model) and to construct some of these primaries. To count the spectrum of primaries, we use group theoretic techniques to obtain character formulas for any number of elds as representations of SO(4; 2). More precisely, we will construct generating functions that can be expanded to any order in the conformal scaling dimension to yield the complete spectrum of primaries constructed out of n scalar elds. We also develop e cient methods to construct these primaries by using a polynomial description. Finally, these primary operators, which are higher spin currents in the free conformal vector model, correspond to higher spin gauge elds in Vasiliev higher spin theory through the holographic duality.Item Large N conformal field theory from gauge/gravity duality(2017) Hasina Tahiridimbisoa, Nirina MauriceIn this dissertation we exploit the Ads slash CFT correspondence to describe a system of strings suspended between giant gravitons. The strings can be in an excited state. The excitations of the strings can be given a particle-like description and are known as magnons. The proposed gauge invariant operators used to construct a complete description of this system belong to the su(2) sector of the N = 4 SYM. Using an open spin chain description of the suspended strings, the states of the system we consider enjoy an SU 2j2)2 symmetry. By making use of this symmetry, we compute the all loop anomalous dimensions of these operators. The spectrum of the dilatation operator in the su(2) sector of the theory is reproduced in the dual gravity description. In the dual theory, the energies of the magnons are computed using strings in a background LLM geometry and the results are in complete agreement with the anomalous dimensions of the operators we have considered. Using the symmetries enjoyed by our system we achieve a complete determination - up to an overall phase - of the reection/scattering matrix between a boundary magnon and a bulk magnon. Thus, although the open boundary conditions of the spin chain spoil integrability. The two-loop subleading correction to the dilatation operator is also explored. This subleading term corresponds to a correction of the magnon energies. The computation of this subleading term requires consideration of the giant's backreaction on their excitations. We nd that this backreaction implies a nontrivial mixing of the dual operators and this mixing is characterized completely.Item Spectra of the excited giant gravitons from the two loop dilatation operator(2016-09-19) Ali, Abdelhamid Mohamed AdamThe AdS/CFT correspondence is a conjectured exact duality between type IIB string theory on the AdS5 S5 background and N = 4 Super Yang-Mills theory, a conformal eld theory (CFT), on the boundary of the AdS space. A speci c observable of the CFT, which can be read from the two point correlation function, is the anomalous dimension. In this dissertation we will compute spectra of anomalous dimensions of excited giant gravitons up to two loops in a speci c limit. We are interested in the anomalous dimensions because the AdS/CFT correspondence associates them with energies of states in quantum gravity. We study operators constructed using n Z elds and m Y elds with n << m: In this case m n is a small parameter. At the leading order in m n and at large N, the problem of determining the anomalous dimensions can be mapped into the dynamics of m non-interacting magnons. The subleading terms at two loops, computed for the rst time in this dissertation, induce interactions between the magnons. Even after including this new correction, we nd the BPS operators remain BPS.Item A non-perturbative theory of giant gravitons using AdS/CFT(2015-05-07) Kemp, Garreth JamesWe explore the non-perturbative physics of giant gravitons in type IIB string theory on the AdS5 ⇥ S5 background in this thesis. The gauge theory dual is N = 4 super Yang-Mills theory with a U(N) gauge group. We diagonalise the one and two-loop dilatation operators acting on the restricted Schur polynomial basis. These operators are dual to a system of giant gravitons with strings attached. Hence, we present evidence for integrability in certain non-planar sectors of the gauge theory. In the second half of the thesis, we turn our focus to N = 4 super Yang-Mills theory with an SO(N) gauge group. In this case, the geometry of the dual gravity theory is AdS5 ⇥RP5. The non-planar physics of the SO(N) theory is distinct from that of the U(N) theory. To pursue the goal of searching for non-planar integrability in the SO(N) gauge theory, one might try to generalise the restricted Schur basis to the SO(N) case. We propose such a basis and evaluate their two-point functions exactly in the free theory. Further, we develop techniques to compute correlation functions of multi-trace operators involving two scalar fields exactly. Lastly, we extend these results to the theory with an Sp(N) gauge group.Item Non-perturbative string theory from the gauge/gravity correspondence(2015-01-29) Graham, StuartABSTRACT In this dissertation we study the action of the one loop dilatation operator on operators with a classical dimension of order N. We consider the su(3) and su(2) sectors. The operators in the su(3) sector are constructed using three complex fields X, Y and Z, while operators in the su(2) sector are constructed from only the two complex fields Y and Z. For the operators in these sectors non-planar diagrams contribute already at the leading order in N and the planar and large N limits are distinct. Although the spectrum of anomalous dimensions in su(3) has been computed for this class of operators, previous studies have neglected certain terms which were argued to be small. After dropping these terms diagonalizing the dilatation operator reduces to diagonalizing a set of decoupled oscillators. In this dissertation we explicitly compute the terms which were neglected previously and show that diagonalizing the dilatation operator still reduces to diagonalizing a set of decoupled oscillators. In the su(2) sector the action of the one loop and the two loop dilatation operator reduces to a set of decoupled oscillators and factorizes into an action on the Z fields and an action on the Y fields. Direct computation has shown that the action on the Y fields is the same at one and two loops. In this dissertation, using the su(2) symmetry algebra as well as structural features of field theory, we give compelling evidence that the factor in the dilatation operator that acts on the Y s is given by the one loop expression, at any loop order. I hereby declare that the content of this dissertation is based on my following original works: • R. de Mello Koch, S. Graham and W. Mabanga, “Subleading corrections to the Double Coset Ansatz preserve integrability” (2013) [arXiv:1312.6230v1 [hep-th]] • R. de Mello Koch, S. Graham and I. Messamah, “Higher Loop Nonplanar Anomalous Dimensions from Symmetry” (2013) [arXiv:1312.6227v1 [hep-th]].Item The large-N limit of matrix models and AdS/CFT(2014-06-12) Mulokwe, MbavhaleloRandom matrix models have found numerous applications in both Theoretical Physics and Mathematics. In the gauge-gravity duality, for example, the dynamics of the half- BPS sector can be fully described by the holomorphic sector of a single complex matrix model. In this thesis, we study the large-N limit of multi-matrix models at strong-coupling. In particular, we explore the significance of rescaling the matrix fields. In order to investigate this, we consider the matrix quantum mechanics of a single Hermitian system with a quartic interaction. We “compactify” this system on a circle and compute the first-order perturbation theory correction to the ground-state energy. The exact ground-state energy is obtained using the Das-Jevicki-Sakita Collective Field Theory approach. We then discuss the multi-matrix model that results from the compactification of the Higgs sector of N = 4 SYM on S4 (or T S3). For the radial subsector, the saddle-point equations are solved exactly and hence the radial density of eigenvalues for an arbitrary number of even Hermitian matrices is obtained. The single complex matrix model is parametrized in terms of the matrix valued polar coordinates and the first-order perturbation theory density of eigenstates is obtained. We make use of the Harish-Chandra- Itzykson-Zuber (HCIZ) formula to write down the exact saddle-point equations. We then give a complementary approach - based on the Dyson-Schwinger (loop) equations formalism - to the saddle-point method. We reproduce the results obtained for the radial (single matrix) subsector. The two-matrix integral does not close on the original set of variables and thus we map the system onto an auxiliary Penner-type two matrix model. In the absence of a logarithmic potential we derive a radial hemispherical density of eigenvalues. The system is regulated with a logarithm potential, and the Dobroliubov-Makeenko-Semenoff (DMS) loop equations yield an equation of third degree that is satisfied by the generating function. This equation is solved at strong coupling and, accordingly, we obtain the radial density of eigenvalues.