ETD Collection
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Item A bilocal description of the conformal algebra at the critical point in 3 dimensions(2021) Johnson, Celeste IreneKlebanov and Polyakov conjectured in 2002 a duality between O(N) vector models and Vasiliev higher spin theory, providing a useful probe of the AdS/CFT Correspondence[1]. When considering this duality in d = 2 + 1 dimensions, the free 3d O(N) vector model is dual to type A minimal Vasiliev higher spin gauge theory with a = 1 scalar at the boundary, and the critical 3d O(N) vector model is dual to type A minimal Vasiliev higher spin theory with scalar having = 2. In particular, in the presence of a relevant interaction, the theory ows from an unstable UV xed point with a scalar of dimension = 1 to an IR xed point where = 2. Strong evidence for this conjectured duality was provided by Giombi and Yin around 2010 where they performed explicit tree level and one-loop calculations. For example they showed that at tree level, the three-point functions on both sides of the conjecture match for both = 1 and = 2, and that the one-loop 3-sphere free energy on both sides of the conjecture agree[2, 3, 4]. Further evidence has been the reconstruction of the bulk by de Mello Koch, Jevicki, Jin and Rodrigues[5] (for another construction see Ref. [6]). In 2018 Mulokwe and Rodrigues showed that, making use of a bilocal eld approach (taking a 1=N expansion) to investigate a three-dimensional O(N) invariant bosonic model with N (aa)2 interaction at the critical point / infrared xed point, there was indeed a state identi ed to correspond to a = 2 scalar state; the = 1 state was found to vanish from the spectrum in agreement with Polyakov and Klebanov[7]. In this thesis, we make use of Collective eld theory, where bilocals are used to encode the invariance of the theory explicitly so that it is the invariant variables which are described by the theory[8]. We build on the constructive approach developed by de Mello Koch, Jevicki, Jin, Rodrigues and Yoon between 2010 and 2015 in both the light-cone and temporal gauge for the free theory, wherein an explicit map between the conformal eld theory in d = 2 + 1 dimensions and the higher spin theory in AdS4 S1 was established[5, 9, 10]. In the Hamiltonian approach, the 1 + 2 + 2 = 5 coordinates of the equal time bilocals map (in phase space) to the coordinates of AdS4 S1, and in this thesis we investigate the applicability of this map to the interacting theory. Using the Hamiltonian approach in time-like gauge, it is found that the quartic interaction contributes linearly in the bilocal elductuation equations so that the spectrum problem is that of a potential scattering problem. The scattering state solution takes a universal form at the critical point. Using a change of variables from bilocal momenta to bulk momenta (as dictated by the map), and instituting a eld rede nition to de ne bulk higher spin elds, we are able to obtain a bulk description of these boundary scattering states. It is rather remarkably shown directly in the bulk that for the interacting theory, the s = 0 state (corresponding to the = 0 + 1 = 1 state) is precisely removed in agreement with Ref. [7]. It is then shown that this approach is equivalent to that taken in Ref. [10], by showing that the bulk algebra agrees with that found in [10] both at the free and interacting critical point.Item Construction of the emergent Yang-Mills theory(2020) De Carvalho, ShaunIn this thesis, we focus on the construction of the emergent Yang-Mills theory that is expected to arise at low energy on the world volume of a giant graviton. Our basic approach is to study the operators in N = 4 super Yang-Mills theory dual to excited giant graviton states. The system we work with consists of giant gravitons and open strings connecting between them. They are described in the language of both restricted Schur polynomials and Gauss graph operators, and we study the action of the one loop dilatation operator at large N, including the leading and rst subleading terms, in the SU(3) sector. The construction of these operators and computations with them requires sophisticated methods from group representation theory, as well as basic ideas from quantum eld theory. Consequently the thesis begins with a careful review of exactly the background that is required. We will consider operators that are a small deformation of a 1 2-BPS multi-giant graviton state. Our rst novel result proves that the subleading matrix elements of the dilatation operator can be interpreted in terms of bosons hopping on a lattice. In this way, we are able to diagonalize the dilatation operator at subleading order. The description and computations are technically challenging, which motivated us to pursue a symmetry based approach to the problem. The second novel result achieved in this thesis, enables us to decompose the state space of excited gaint graviton branes (the Gauss graph operators) into irreducible representations of the su(2j2) global symmetry. As explained by Beisert, this algebra admits central extensions. We argue that in our non-planar setting the global symmetry again is centrally extended, with the charges naturally describing gauge transformations of the emergent gauge theory. Gauge invariance forces these charges to vanish so that we end up with the physically expected result: the full global su(2j2) symmetry is not centrally extended.Item Algebraic structures in the counting and construction of primary operators in free conformal field theory(2018) Rabambi, Phumudzo TeflonThe AdS/CFT correspondence relates conformal eld theories in d dimensions to theories of quantum gravity, on negatively curved spacetimes in d+1 dimensions. The correspondence holds even for free CFTs which are dual to higher spin theories. Motivated by this duality, we consider a systematic study of primary operators in free CFTs. We devise an algorithm to derive a general counting formula for primary operators constructed from n copies of a scalar eld in a 4 dimensional free conformal eld theory (CFT4). This algorithm is extended to derive a counting formula for fermionic elds (spinors), O(N) vector models and matrix models. Using a duality between primary operators and multi-variable polynomials, the problem of constructing primary operators is translated into solving for multi-variable polynomials that obey a number of algebraic and di erential constraints. We identify a sector of holomorphic primary operators which obey extremality conditions. The operators correspond to polynomial functions on permutation orbifolds. These extremal counting of primary operators leads to palindromic Hilbert series, which indicates they are isomorphic to the ring of functions de ned on speci c Calabi-Yau orbifolds. The class of primary operators counted and constructed here generalize previous studies of primary operators. The data determining a CFT is the spectrum of primary operators and the OPE coe cients. In this thesis we have determined the complete spectrum of primary operators in free CFT in 4 dimensions. This data may play a role in attempts to give a derivation of a holographic dual to CFT4. Another possible application of our results concern recent studies of the epsilon expansion, which relates explicit data of the combinatorics of primary elds and OPE coe cients to anomalous dimensions of an interacting xed pointItem Emergent Yang-Mills theory(2017) De Carvalho, ShaunIn this dissertation we tackle the question: is there an emergent Yang-Mills theory coming from the low energy description of branes and open strings? This new Yang-Mills theory has no connection to the original gauge symmetry of the CFT. We thus explore a large N but non-planar limit of the theory. This is done with new methods developed in group representation theory. A study the dilatation operator D in N = 4 SYM theory is done since its eigenvalue, the anomalous dimension, is mapped to the energy of the open string in the IIB string theory. The construction of the spherical harmonics from the harmonic expansion on the 3-sphere, S3, is done to understand the theory of the giant graviton's worldvolume. The light-front parton picture is examined, since it explains how one can \glue" single momentum modes together to obtain higher momentum modes, and we believe that this procedure is described dynamically using magnon bound states. Following from this, we work on determining the exact magnon bound state spectrum. Finally, we test our hypothesis and see if the spectrum of the bound states matches the harmonic spectrum from the harmonic expansion on the 3-sphere, S3. A non-trivial check is also performed to show that the bound state spectrum does indeed match the spectrum coming from N = 4 SYM.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 Bilocal bosonization of nonrelativistic fermions in d dimensions(2016-08-18) Braemhoej, Juliet Diana