<p>In this thesis we study aspects of (0,2) superconformal field theories (SCFTs), which are suitable for compactification of the heterotic string. In the first part, we study a class of (2,2) SCFTs obtained by fibering a Landau-Ginzburg (LG) orbifold CFT over a compact K\"ahler base manifold. While such models are naturally obtained as phases in a gauged linear sigma model (GLSM), our construction is independent of such an embedding. We discuss the general properties of such theories and present a technique to study the massless spectrum of the associated heterotic compactification. We test the validity of our method by applying it to hybrid phases of GLSMs and comparing spectra among the phases. In the second part, we turn to the study of the role of accidental symmetries in two-dimensional (0,2) SCFTs obtained by RG flow from (0,2) LG theories. These accidental symmetries are ubiquitous, and, unlike in the case of (2,2) theories, their identification is key to correctly identifying the IR fixed point and its properties. We develop a number of tools that help to identify such accidental symmetries in the context of (0,2) LG models and provide a conjecture for a toric structure of the SCFT moduli space in a large class of models. In the final part, we study the stability of heterotic compactifications described by (0,2) GLSMs with respect to worldsheet instanton corrections to the space-time superpotential following the work of Beasley and Witten. We show that generic models elude the vanishing theorem proved there, and may not determine supersymmetric heterotic vacua. We then construct a subclass of GLSMs for which a vanishing theorem holds.</p> / Dissertation

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016. / The convenient pictorial descriptions of the half-BPS and near-BPS sectors of the AdS=CFT equivalent theories of N = 4, D = 4 super Yang-Mills and D = 10 Type IIB superstring theory on AdS5 S5 are exploited in this thesis by using Schur polynomials labelled by Young diagrams as a basis for the gauge invariant operators in the eld theory. We use a \Fourier transform" on these operators to construct asymptotic eigenstates of the dilatation operator, the spectrum of which agrees precisely with the rst two leading order terms in the smallcoupling expansion of the exact result determined by symmetry. Motivated by the geometric description of the systems of open strings with magnon excitations to which the operators are dual, we propose a simple and minimal all-loop expression that interpolates between anomalous dimensions computed in the gauge theory and energies computed in the string theory. The connection to the string theory result provides the insight necessary to understand the interpretation of our Gauss graphs in the magnon language. Symmetry determines the two-body scattering matrix for the magnons up to a phase, and it is demonstrated that integrability is spoiled by the boundary conditions on the open strings. The Schur polynomial construction is then applied to the study of closed strings on a class of half- BPS excitations of the AdS5 S5 background. The string theory predictions for the magnon energies are again reproduced by calculating the anomalous dimensions of particular linear combinations of our operators. Group theoretic quantities which can be read o the Young diagram labels provide the correct modi cation of terms in the dilatation action to account for the energies of magnons at di erent radii on the LLM plane. The representation theory implies a natural splitting of the full symmetry group - the distinction between what is the background and what is the excitation is accomplished in the choice of the subgroup and representations used to construct the operator. Connecting the descriptions utilised in obtaining these results is expected to allow the construction of operators dual to general open string con gurations on the class of backgrounds considered. / GR 2016

In this thesis we explore the finite N spectrum of BPS operators in four-dimensional supersymmetric conformal field theories (CFT), which have dual AdS gravitational descriptions. In the first part we analyze the spectrum of chiral operators in the free limit of quiver gauge theories. We find explicit counting formulas at finite N for arbitrary quivers, construct an orthogonal basis in the free inner product, and derive the chiral ring structure constants. In order to deal with arbitrarily complicated quivers, we develop convenient diagrammatic techniques: the results are expressed by associating Young diagrams and Littlewood-Richardson coefficients to modifications of the original quiver. We develop the notion of a "quiver character", which is a generalization of the symmetric group character, obeying analogous orthogonality properties. In the second part we analyze how the BPS spectrum changes at weak coupling, focusing on the N = 4 supersymmetric Yang-Mills. We find a formal expression for the complete set of eighth-BPS operators at finite N, and use it to derive corrections to a near-BPS operator. In the third part of this thesis we move on to the strong coupling regime, where the dual gravitational description applies. The BPS spectrum on the gravity side includes D3-branes wrapping arbitrary holomorphic surfaces, a generalization of the spherical giant gravitons. Quantizing this moduli space gives a Hilbert space, which, via duality and nonrenormalization theorems, should map to the space of BPS operators derived in the weak coupling regime. We apply techniques from fuzzy geometry to study this correspondence between D3-brane geometries, quantum states, and BPS operators in field theory.

Quiver gauge theories are widely studied in the context of AdS/CFT, which establishes a correspondence between CFTs and string theories. CFTs in turn offer a map between quantum states and Gauge Invariant Operators (GIOs). This thesis presents results on the counting and correlators of holomorphic GIOs in quiver gauge theories with flavour symmetries, in the zero coupling limit. We first give a prescription to build a basis of holomorphic matrix invariants, labelled by representation theory data. A fi nite N counting function of these GIOs is then given in terms of Littlewood-Richardson coefficients. In the large N limit, the generating function simpli fies to an in finite product of determinants, which depend only on the weighted adjacency matrix associated with the quiver. The building block of this product has a counting interpretation by itself, expressed in terms of words formed by partially commuting letters associated with closed loops in the quiver. This is a new relation between counting problems in gauge theory and the Cartier-Foata monoid. We compute the free fi eld two and three point functions of the matrix invariants. These have a non-trivial dependence on the structure of the operators and on the ranks of the gauge and flavour symmetries: our results are exact in the ranks, and their expansions contain information beyond the planar limit. We introduce a class of permutation centraliser algebras, which give a precise characterisation of the minimal set of charges needed to distinguish arbitrary matrix invariants. For the two-matrix model, the relevant non-commutative algebra is parametrised by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators. The structure of the algebra, notably its dimension, its centre and its maximally commuting sub-algebra, is related to Littlewood-Richardson numbers for composing Young diagrams.

In this thesis we exploit the setup of AdS3/CFT2 holography, and in particular the D1D5 two-dimensional CFT, to describe states dual to geometries relevant for the \fuzzball" proposal for the description of six-dimensional black hole microstates. Precise holographic dualities between CFT and bulk geometric objects are established and checked, both for 2 and 3-charge states. In particular, VEVs of CFT operators of small conformal dimension are checked to encode deviations from AdS3 geometry near the spacetime boundary. 4-point functions of the \heavy-heavy-light-light" type are also considered and matching is found between CFT and bulk computations via the usual AdS/CFT prescription, with the heavy states being dual to (simple) microstate geometries. In this context, the issue of the presence of spurious singularities at leading order in the large N limit is assessed and cancellations are found even without considering sub-leading corrections, at the cost of considering the full detail of the D1D5 CFT (i.e. including the Virasoro blocks of operators of small dimension charged under the internal SU(2)L SU(2)R R-symmetry group). Finally, more complicated 4-point functions, involving operators in the twisted sector of the CFT, are computed and the results are checked against known results in the literature with the aim of verifying the robustness of the (new) techniques used. Supersymmetric Ward identities are also derived, and checked for some cases, between correlators written in terms of bosons and in terms of fermions.

This thesis considers various extensions to the fluid/gravity correspondence as well as problems fundamental to the study of fluid dynamics. The fluid/gravity correspondence is a map between the solutions of the Navier-Stokes equations of fluid dynamics and the solutions of the Einstein equations in one higher spatial dimension. This map arose within the context of string theory and holography and is a specific realisation of a much wider class of dualities known as the Anti de Sitter/Conformal Field Theory (AdS/CFT) correspondence. The first chapter is an introduction; the second chapter reviews the fluid/gravity correspondence. The next two chapters extend existing work on the fluid/gravity map. Our first result concerns the fluid/gravity map for forced fluid dynamics in arbitrary spacetime dimensions. Forced fluid flows are of particular interest as they are known to demonstrate turbulent behaviour. For the case of a fluid with a dilaton-dependent forcing term, we present explicit expressions for the dual bulk metric, the fluid dynamical stress tensor and Lagrangian to second order in boundary spacetime derivatives. Our second result concerns fluid flows with multiple anomalous currents in the presence of external electromagnetic fields. It has recently been shown using thermodynamic arguments that the entropy current for such anomalous fluids contains additional first order terms proportional to the vorticity and magnetic field. Using the fluid/gravity map, we replicate this result using gravitational methods. The final two chapters consider questions related to the equations of fluid dynamics themselves; these chapters do not involve the fluid/gravity correspondence. The first of these chapters is a review of the various constraints that must be satisfied by the transport coefficients. In the final chapter, we derive the constraints obtained by requiring that the equilibrium fluid configurations are linearly stable to small perturbations. The inequalities that we obtain here are slightly weaker than those found by demanding that the divergence of the entropy current is non-negative.

No description available.

In the further development of the string theory, one needs to understand 3 or 4-dimensional volume minimizing subvarieties in 7 or 8-dimensional manifolds. As one example, one would like to understand 4-dimensional volume minimizing cycles in a torus T8. The Cayley calibration form can be used to find all volume minimizing cycles in each homology class of T8. In order to apply the Cayley form to 8-dimensional tori, we need to understand the finite symmetry of the Cayley form, which has a continuous symmetry group Spin(7). We have found one finite symmetry group of order eight generated by three elements. We have also studied the symmetry groups of tori based on the results of H.S.M. Coxeter, and have had a simple description of the four crystallographic groups in O(8). They can be used to classify all finite symmetry groups of the Cayley form.

We determine several families of so-called associative 3-dimensional manifolds in R7. Such manifolds are of interest because associative 3-cycles in G2 holonomy manifolds such as R6 × S1, whose universal cover is R7, are candidates for representations of fundamental particles in String Theory. We apply the classic results of Harvey and Lawson to find 3-manifolds which are graphs of functions f : Im H → H and which are invariant under a particular 1-parameter subgroup of G2, the automorphism group of the Cayley numbers, O. Systems of PDEs are derived and solved, some special cases of a classic theorem of Harvey and Lawson are investigated, and theorems aiding in the classification of all such manifolds described here are proven. It is found that in most of the cases examined, the resulting manifold must be of the form of the graph of a holomorphic function crossed with R. However, some examples of other types of graphs are also found.

“Eclipses” orchestrates the movements of human beings as celestial bodies—as wandering beings that emit waves, cycle in orbit, and create shadows. Subtle microtonal movements, or retunings of the same pitch, tell the story of gradual planetary motion leading to the drama of the eclipse. In the dimmed light of a solar eclipse, even the most familiar surroundings seem unfamiliar and strange. The harmonic material of “Eclipses” was composed using the Rainbow Harmony Matrix, a MaxMSP-based mouse and keyboard interface designed for real-time interaction with a 2D colorized pitch space defined by Just Intonation ratios. The most difficult coding in this patch was completed thanks to the generous help of Jean-François Charles. The spatialization and color-coding of Just Intonation pitches provided by the Rainbow Harmony Matrix proved invaluable to organizing harmonic and melodic material. Elements of rhythm, timbre, and articulation were largely left to the ear’s intuition. “Penumbra” magnifies the shadow cast when one celestial body partially obscures the light from another. Moments from “Eclipses” are revisited through a telescopic lens, such that the small whole number relations hiding in the harmonies are magnified, into the field of rhythm. Small gestures orbit each other in wandering motion, building a web of cause and effect. The harmonic material from “Penumbra” is extracted from two chords occurring in measures 41-43 of “Eclipses.” The small whole number frequency ratios governing the cross-relations of these two chords are magnified and modified to create polyrhythms that cycle over several minutes.