Probing space-time geometry using young diagrams

A thesis submitted to the University of the Witwatersrand, Faculty of Science in fulfillment of the academic requirements of the degree of Doctoral of Philosophy.Quantum field theories and theories of gravity play an essential role in understanding nature. A dramatic recent development has been the discovery that quantum field theories are equivalent or dual to theories of quantum gravity on negatively curved spacetime. This duality goes under the name of the AdS/CFT correspondence. Sometimes the computation of certain observables in field theory are more difficult than the computation of the same observables in the theory of gravity and the opposite is also true. This makes the correspondence a powerful tool, that might provide an approach to strong coupling dynamics. We explore the AdS/CFT correspondence between type IIB string theory on asymptotically AdS5×S5 backgrounds as our gravity theory and N = 4 super Yang-Mills as our conformal field theory. We study BPS operators with bare dimension of order N2 in the field theory and identify them with BPS geometries on the gravity side of the correspondence. The dynamics of 1/2 BPS geometries are identified with gauge invariant operators constructed using a single field in the field theory, while the dynamics of 1/4 BPS geometries are identified with gauge invariant operators constructed using two fields. We find a sector of the two matrix model defined by the SU(2) sector in the field theory, that can be reduced to eigenvalue dynamics. The BPS operators in this sector are associated to solutions on the gravity side of the correspondence. We also identify the gauge invariant operators with bare dimension of order N, constructed using three fields, with 1/8 BPS giant graviton states. We count these gauge invariant operators constructed using three fields in the field theory and show that the counting of these operators is in agreement with the number of giant graviton states. We also demonstrate a correspondence between correlation functions of the field theory and the overlaps of the giant graviton wave functions. By working in terms of the eigenvalues we have managed to go from the matrix, which contains O(N2) degrees of freedom, to the eigenvalues which are O(N) degrees of freedom. Thus our work points to a significant simplification of the dynamics, something that deserves to be understood better. Another concrete result that we have achieved, is a proposal for some of the operators that are dual to the 1/4 BPS geometries. This is a genuine two matrix problem so it represents a novel extension of the understanding achieved by LLM of the 1/2 BPS geometries, constructed using a single matrix. The observables dual to new geometries have a bare dimension of O(N2). We have also considered operators with a bare dimension O(N), which are dual to 1/4 BPS giant gravitons. In this case too, we demonstrate that the eigenvalue description is useful. Almost all of the studies of the large N limit of CFT have focused on the planar limit. Here, since the operator dimensions scale as we take N →∞, we are considering large N but non-planar limits of the CFT. In these limits non-planar diagrams are not suppressed and the problem is considerably more difficult. The fact that we are able to explore this limit is concrete evidence for the power of the eigenvalue description and it suggests that a systematic treatment of large N but non-planar limits is possible.XL201

The simplest gauge-string duality

A dissertation submitted to the University of the Witwatersrand, Faculty of Science in ful lment of the academic requirements of the degree of Master of Science. Johannesburg, 2015.The gauge/gravity correspondence is a conjectured exact duality between quantum eld theories and theories of quantum gravity. A very simple gauge/string duality, claims an equivalence between the Gaussian matrix model and the topological A-model string theory on P1. In this dissertation we study this duality, proposing concrete operators in the matrix model that are dual to gravitational descendants of the puncture operator of the topological string theory. We test our proposal by showing that a large number of matrix model correlators are in complete agreement with correlators in the dual topological string theory. Contact term interactions, as proposed by Gopakumar and Pius, play an interesting and non-trivial role in the duality

From Gauss Graphs to Giants

We identify the operators in ${\cal N}=4$ super Yang-Mills theory that correspond to ${1\over 8}$-BPS giant gravitons in AdS$_5\times$S$^5$. Our evidence for the identification comes from (1) counting these operators and showing agreement with independent counts of the number of giant graviton states, and (2) by demonstrating a correspondence between correlation functions of the super Yang-Mills operators and overlaps of the giant graviton wave functions.Comment: 16 pages; v2: matches published versio

Topological String Correlators from Matrix Models

We discuss how to compute connected matrix model correlators for operators related to the gravitational descendants of the puncture operator, for the topological A model on P^1. The relevant correlators are determined by recursion relations that follow from a systematic 1/N expansion of well chosen Schwinger-Dyson equations. Our results provide further compelling evidence for Gopakumar's proposed "simplest gauge string duality" between the Gaussian matrix model and the topological A model on P^1.Comment: 25 pages; v2. refs adde