Counting paths with Schur transitions

In this work we explore the structure of the branching graph of the unitary group using Schur transitions. We find that these transitions suggest a new combinatorial expression for counting paths in the branching graph. This formula, which is valid for any rank of the unitary group, reproduces known asymptotic results. We proceed to establish the general validity of this expression by a formal proof. The form of this equation strongly hints towards a quantum generalization. Thus, we introduce a notion of quantum relative dimension and subject it to the appropriate consistency tests. This new quantity finds its natural environment in the context of RCFTs and fractional statistics; where the already established notion of quantum dimension has proven to be of great physical importance.Comment: 30 pages, 5 figure

A non-perturbative theory of giant gravitons using AdS/CFT

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. February 2015.We 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

From Large N Nonplanar Anomalous Dimensions to Open Spring Theory

In this note we compute the non-planar one loop anomalous dimension of restricted Schur polynomials that have a bare dimension of O(N). This is achieved by mapping the restricted Schur polynomials into states of a specific U(N) irreducible representation. In this way the dilatation operator is mapped into a u(n) valued operator and, as a result, can easily be diagonalized. The resulting spectrum is reproduced by a classical model of springs between masses.Comment: 13+1 pages, 3 figure

Nonplanar integrability at two loops

In this article we compute the action of the two loop dilatation operator on restricted Schur polynomials that belong to the su(2) sector, in the displaced corners approximation. In this non-planar large N limit, operators that diagonalize the one loop dilatation operator are not corrected at two loops. The resulting spectrum of anomalous dimensions is related to a set of decoupled harmonic oscillators, indicating integrability in this sector of the theory at two loops. The anomalous dimensions are a non-trivial function of the 't Hooft coupling, with a spectrum that is continuous and starting at zero at large N, but discrete at finite N.Comment: version to appear in JHE