A study of giant graviton dynamics in the restricted schur polynomial basis

MSc., Faculty of Science, University of the Witwatersrand, 2011Anomalous dimensions are calculated for a certain class of operators in the restricted Schur polynomial basis in the large N limit. A new computationally simple form of the dilatation operator is derived and used in this dissertation. The class of operators investigated have bare dimension of O(N). Thus the calculation necessarily sums non-planar Feynmann diagrams as the planar approximation has broken down for operators of this size. The operators investigated have two long columns and the operators mix under the action of the dilatation operator, however the mixing of operators having a different number of columns is suppressed and can be neglected in the large N limit. The action of the one loop dilatation operator is explicitly calculated for the cases where the operators have two, three and four impurities and it is found that in a particular limit the action of the one loop dilatation operator reduces to that of a discrete second derivative. The lattice on which the discretised second derivative is defined is provided by the Young tableaux itself. The one loop dilatation operator is diagonalised numerically and produces a surprisingly simple linear spectrum, with interesting degeneracies. The spectrum can be understood in terms of a collection of harmonic oscillators. The frequencies of the oscillators are all multiples of 8g2Y M and can be related to the set of Young tableaux acted upon by the dilatation operator. This equivalence to harmonic oscillators generalises on previously found results in the BPS sector, and suggests that the system is integrable. The work presented here is based primarily on research carried out by R.de Mello Koch, V De Comarmond, and K. Jefferies in [1]

Exploring commitment of secondary teachers in Seychelles

This thesis reports on an investigation into teacher commitment in secondary schools in Seychelles. The overarching aim was to gain an insight into the experiences and perceptions of teacher commitment in order to get a better understanding of teachers’ career trajectories and issues relating to teacher retention. Another aim was to explore the experiences and perceptions of the participating teachers, headteachers and policymakers on the factors that influence commitment and trajectories of secondary teachers at the different stages of their teaching careers. In order to achieve these aims a qualitative methodology was chosen with a combination of three different approaches: phenomenography, phenomenology and multiple case studies. The use of multiple-approaches was considered appropriate in order to enhance the results of the investigation of such a complex phenomenon like teacher commitment. The case studies focused on four teacher groups representing newly qualified teachers, mid-career teachers, experienced teachers and teachers who had left the profession. Data were sought from different participant groups in relation to teacher commitment, experiences and career trajectories. The exploration involved semi-structured interviews with secondary teachers, headteachers and policymakers. The findings show that participants describe teacher commitment in relation to altruism, personal qualities, pedagogical content knowledge and connectedness. The ideas of what constitutes a committed teacher for these participants reveal complexity in the phenomenon of teacher commitment. Personal, organisational and contextual factors are found to influence these participants’ understandings. The findings identify a complex interplay of personal and contextual spheres of influence on teacher commitment. Another level of complexity that the findings revealed relate to the interconnection between teacher commitment, teachers’ career stages and retention. The commitment of beginning teachers is found to be more at risk than that of mid-career and experienced teachers. Education stakeholders hold different views to those of teachers on the factors that impact on teacher commitment and retention. The study concludes by proposing a conceptual model for teacher commitment that illustrates its complex nature. Teacher commitment is multifaceted and the nature and level of commitment held by teachers involves the constant negotiation between these different factors. The findings of the study contribute to a nuanced understanding of teacher commitment and have the potential to generate more in-depth and extensive studies of this phenomenon. These findings may inform policymakers both in Seychelles and in other national and international contexts about issues relating to teacher recruitment, development and retention, which are worldwide concerns

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

Orthogonal Bases of Invariants in Tensor Models

Representation theory provides a suitable framework to count and classify invariants in tensor models. We show that there are two natural ways of counting invariants, one for arbitrary rank of the gauge group and a second, which is only valid for large N. We construct bases of invariant operators based on the counting, and compute correlators of their elements. The basis associated with finite N diagonalizes the two-point function of the theory and it is analogous to the restricted Schur basis used in matrix models. We comment on future lines of investigation.Comment: Two overlapping but independent results are merged to a joint work. 16 pages, 1 tabl

Correlation functions and representation bases in free N=4 Super Yang-Mills

We study exact correlation functions of N=4 SYM at zero coupling. It has been known that it is convenient to label local gauge invariant operators by irreducible representations of symmetric groups/Brauer algebras. We first review the construction of representation bases from the viewpoint of the enhanced symmetry structure of the free theory. We present a basis of multi-matrix models using elements of Brauer algebras, generalising our previous construction for two matrices. We will compute multi-point functions of the basis with the exact N-dependence. In particular we study three-point functions of a class of BPS operators, and we find that they are given by a branching rule of the Brauer algebra. The three-point functions take a factorised form if representations on the operators satisfy a relation.Comment: 28 pages, typos correcte

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

A double coset ansatz for integrability in AdS/CFT

We give a proof that the expected counting of strings attached to giant graviton branes in AdS_5 x S^5, as constrained by the Gauss Law, matches the dimension spanned by the expected dual operators in the gauge theory. The counting of string-brane configurations is formulated as a graph counting problem, which can be expressed as the number of points on a double coset involving permutation groups. Fourier transformation on the double coset suggests an ansatz for the diagonalization of the one-loop dilatation operator in this sector of strings attached to giant graviton branes. The ansatz agrees with and extends recent results which have found the dynamics of open string excitations of giants to be given by harmonic oscillators. We prove that it provides the conjectured diagonalization leading to harmonic oscillators.Comment: 33 pages, 3 figures; v2: references adde

Beyond the Planar Limit in ABJM

In this article we consider gauge theories with a U(N)X U(N) gauge group. We provide, for the first time, a complete set of operators built from scalar fields that are in the bi fundamental of the two groups. Our operators diagonalize the two point function of the free field theory at all orders in 1/N. We then use this basis to investigate non-planar anomalous dimensions in the ABJM theory. We show that the dilatation operator reduces to a set of decoupled harmonic oscillators, signaling integrability in a nonplanar large N limit.Comment: v2: minor revisison

Surprisingly Simple Spectra

The large N limit of the anomalous dimensions of operators in ${\cal N}=4$ super Yang-Mills theory described by restricted Schur polynomials, are studied. We focus on operators labeled by Young diagrams that have two columns (both long) so that the classical dimension of these operators is O(N). At large N these two column operators mix with each other but are decoupled from operators with $n\ne 2$ columns. The planar approximation does not capture the large N dynamics. For operators built with 2, 3 or 4 impurities the dilatation operator is explicitly evaluated. In all three cases, in a certain limit, the dilatation operator is a lattice version of a second derivative, with the lattice emerging from the Young diagram itself. The one loop dilatation operator is diagonalized numerically. All eigenvalues are an integer multiple of $8g_{YM}^2$ and there are interesting degeneracies in the spectrum. The spectrum we obtain for the one loop anomalous dimension operator is reproduced by a collection of harmonic oscillators. This equivalence to harmonic oscillators generalizes giant graviton results known for the BPS sector and further implies that the Hamiltonian defined by the one loop large $N$ dilatation operator is integrable. This is an example of an integrable dilatation operator, obtained by summing both planar and non-planar diagrams.Comment: 34 page

Non-planar Anomalous Dimensions in the sl(2) Sector

In this note we compute the non-planar one loop anomalous dimension of restricted Schur polynomials that belong to the sl(2) sector of N=4 super Yang-Mills theory and have a bare dimension of order N. Although the details are rather different, ultimately the problem of diagonalizing the dilatation operator in the sl(2) sector can be reduced to the su(2) sector problem. In this way we establish the expected dynamical emergence of the Gauss Law for giant gravitons and further show that the dilatation operator reduces to a set of decoupled harmonic oscillators.Comment: 13 + 1 page