Instantons on various spaces can be constructed via a generalization of the Fourier transform called the ADHM-Nahm transform. An explicit use of this construction, however, involves rather tedious calculations. Here we derive a simple formula for instantons on a space with one periodic direction. It simplifies the ADHM-Nahm machinery and can be generalized to other spaces.

A dissertation submitted in fulfillment of the requirements for the degree of Master of Science. August 2016. / The six scalar fields in N = 4 super Yang-Mills theory enjoy a global SO(6) symmetry, and large N but non-planar limits of this theory are well-described by adopting a group representation approach. Studies have shown that the one-loop dilatation operator is highly determined by the action of the su(2)=su(3) subalgebras on restricted Schur polynomials. These actions involve the traces of products of projection operators. In this dissertation, exact analytical formulae for these traces are found which in turn are used to find the exact action of these algebras on restricted Schur polynomials. The potential of the su(2) algebra to determine the one-loop dilatation operator is also explored. This is done by exploiting necessary symmetry conditions and moving to a continuum limit in order to derive a number of partial differential equations which determine the dilatation operator. The ultimate goal of this work is to provide tools to find the exact one-loop dilatation operator in the non-planar limit. / LG2017

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. 14 May 2013 / We study the action of the dilatation operator on restricted Schur polynomials labeled by Young diagrams with p long columns or p long rows. A new version of Schur-Weyl duality provides a powerful approach to the computation and manipulation of the symmetric group operators appearing in the restricted Schur polynomials. Using this new technology, we are able to evaluate the action of the one loop dilatation operator. The result has a direct and natural connection to the Gauss Law constraint for branes with a compact world volume. Generalzing previous results, we find considerable evidence that the dilatation operator reduces to a decoupled set of harmonic oscillators. This strongly suggests that integrability in N = 4 super Yang-Mills theory is not just a feature of the planar limit, but extends to other large N but non-planar limits.

In this thesis we explore certain limits of the AdS/CFT correspondence for integrability. This is done by calculating the action of the dilatation operator on operators known as restricted Schur polynomials, which are AdS/CFT dual to D3-branes known as giant gravitons. We focus on operators in N = 4 super-Yang-Mills theory, which is dual to type IIB string theory on an AdS5×S5 background. We find that, in various cases, this theory is integrable in a large N non-planar limit.

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.

曾肖波. / "2006年8月" / 論文(哲學碩士)--香港中文大學, 2006. / 參考文獻(leaves 148-157). / "2006 nian 8 yue" / Abstracts in Chinese and English. / Zeng Xiaobo. / Lun wen (zhe xue shuo shi)--Xianggang Zhong wen da xue, 2006. / Can kao wen xian (leaves 148-157). / Chapter 1 --- 緒言 --- p.1 / Chapter 2 --- 秦漢文獻所記商鞅生平及評價 --- p.5 / Chapter 2.1 --- 史部文獻 --- p.5 / Chapter 2.2 --- 子部文獻 --- p.11 / Chapter 3 --- 《商君書》真僞及成書年代硏究 --- p.16 / Chapter 3.1 --- 《商君書》的流傳情況 --- p.16 / Chapter 3.1.1 --- 釋名 --- p.16 / Chapter 3.1.2 --- 篇數 --- p.18 / Chapter 3.2 --- 《商君書》歷代硏究成果述評 --- p.21 / Chapter 3.2.1 --- 真僞問題綜述 --- p.21 / Chapter 3.2.2 --- 各篇撰寫年代述評 --- p.26 / Chapter 3.3 --- 《商君書》各篇撰寫年代考析依據 --- p.42 / Chapter 3.3.1 --- 字詞解釋 --- p.43 / Chapter 3.3.1.1 --- 術 --- p.43 / Chapter 3.3.1.2 --- 勢 --- p.46 / Chapter 3.3.1.3 --- 丞、監 --- p.49 / Chapter 3.3.2 --- 關聯詞語及相關句式 --- p.51 / Chapter 3.3.2.1 --- 而、則 --- p.51 / Chapter 3.3.2.2 --- 然、然而 --- p.56 / Chapter 3.3.2.3 --- 然則 --- p.56 / Chapter 3.3.3 --- 文本對讀 --- p.58 / Chapter 3.3.3.1 --- 書中各篇文字重出情況 --- p.58 / Chapter 3.3.3.1.1 --- 〈算地〉與〈徠民〉 --- p.58 / Chapter 3.3.3.1.2 --- 〈弱民〉、〈去彊〉與〈錯法〉 --- p.60 / Chapter 3.3.3.2 --- 《商君書》與其他文獻文字重出情況 --- p.62 / Chapter 3.3.3.2.1 --- 《商君´書Ø弱民》與《荀´子Ø議兵》 --- p.63 / Chapter 3.3.3.2.2 --- 《商君´書Ø靳令》與《韓非´子Ø飭令》 --- p.66 / Chapter 3.3.3.2.3 --- 《商君´書Ø更法》與《戰國´策Ø趙靈王平晝閑居》 --- p.69 / Chapter 3.3.4 --- 出土文獻 --- p.75 / Chapter 4 --- 《商君書》思想論析 --- p.79 / Chapter 4.1 --- 《商君書》的法 --- p.79 / Chapter 4.1.1 --- 法、刑、律之義 --- p.79 / Chapter 4.1.2 --- 法治之義 --- p.80 / Chapter 4.1.3 --- 法家之義 --- p.81 / Chapter 4.2 --- 商鞅思想形成背景 --- p.82 / Chapter 4.3 --- 《商君書》思想內容槪要 --- p.84 / Chapter 4.4 --- 《商君書》與法家學派思想比較 --- p.86 / Chapter 4.4.1 --- 與《管子》比較 --- p.89 / Chapter 4.4.2 --- 與《慎子》比較 --- p.91 / Chapter 4.4.3 --- 與《申子》比較 --- p.93 / Chapter 4.5 --- 《商君書》與儒家學派思想比較 --- p.95 / Chapter 4.5.1 --- 《商君書》與儒家學派思想的異同 --- p.95 / Chapter 4.5.1.1 --- 相異之處 --- p.95 / Chapter 4.5.1.2 --- 相同之處 --- p.96 / Chapter 4.5.2 --- 《商君書》與儒家學派思想的關係 --- p.97 / Chapter 4.5.2.1 --- 子夏之儒 --- p.97 / Chapter 4.5.2.2 --- 荀子 --- p.98 / Chapter 4.6 --- 《商君書》與《呂氏春秋》比較 --- p.100 / Chapter 5 --- 《商君書》特有用語闡釋 --- p.105 / Chapter 5.1 --- 六蝨 --- p.105 / Chapter 5.2 --- 壹 --- p.109 / Chapter 5.3 --- 宿治 --- p.114 / Chapter 5.4 --- 以刑去刑 --- p.115 / Chapter 5.5 --- 刑賞 --- p.116 / Chapter 6 --- 《商君書》對戰國以後的影響 --- p.120 / Chapter 6.1 --- 從睡虎地秦簡看《商君書》對秦國的政治影響 --- p.120 / Chapter 6.1.1 --- 強國政策 --- p.121 / Chapter 6.1.1.1 --- p.121 / Chapter 6.1.1.2 --- p.122 / Chapter 6.1.2 --- 管治政策 --- p.123 / Chapter 6.1.2.1 --- 以法爲本 --- p.123 / Chapter 6.1.2.2 --- 廢逆旅，嚴禁游士 --- p.124 / Chapter 6.1.3 --- 經濟政策 --- p.125 / Chapter 6.1.3.1 --- 打擊商人 --- p.125 / Chapter 6.1.3.2 --- 壹山澤，貴酒肉之價 --- p.127 / Chapter 6.1.4 --- 刑律政策 --- p.129 / Chapter 6.1.4.1 --- 厲行治吏 --- p.129 / Chapter 6.1.4.2 --- 治民方面，重刑連罪 --- p.131 / Chapter 6.2 --- 從張家山漢簡看《商君書》對西漢初年的政治影響 --- p.132 / Chapter 6.2.1 --- 法律制度 --- p.134 / Chapter 6.2.2 --- 封爵制度 --- p.136 / Chapter 6.3 --- 商鞅思想對韓非子的影響 --- p.137 / Chapter 6.4 --- 商鞅思想對西漢以後法家的影響 --- p.142 / Chapter 7 --- 結語 --- p.144 / 參考書目 --- p.148 / 附錄 --- p.158 / Chapter 附錄1 --- 《史´記Ø商君列傳》及《鹽鐵´論Ø非鞅》全文 --- p.158 / Chapter 附錄2 --- 秦漢典籍記商鞅資料彙編 --- p.165 / Chapter 附錄3 --- 《商君書》內文互見對讀表 --- p.180 / Chapter 附錄4 --- 《商君書》與其他文獻異文對讀表 --- p.184 / Chapter 附錄5 --- 《商君書》「而」、「則」出現次數及「然」字用例彙編 --- p.197

Planar maximally supersymmetric Yang-Mills theory (N = 4 SYM) is a special quantum fi eld theory. A few of its remarkable features are conformal symmetry at the quantum level, evidence of integrability and, moreover, it is a prime example of the AdS/CFT duality. Triggered by Witten's twistor string theory [1], the past 15 years have witnessed enormous progress in reformulating this theory to make as many of these special features manifest, from the choice of convenient variables to recursion relations that allowed new mathematical structures to appear, like the Grassmannian [2]. These methods are collectively referred to as on-shell methods. The ultimate hope is that, by understanding N = 4 SYM in depth, one can learn about other, more realistic quantum fi eld theories. The overarching theme of this thesis is the investigation of how on-shell methods can aid the computation of quantities other than scattering amplitudes. In this spirit we study form factors and correlation functions, said to be partially and completely off-shell quantities, respectively. More explicitly, we compute form factors of half-BPS operators up to two loops, and study the dilatation operator in the SO(6) and SU(2j3) sectors using techniques originally designed for amplitudes. A second part of the work is dedicated to the study of scattering amplitudes beyond the planar limit, an area of research which is still in its infancy, and not much is known about which special features of the planar theory survive in the non-planar regime. In this context, we generalise some aspects of the on-shell diagram formulation of Arkani-Hamed et al. [3] to take into account non-planar corrections.

O objetivo desta dissertação é estudar o problema de autovalor de Dirichlet para variedades riemannianas completas. Mais precisamente, pretendemos estudar uma cota por baixo para o -ésimo autovalor de um domínio limitado em uma variedade riemanniana completa. Tal cota é obtida fazendo-se uso de uma fórmula de recorrência de Cheng e Yang e um teorema de Nash. Ademais, pretendemos estudar uma desigualdade universal para os autovalores no espaço hiperbólico. / The goal of this dissertation is to study the Dirichlet eigenvalue problem for a complete riemannian manifold. More accurately, we intend to investigate a lower-bound for the -ℎ eigenvalue on a bounded domain in a complete riemannian manifold. Such a bound is obtained by making use of a recursion formula of Cheng and Yang and Nash’s Theorem. Furthermore, we study a universal inequality for eigenvalues of the Dirichlet eigenvalue problem on a bounded domain in a hyperbolic space (−1).

Orientador: Abraham Hirsz Zimerman / Banca: Waldyr Alves Rodrigues Junior / Banca: Cesar Augusto Linhares / Banca: Juan Alberto Mignaco / Banca: Ruben Aldrovandi / Doutor

This research aims to suggest a means of creating a closer relationship between technological artwork and its viewers through applying 'naturalising' technology in art using the theory of Yin-Yang O-Hang. The questions are raised : 1/. What are the vital factors of technological artworks? 2/. What sorts of methods can make using technology in arts more effective? 3/. How to overcome the difficulties in getting an intimate relationship between viewers and technological artworks? In order to optimise the aesthetic use of technology in art, the research is thus focused on 'naturalising' technology as a possible solution to overcome the negative aspects of technological artworks through the application of the theory of Yin-Yang O-Hang.Throughout this research, the author has discussed the fundamental concerns of the theory and its applications in Western and Eastern art. Furthermore, in investigating a number of artworks, the trends have been explored in technological art which are compatible with nature and humanity. The author has tried to find a way of applying the theory of Yin-Yang O-Hang to make technological art more intimate and accessible to viewers / Master of Arts (Hons)