Supergravity and field theory solutions in AdS/CFT correspondence

现代理论物理学在解决长期存在的基本问题方面迈出了很大的一步。像QCD中夸克的禁闭、黑洞信息悖论、宇宙学常数问题和暗物质等问题现在都在寻找单一的统一理论-弦理论的答案。在处理这些问题方面最新的，也是最有希望的发展是ADS/CFT猜想。这个猜想将超共形场理论与定义在时空上的弦理论或M-理论联系起来，这些理论是反de Sitter空间的乘积，比相应的超共形场理论定义的空间高一维，乘以一个紧致空间的总数为10或11维。最有趣的方面之一，也是证明这一猜想的主要困难之一，就是它是一个弱-强耦合对应。定义在不同维度的时空上的共形场论为我们提供了一种可行的描述有趣的现象，如临界现象、相变、重整化群流，描述了重整化群下质量量子场论流向哪里。共形场理论已经被证明是主要感兴趣的领域是广泛的，从量子霍尔效应到经济学研究中相关的网络模型。为了真正利用共形场理论描述，人们应该能够在出现的各种耦合常数的所有区域进行计算，特别是关联函数。一般来说，微扰理论的系统论只适用于任何场论的弱耦合区域，而有趣的现象，如QCD限制，发生在强耦合区域，这是标准微扰技术和方法所不能达到的。在对偶的另一边，相应的弦理论的弱耦合描述原则上是容易处理的，并且可以利用微扰理论来从中提取定量的答案。ADS/CFT对应为我们研究强耦合共形场理论提供了一个很有价值的工具。ADS/CFT猜想还没有完全被证明，在这个阶段，证据的收集将导致它的证明或反驳是相当关键的。我的研究将部分地集中在对对应双方的各种不平凡的调查上，对于这些对应，猜想的词典表明，我们对同一物理系统有不同的描述。我要考虑的一个例子是研究超对称威尔逊环算符的超引力对偶，这些算子存在于超保角规范理论中，即N=4的超杨-米尔斯理论在四个时空维度上。人们对这些算子进行了广泛的研究，并研究了它们的各种性质。由于ADS/CFT预测每个算符都应该有一个弦理论对偶，所以证明它们的弦理论描述的存在以及它预测的性质与场论中发现的性质完全相同是相当关键的。我的第二部分研究将集中在正在进行的具有弦理论描述的超共形场论的分类计划中。通常，弦理论的共形不变性要求用于研究该理论的背景应该是相应的低能量极限的解，即超引力理论。这类背景的一种特殊类别将是与ADS/CFT相关的背景，即在几何体中包含全局反De Sitter因子的背景。然后，问题归结为描述哪些紧致空间被允许作为乘积空间的第二个因素。

Modern theoretical physics has made big steps in addressing longstanding, fundamental questions. Problems like the confinement of quarks in QCD, the black hole information paradox, the cosmological constant problem and dark matter are now looking for answers in a single, unified theory, string theory.The most recent, and also one of the most promising, development in approaching a set of these issues is the AdS/CFT conjecture. This conjecture relates superconformal field theories to string theory or M-theory defined on spacetimes which are products of Anti de Sitter space, of one dimension higher than the corresponding super conformal field theory is defined on, times a compact space summing up to a total number of ten or eleven dimensions. One of the most interesting aspects, and also one of the main difficulties proving the conjecture, is that it is a weak-strong coupling correspondence.Conformal field theories defined on spacetimes of various dimensions provide us with a viable description of interesting phenomena like critical phenomena, phase transitions, renormalization group flows describing where massive quantum field theories flow to under the renormalization group. The range of areas that conformal field theories have been proved to be of primary interest is vast, varying from the Quantum Hall Effect to Network Models relevant in economics studies.In order to actually take advantage of the conformal field theoretical description one should be able to perform calculations, in particular the correlation functions, at all regimes of the various coupling constants that appear. In general, the systematics of perturbation theory is applicable only in the weak coupling regime of any field theory whereas interesting phenomena, like the QCD confinement, happen at the strong coupling regime which is inaccessible with the standard perturbative techniques and methodology. On the other side of the duality, the weak coupling description of the corresponding string theory is, in principle, tractable and perturbation theory can be employed to extract quantitative answers from it. The AdS/CFT correspondence provides us with an invaluable tool in studying strongly coupled conformal field theories.The AdS/CFT conjecture has not been proved at full generality yet and the collection of evidence that would lead to proving, or disproving it, is rather crucial at this stage. My research will partially focus in various non-trivial investigations on both sides of the correspondence for which the dictionary of the conjecture suggests that we have a different description of the same physical system. One of the examples that I want to consider is the study supergravity dual of supersymmetric Wilson loop operators present in superconformal gauge theories as N=4 super Yang-Mills theory in four spacetime dimensions. These operators have been extensively studied and various properties of them have been investigated. Since AdS/CFT predicts that every operator should have a string theory dual, it is rather crucial to prove that a string theory description of them exists and that it predicts exactly the same properties with the ones that have been discovered from field theory considerations.The second part of my research will focus on the ongoing program of classification of superconformal field theories that have a string theory description.In general, conformal invariance of string theory requires that the backgrounds used to study the theory on should be solutions of the corresponding low energy limit, a supergravity theory. A special class of such backgrounds would be the ones relevant for AdS/CFT, namely backgrounds that contain a global Anti de Sitter factor in the geometry. The question then reduces to describing which compact spaces are allowed to be the second factor of the product space.