String theory approach to problems of strongly coupled gauge theories

大多数现代量子场论是建立在杨和米尔斯著名的框架之上的，他们使用也发生在计量学中的结构来描述基本粒子的动力学。然而，尽管这一理论的预测已经在实验上得到了严格的检验，但它的数学基础仍然没有完全被理解。也就是说，如何解析地计算该理论在强耦合下的谱是一个重要的公开问题。尽管有计算机模拟再现了观测到的光谱，但对它的理论理解仍然缺乏。计算杨-米尔斯理论的光谱的主要障碍是，人们需要理解一个处于强耦合状态的系统；存在非自然的小参数，这将允许标准的微扰方法。大约十年前，马尔达塞纳就如何解决这个长期存在的问题提出了一个巧妙的猜想。这个想法非常简单，尽管在概念上具有挑战性：他建议在更高维度的弯曲空间中考虑弦理论，而不是在我们世界的四个时空维度中分析该理论。他推测，这个高维的弦理论等价于杨和米尔斯的低维理论，并在两者之间提出了一个具体的映射。这张图的关键点在于它颠倒了两种理论之间的耦合：因此，一种理论中的强耦合(且难以处理)现象被映射到另一种理论中的弱耦合(且易于计算)现象。尽管到目前为止，该猜想已经在特定的限度内得到了检验，但它的证明仍然缺乏。最近关于极大超对称杨-米尔斯理论(N=4SYM)的可积结构的发现为如何证明这一猜想提供了新的思路。我的研究的主要目的是利用这些见解来证明弦/规范理论的一致性。将使用的方法需要结合来自可积性、量子场论、弦理论和群论的几个交叉学科的技术。因此，这个项目非常具有挑战性，证明的构建可能会导致数学和物理的新发展。尽管最初的猜想是针对杨-米尔斯理论(N=4 SYM理论)的一个非常特殊的情况而提出的，但有强烈的迹象表明，这些想法应该更普遍地适用。挑战猜想的极限并理解它在哪里被打破是我研究的另一个目标。由于这一猜想开辟了一种看待场论和引力(即几何)现象的概念上的新方法，理解它可以应用于哪种系统将使我们了解杨-米尔斯理论的基本性质。

Most of modern quantum field theory is based on the remarkableframework of Yang and Mills, who used structures that also occur ingeometry to describe the dynamics of elementary particles. However,despite the fact that the predictions of this theory have beenrigorously tested experimentally, its mathematical foundation is stillnot fully understood. Namely, an important open problem exists on howto analytically compute the spectrum of this theory at strong coupling. Though there are computer simulations which reproduce the observed spectrum,a theoretical understanding of it is still missing.The main obstacle in computing the spectrum of Yang-Mills theory, isthat one needs to understand a system at strong coupling; nonatural small parameter exists which would allow for a standardperturbative approach. Some ten years ago, an ingenious conjecturewas proposed by Maldacena on how to address this long-standingproblem. The idea is very simple, though conceptually challenging:instead of analysing the theory in four space-time dimensions of ourworld, he proposed to consider string theory in a higher-dimensional,curved space. He conjectured that this higher-dimensional stringtheory is equivalent to the lower-dimensional theory of Yang and Millsand proposed a specific map between the two. The key point of thismap is that it inverts the coupling between the two theories: hencestrongly coupled (and hard to address) phenomena in one theory aremapped to weakly coupled (and easy to compute) phenomena in the othertheory.Though the conjecture has so far been checked in specific limits, its proof isstill lacking. Recent discoveries of integrable structures inmaximally supersymmetric Yang-Mills theory (N=4 SYM)have introduced new ideas on how one could prove theconjecture. The main goal of my research is to work towards a proof ofthe string/gauge theory correspondence using these insights.The methods which will be used require a combination of severalinterdisciplinary techniques coming from integrability, quantum fieldtheory, string theory and group theory. As such, the project is verychallenging and a construction of the proof is likely to lead to newdevelopments in mathematics and physics.Although the initial conjecture was formulated for a very specificcase of Yang-Mills theory (the N=4 SYM theory) there are strongindications that the ideas should hold much more generally. Pushingthe limits of the conjecture and understanding where it breaks isanother goal of my research. Since the conjecture opened up a conceptuallynew way of looking at field theory and gravitational(i.e. geometrical) phenomena, understanding to what kind of systems itcan be applied will teach us about fundamental properties ofYang-Mills theories.