String theory approach to problems of strongly coupled gauge theories

大多数现代量子场论都是基于杨和米尔斯的非凡框架，他们使用同样出现在几何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经过了严格的实验检验，但其数学基础仍未被完全理解。即，如何在强耦合下解析计算该理论的谱是一个重要的开放性问题。虽然有计算机模拟可以再现观测到的光谱，但对它的理论理解仍然缺失。计算Yang-Mills理论谱的主要障碍是需要理解强耦合的系统；存在非自然的小参数，允许采用标准的微扰方法。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：他没有在我们世界的四个时空维度中分析弦理论，而是提出在更高维度的弯曲空间中考虑弦理论。他推测这种高维弦理论相当于Yang和mills的低维弦理论，并提出了两者之间的特定映射。这个图的关键点是它颠倒了两种理论之间的耦合：因此，一个理论中的强耦合（难以处理）现象被映射到另一个理论中的弱耦合（易于计算）现象。虽然这个猜想到目前为止已经在特定的范围内得到了检验，但它的证据仍然缺乏。最近关于极大超对称杨-米尔斯理论（N=4 SYM）的可积结构的发现为如何证明这一猜想提供了新的思路。我研究的主要目标是利用这些见解来证明弦/规范理论的对应关系。将使用的方法需要结合来自可积性，量子场论，弦理论和群论的几种跨学科技术。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学的新发展。虽然最初的猜想是为杨-米尔斯理论（N=4 SYM理论）的一个非常特殊的情况而制定的，但有强烈的迹象表明，这些想法应该更普遍。突破这个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。由于这个猜想在概念上开辟了一种看待场论和引力的新方法。几何现象，理解它可以应用于什么样的系统将教会我们杨-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.