Twistors in Field and String Theories

在自然界中，我们观察到两种基本粒子：玻色子和费米子。所有已知的物质都是由费米子（如电子和夸克）组成的，而物质之间的相互作用——电磁、弱力和强力——是由玻色子（如光子和胶子）介导的。物理学家已经发展出一种描述粒子动力学的奇妙理论：量子场论（QFT）。粒子之间存在着许多对称性，对对称性的探索是理解自然的关键之一。一种有趣的对称是超对称，它将玻色子和费米子联系起来，反之亦然。人们很快就清楚地认识到，它可能在粒子物理学中扮演着重要的角色。尽管QFT取得了成功，但它似乎与引力不相容。当你希望统一自然界中的所有力量时，你寻求替代方法。难以包含重力的原因是在量子傅立叶变换中粒子是点状的。考虑到这一点，人们开始研究将粒子视为扩展物体的理论。这导致了弦理论的诞生。弦理论实际上把所有的粒子都看作是微小弦的激发模式。它还自动包含一种叫做引力子的粒子：引力的中介。它在理论上的成功使弦理论成为最有希望成为万有理论的候选者。因此，它必须产生从QFT得到的所有结果。这个问题成为最重要的课题之一。十年前，人们认识到某个弦理论与某个QFT对偶：反德西特（ad）空间上的IIB型超弦理论和N=4超对称杨-米尔斯（SYM）理论。最值得注意的是，这种对偶，被称为AdS/CFT对偶，将一个有引力的理论与一个没有引力的理论联系起来。虽然已经发现了这种对偶性的有力证据，但其证据仍有待证实。此外，最近观察到SYM理论与另一种弦理论对偶：扭转弦理论。它是在扭曲空间上表述的，扭曲空间对讨论物理理论的性质非常有帮助。最有趣的是，扭曲弦表明了在现象学上有趣的量的明显的简单性，比如在[S]YM理论中粒子间的散射振幅——物理学家在粒子对撞机上测量的量。简而言之，提出的研究方案涉及刚才提到的理论的扭曲空间描述。我将探索某些引力理论，称为爱因斯坦超引力，并将它们转化为扭曲空间。扭曲理论的强大之处在于，一切都是用扭曲空间的几何数据来编码的，因此，被简化为几何中的经典问题——更容易回答的问题。我将研究超重力的扭弦公式。像SYM理论一样，它们似乎也承认扭曲或弦的形式。这些考虑将阐明超引力的量子方面，特别是最大超对称爱因斯坦引力的微扰有限性问题。如果存在一致的扭弦，它就为量子引力提供了一种新的方法。我还将研究经典和量子N=4 SYM理论，通过扭转。我将研究由某些非局部对称性产生的可积性。这部分项目将深入了解可积性的起源，这是AdS/CFT对偶中迫切需要解决的问题之一。鉴于此，我还将研究AdS弦的可积性。由于N=4 SYM理论不仅与此对偶，而且与扭弦对偶，因此必须理解这两种弦理论之间的联系。这个问题和扭弦理论中的可积性问题也将被研究。综上所述，所获得的结果将有助于我们理解场理论和弦理论之间的关系，它们也将影响规范理论和重力理论中的微扰问题[例如散射振幅计算的新方法]。

In nature, we observe two species of elementary particles: bosons and fermions. All known matter consists of fermions such as electrons and quarks while interactions - electromagnetism, weak and strong forces - between the matter is mediated by bosons such as photons and gluons. Physicists have developed a wonderful theory to describe the particle dynamics: quantum field theory [QFT]. Among the particles there are many symmetries and the latters search is one of the keys to understanding nature. One intriguing symmetry is supersymmetry which relates bosons to fermions and vice versa. It soon became clear that it may play a fundamental role in particle physics. Despite its success, QFT seems to be incompatible with gravity. As one wishes to unify all forces in nature, one seeks for alternative approaches. The reason why it is difficult to include gravity is that in QFT particles are point-like. With this in mind, people started studying theories that view particles as extended objects. This led to the birth of string theory. String theory in fact treats all particles as excitation modes of tiny strings. It also automatically contains a particle called graviton: the mediator of gravity. Its theoretical success made string theory the most promising canditate for a theory of everything. As such it must yield all results obtained from QFT. This issue became one of the most important subjects. A decade ago it was realised that a certain string theory is dual to a certain QFT: type IIB superstring theory on an Anti-de Sitter [AdS] space and N=4 supersymmetric Yang-Mills [SYM] theory. Most remarkably, this duality, known as AdS/CFT duality, relates a theory with gravity to one without. Whilst strong evidence for this duality has been found, its proof remains open. In addition, it was recently observed that SYM theory is dual to another string theory: twistor string theory. It is formulated on twistor space, a space that is extremely helpful for discussing properties of physical theories. Most interestingly, twistor strings make manifest the apparent simplicity of phenomenologically interesting quantities such as scattering amplitudes between particles in [S]YM theory - quantities that physicists measure at particle colliders. In short, the proposed research programme deals with the twistor space description of the theories just mentioned. I will explore certain gravity theories, called Einstein supergravities, and translate them to twistor space. The power of twistor theory is that everything is encoded in terms of geometric data on twistor space and thus, reduced to classical questions in geometry - questions which are easier to answer. I will study the twistor string formulation of supergravities. Like SYM theories, they also seem to admit a twistor string formulation. These considerations will shed light on quantum aspects of supergravities and in particular on the question of perturbative finiteness of maximally supersymmetric Einstein gravity. If a consistent twistor string exists, it provides a new approach to quantum gravity. I will also study classical and quantum N=4 SYM theory by means of twistors. I will investigate integrability properties that result from certain non-local symmetries. This part of the project will give insight into the origin of integrability, which is one of the pressing questions in the AdS/CFT duality. In of view that, I will also study integrability aspects of the AdS string. Since N=4 SYM theory is dual not only to this but also to the twistor string, one has to understand the connection between these two string theories. This issue and the question of integrability in twistor string theory will also be investigated. Summarising, the results obtained will contribute to our understanding of the relation between field and string theories, and they will also impact on perturbative issues [e.g. new methods for the computation of scattering amplitudes] in gauge and gravity theories.