The foundations of twistor-string theory

罗杰·彭罗斯的旋风计划被设想为量子引力的一种方法。在爱因斯坦的广义相对论中，时空的几何是动态的，并产生了引力场。为了量化它，人们必须考虑，如果几何学本身受到量子涨落的影响，什么样的背景才是理所当然的。TWSTOR计划是将TWSTOR空间作为给定的背景和舞台，在这里将量子理论和引力相结合的理论最自然地表达出来。如果这要奏效，就必须有可能用旋风空间的结构来重新表述所有的基本物理学。早期的成功不仅是对线性无质量场的编码，也是对具有右旋圆偏振的非线性规范场和引力场的编码。然而，直到2003年底，这个程序不仅停留在编码杨-米尔斯和引力的完全非线性结构(当它们不是圆极化时)的问题上，而且还停留在系统地结合量子场论的问题上。Witten引入扭曲弦理论是向前迈出的重要一步，它现在清楚地表明了如何克服这些长期存在的困难，至少在微扰理论的背景下是这样。Witten的论文的重点不是扭曲程序或量子引力，而是寻找新的数学技术来研究规范理论，规范理论是粒子物理标准模型的力定律的理论基础。规范理论散射幅度的微扰计算特别具有挑战性，目前的分析和数值技术在低于即将在欧洲核子研究中心的大型强子对撞机上进行的实验所需的点上失去了动力。Witten的出发点是Parke和Taylor关于所谓的最大螺旋度破坏(MHV)规范理论振幅的引人注目的公式。尽管这些公式是许多费曼图的总和，但它们特别紧凑。在20世纪80年代末，Nair在超对称的背景下发现了这些振幅的一个显著的解释，即在超扭曲空间中的全纯曲线空间上的积分，超扭曲空间是彭罗斯原始扭曲空间的超对称扩展。Witten的推广是将一般规范理论的振幅表示为所有代数曲线空间上的积分，但现在是在超扭曲空间中的任意亏格和次数。这些公式在很大程度上得到了验证(至少在树的水平上)，并对微扰规范理论产生了实质性的影响。然而，这种影响目前受到对扭弦理论基础的正确理解的限制，以及现有的扭弦理论自动包含共形超引力的事实，这是一种必然破坏量子计算的非物理理论。在PI及其合作者随后的工作中，共形超引力的存在被视为一个机会，因为它包含了爱因斯坦引力。建立了扭弦理论，其中的引力自由度正好是N=4和8的爱因斯坦超引力的自由度。然而，这些还需要进一步的研究，因为尚不清楚它们是否预测了正确的散射幅度。这项提议的目的是探索潜在的几何结构，为该课题提供数学基础，并发现和研究能够描述爱因斯坦引力的新的扭曲弦理论，甚至仅仅是规范理论本身。已经发现，理解这些理论的正确数学框架包括来自代数和微分几何的令人兴奋的新想法，例如手性代数的鞘和广义的复杂结构。最终的目标是发展扭曲弦理论，将其作为研究量子规范理论和研究量子引力的工具。

The twistor programme of Roger Penrose was conceived as an approach to quantum gravity. In Einstein's theory of general relativity, the geometry of space-time is dynamical and gives rise to the gravitational field. In order to quantize it, one has to consider what kind of background can be taken for granted if geometry itself is to be subject to quantum fluctuations. The twistor programme is to take twistor space to be the given background and the arena in which the theory that unites quantum theory and gravity is most naturally expressed. If this is to work, it must be possible to reformulate all basic physics in terms of structures on twistor space. Early successes were the encoding not only of linear massless fields, but also non-linear gauge fields and gravitational fields with right handed circular polarization. However, until late 2003, this programme had been stuck not only on the problem of encoding the full non-linear structure of Yang-Mills and gravity when they are not circularly polarized, but also of the systematic incorporation of quantum field theory. Witten's introduction of twistor-string theory was a major step forward that now gives a clear idea as to how these longstanding difficulties might be overcome, at least in the context of perturbation theory.The focus of Witten's paper was not on the twistor programme nor quantum gravity, but on finding new mathematical techniques to study gauge theories, the class of theories underlying the force laws of the standard model of particle physics. The perturbative calculation of gauge theory scattering amplitudes is particularly challenging and current analytical and numerical techniques run out of steam at a point below that required by upcoming experiments at the Large Hadron Collider at CERN. Witten's starting point was the remarkable formulae due to Parke and Taylor for so called `maximal helicity violating' (MHV) gauge theory amplitudes. Despite being a sum of many many Feynman diagrams, these formulae are particularly compact. In the late 1980s, Nair had found a remarkable interpretaion of these amplitudes in a supersymmetric context as integrals over a space of holomorphic curves in super twistor space, a supersymmetric extension of Penrose's original twistor space. Witten's generalisation was to express general gauge theory amplitudes as an integral over the space of all algebraic curves, but now of arbitrary genus and degree, in super-twistor space. These formulae have been largely verified (at least at tree level) and have had a substantial impact on perturbative gauge theory. However, this impact is currently limited by a lack of proper understanding of the foundations of twistor-string theory and by the fact that existing twistor-string theories automatically incorporate conformal supergravity, an unphysical theory that necessarily corrupts quantum calculations.In subsequent work of the PI and collaborators, the existence of conformal supergravity was seen as an opportunity because it contains Einstein gravity. Twistor-string theories were constructed in which the gravitational degrees of freedom are precisely those of N=4 and 8 Einstein supergravity. However, these require further investigation as it is not clear whether these predict the correct scattering amplitudes.The aims of this proposal are to explore the underlying geometry, to provide mathematical foundations for the subject and to find and investigate new twistor string theories that can describe Einstein gravity, or even just gauge theories on their own. It has emerged that the correct mathematical framework for understanding these theories involves exciting new ideas from algebraic and differential geometry such as sheaves of chiral algebras and generalised complex structures. The eventual aims are to develop twistor-string theory as a tool for studying quantum gauge theories, and for studying quantum gravity.