Fundamental Implications of Fields, Strings and Gravity

在20世纪，广义相对论和量子场论分别作为非常成功的理论出现在非常大的和非常小的尺度上描述物理学。然而，当考虑到非常大但非常小的物体，如黑洞奇点和宇宙的开始时，它们是不相容的。新的物理学需要提供一个统一的理论，弦理论是最有希望的候选者，因为它可以用来从量子系统中获得爱因斯坦方程。它为理解霍金预言的黑洞蒸发的各个方面提供了新的方法，以及新的“全息”技术，导致了用几何方法计算的量与迄今难以计算的强耦合量子场论中的可观量之间的显著关系。我们的建议集中在发展新的几何和代数技术，以研究与弦理论相关的量子理论和几何解决方案的关键方面。在全息学中，我们将研究几何是如何从矩阵量子力学中产生的，并利用这一点来探索黑洞在蒸发过程中的物理性质如何变化，以及相关的量子态如何对应于几何解。我们的团队在机器学习和数值模拟技术方面拥有专业知识，这将在本分析中使用。我们还将利用我们在可积性方面的专业知识来详细研究量子态和几何解之间的全息对偶。可积性通过将解决方案从有限的物理参数范围扩展到更一般的情况，为解决某些量子系统提供了强大的工具。我们将开发新的方法来分析弦理论中的无质量量子态，从而对这些理论有更全面的理解。在应用“高等几何”方法研究振幅中隐藏的几何结构方面也将取得进一步的进展。振幅是描述量子态相互作用的关键，在粒子物理实验中也是必需的，然而振幅的计算通常是非常复杂的。利用更高的几何代数结构，我们将构造新的几何和代数方法来计算振幅。在几何方面，我们将发展新的方法来分类与十维和十一维超引力中的De-Sitter解有关的代数结构，这是弦理论的低能量极限。这样的解与宇宙学有关，在某些黑洞视界附近的几何学中也出现了De-Sitter几何，并且已经发展了全息方法来理解与这些几何相关联的量子态。值得注意的是，最近弦理论几何学和流体力学方程之间也建立了联系。某些引力解，例如与黑洞视界有关的几何形状，产生了流体力学的纳维-斯托克斯方程。我们将研究这种结构是否可以推广，利用特殊的彭罗斯坐标，看看流体动力学方程是否是超重力解的更一般的性质。我们还将研究在弦理论中开发的“高等几何”方法是否可以适用于描述与大气锋面研究相关的流体力学解，如涡旋。这些相互关联的项目将在理论物理方面产生重大影响，在实验粒子物理、机器学习技术和黑洞性质方面具有潜在的现实应用。我们将利用我们广泛的国内和国际合作来推进这项工作。它将有助于回答STFC支持的科学挑战的重要关键要素，涉及引力和时空的真实性质，以及什么是基本定律、粒子、对称性和物理学领域。

In the 20th century, General Relativity and Quantum Field Theory emerged as extraordinarily successful theories used to describe physics on very large, and very small scales respectively. They are however incompatible when considering very massive, but very small objects, such as black hole singularities, and the beginning of the universe. New physics is required to provide a unified theory, and String Theory is the most promising candidate, as it can be used to obtain Einstein's equations from a quantum system. It has produced new ways to understand aspects of black hole evaporation, predicted by Hawking, as well as novel "holographic" techniques leading to remarkable relationships between quantities computed using geometric methods and observables in strongly coupled quantum field theories which have hitherto been difficult to calculate. Our proposal is focused on developing new geometric and algebraic techniques to investigate key aspects of quantum theories and geometric solutions related to String Theory. In holography, we will investigate how geometry emerges from matrix quantum mechanics, and use this to probe how physical properties of black holes vary during the evaporation process, and also to describe how the related quantum states correspond to geometric solutions. Our group has expertise in machine learning and numerical simulation techniques which will be utilized in this analysis. We will also use our expertise in integrability to examine in detail the holographic duality between quantum states and geometric solutions. Integrability provides powerful tools for solving certain quantum systems by extending the solutions away from a limited range of physical parameters, to more general cases. We will develop new methods for analysing massless quantum states in String Theory, leading to a much more complete understanding of these theories. Further progress will also be made in applying "higher geometry" methods to investigate hidden geometric structures in amplitudes. Amplitudes are crucial for describing quantum state interactions, and are needed in particle physics experiments, however amplitude calculations are usually very complicated. Using higher geometry algebraic structures we will construct new geometric and algebraic methods for amplitude calculation.In terms of geometry, we will develop new methods to classify the algebraic structures associated with de-Sitter solutions in ten and eleven dimensional supergravities, which are the low energy limit of string theory. Such solutions are relevant to cosmology, and de-Sitter geometries also arise in the geometry near to the event horizons of certain black holes, and holographic methods have been developed to understand the quantum states associated with these geometries. Remarkably, there has recently also been a connection made between String Theory geometry, and equations in hydrodynamics. Certain gravitational solutions, such as geometries associated with black hole horizons, give rise to the Navier-Stokes equations of hydrodynamics. We will investigate if this construction can be generalized, utilizing special Penrose co-ordinates, to see if hydrodynamic equations are more generic properties of supergravity solutions. We will also examine if "higher geometry" methods developed in String Theory can be adapted to describe hydrodynamic solutions such as vortices, which are relevant to the study of atmospheric fronts.These interconnected projects will produce results of significant impact in theoretical physics, with potential real-world applications in experimental particle physics, machine learning techniques, and properties of black holes. We will pursue this work using our extensive national and international collaborations. It will help to answer important key elements of the Science Challenges supported by STFC, relating to the true nature of gravity and space-time, and what are the fundamental laws, particles, symmetries and fields of physics.