Fundamental Implications of Fields, Strings and Gravity

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.