Quantum Symmetries in String Theory

If two people on opposite sides of the planet both drop a ball the same thing happens: the ball falls towards the ground with constant acceleration. This is a simple consequence of the rotational symmetry of the Earth's gravity. More generally, the power of symmetry lies in its ability to tell us that the laws of physics are related in situations that may look very different. In this way symmetry has had a profound impact on physics, often making impossible problems possible. The aim of my research is to develop novel symmetry-based methods tackling challenging open problems in theoretical physics.

In the early 20th century two of the most successful theories of modern physics were discovered. The first, general relativity, describes classical gravity at large distances, while the second, quantum mechanics, underlies atomic-scale physics. However, attempting to unify them using standard methods leads to uncontrollable infinities. This long-standing problem suggests that a new framework is needed to develop a quantum theory of gravity.

Starting from the elementary idea of considering fundamental objects that are extended in space, string theory has become a leading candidate for the quantum theory of gravity. Gravitons, the particles that carry the gravitational force, are oscillating closed loops of string. One of the most well-known facts about string theory is that it requires 10 space-time dimensions. Attempts to recover the 4 dimensions of our universe have been remarkably fruitful, leading to a wealth of new mathematics and fundamentally changing how we think about physical theories.

One way to relate string theory to physics in 4 dimensions is through the AdS/CFT correspondence. This correspondence is a remarkable duality between two models. The first is a quantum theory of gravity described by closed strings on a highly symmetric curved space-time. The second is a quantum field theory in 4 dimensions, which has important connections to quantum chromodynamics: the gauge theory of the strong interaction holding protons and neutrons together in the nuclei of atoms.

The power of the duality is that analytic computations in string theory give us new information about the strongly-coupled regime of quantum field theory. Strongly-coupled systems are typically hard to study and underlie many important open questions in science. Indeed, rigorously establishing the existence of a mass gap in Yang-Mills theory is one of the seven Millennium Prize problems. The AdS/CFT correspondence therefore ties together two long-standing problems of theoretical physics: describing a quantum theory of gravity and solving strongly-coupled quantum field theories.

The strings of the AdS/CFT correspondence are not just moving on a curved space-time, but also in background fields. This can be compared to an electron moving in a background magnetic field: the electron interacts with the magnetic field causing it to follow a curved path. Many important string theories are of this type and they are typically difficult to study. Symmetry can provide us with a solution. One of the most striking manifestations of symmetry is integrability, a rich mathematical property of certain physical models. Integrability can be thought of as the presence of a large hidden symmetry that can be used to derive exact results. In special cases the strings of the AdS/CFT correspondence have this remarkable property and using the associated methods these theories can potentially be solved.

The aim of my research is to develop novel symmetry-based methods advancing our knowledge of the role integrability plays in string theory and the AdS/CFT correspondence. This would allow us to solve string theories on non-trivial backgrounds and to construct new examples of gauge/gravity duality, thereby providing new insights into the fundamental nature of quantum gravity and strongly-coupled physics in the real world.

Impact on Physics and Science.

Achieving the primary objectives of the research proposal will have a significant impact on our understanding of quantum gravity and strongly-coupled physics through the study of integrable models and their role in string theory and the AdS/CFT correspondence.

Strongly-coupled systems appear in many areas of science, for example in condensed matter physics and statistical mechanics. Recent indications suggest that the proposed research could provide new insights into the physics of these systems. There are also deep links between integrability and various areas of mathematics, including quantum groups, Hopf algebras and non-commutative geometry. One of the goals of the Fellowship is to develop these interdisciplinary connections having a potential impact on a wide range of fields.

To maximise this impact over the course of the Fellowship I will publish in leading peer-reviewed journals and present at international conferences. Articles will also be posted and cross-listed on the open-access arXiv preprint server. Furthermore, I will develop my website to include new accessible content targeting researchers from other fields within science and mathematics.

Impact on the UK Research Sector.

The Fellowship will establish a new internationally-leading research group in the UK. It will lead to new collaborations, strengthening the ties between Imperial College London and other leading research institutions worldwide. The original nature of the proposed research will further strengthen the reputation of the Theoretical Physics group, attracting accomplished researchers and inspiring talented students. As a result, the Fellowship will have a positive impact on the international competitiveness of the UK research and development sector.

Within the research proposal there are objectives that are suitable for collaboration with young researchers. During the Fellowship, I will look for opportunities to supervise PhD students. Given the analytical nature of the research, this will contribute to the students' professional development, in particular, their high-level quantitative and qualitative reasoning. Such skills will increase their potential to have a significant impact on UK science and technology in the future.

Broader Impact on the Public.

Throughout its history theoretical physics has captured the imagination of young scientists, mathematicians and engineers. The wonders of quantum mechanics and the mysteries of the universe are two of the most inspiring topics in the field. String theory brings them together with the goal of formulating a theory of quantum gravity. Working towards answering such fundamental questions allows the field to continue to stimulate and motivate future generations to study STEM subjects. An important route to achieving this impact is to communicate the ideas and concepts underlying the subject, together with the proposed research and the results of the Fellowship, to the general public and potential future leaders in science, mathematics and engineering.

In recent years I have been exploring opportunities to engage in outreach activities. In July 2015 I visited a secondary school to discuss life as a theoretical physicist with A-level mathematics students. In November 2018 I returned to deliver an interactive 90 minute session on symmetry and the role it plays in theoretical physics, designed with the aim of inspiring and challenging the students. With approximately 30 GCSE students participating enthusiastically, the next session on the role of geometry in theoretical physics is planned for July 2019.

I plan to continue developing this programme during the Fellowship, establishing new ties with other schools in London and further afield. I also intend to explore new avenues for outreach, for example working with charities and museums whose aim is to promote participation in STEM subjects.