Quantum Symmetries in Diverse Dimensions

Symmetries are a powerful tool in science. They relate the behaviour of different experiments in the same physical system. In this way, they can have profound implications, making impossible problems possible. In many situations symmetries are manifest and can be understood intuitively. However, in most interesting systems of relevance to modern physics, they are hidden. Through my research, I aim to uncover, understand and exploit these hidden symmetries. Revealing them, their structure and their applications is a vibrant and active area of exploration. Quantum Field Theory (QFT) is the language we use to describe the physics of our universe, from the behaviour of fundamental particles to effective descriptions of condensed matter systems. Quantum Gravity (QG) is the unification of Quantum Field Theory and classical gravity, and is expected to describe the microscopic behaviour of black holes and the cosmology of the early universe. Two major challenges of modern physics are to (i) solve strongly-coupled QFT and (ii) formulate a theory of QG. Strongly-coupled systems are common in nature, but are hard to model since they cannot be easily approximated by solvable systems. A definitive theory of QG remains elusive, but String Theory is one of the most promising candidates and has changed our understanding of the subject. Gauge/Gravity Duality and Integrable Models are two approaches to tackling these long-standing problems: Gauge/Gravity Duality relates QFT and QG through dualities, such as the AdS/CFT Correspondence. The power of the duality lies in its "weak-strong nature." This means that analytic computations in the weak-coupling regime of String Theory can be used to explore the physics of strongly-coupled QFT. Integrable Models have a large hidden symmetry that gives rise to enough integrals of motion, or conserved quantities, to fully determine the dynamics. This means integrability can be used to conjecture exact classical and quantum results interpolating from weak to strong coupling. These two approaches come together in the integrable 2d worldsheet theories of strings, where, remarkably, integrability can be used to compute the energy spectrum of free strings. Moreover, exciting recent developments in the construction of integrable field theories in higher dimensions means that integrability can potentially give direct access to strongly-coupled QFT in 3d and 4d. The Goal of my UKRI FLF and FLF Renewal is to develop novel symmetry-based methods advancing our understanding of QFT and QG in Diverse Dimensions in the framework of String Theory, the AdS/CFT Correspondence and Integrable Models. The four Research Objectives realising this Goal are: I. To classify integrable field theories, including those appearing in String Theory. II. To uncover the universal algebraic structures of Integrable Models. III. To construct new integrable instances of Gauge/Gravity Duality. IV. To investigate applications beyond high-energy physics. The Research Programme is formed of two Work Packages: WP1: Trigonometric and Elliptic String Integrability. Integrable deformations expose hidden structures in a controlled way. They provide the best candidates to construct new integrable instances of Gauge/Gravity Duality. WP2: Integrability in Higher-Dimensional QFT. Systematically constructing and solving new Integrable Models in higher dimensions will dramatically widen the range of physical phenomena we can model with integrable field theories. Integrability allows us to derive exact results in interacting models. Their dynamics is qualitatively different from the chaotic dynamics of many physical setups, but they provide a valuable and highly non-trivial starting point to explore integrability-breaking systems.