Quantum Symmetries in String Theory

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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.