Quantum field theory (QFT) is the modern theoretical framework that underlies particle physics and results from combining the laws of quantum mechanics with Einstein’s theory of special relativity. Its predictions help us understand nature from the smallest to the largest of scales, ranging from the interactions observed at colliders, to the properties of materials, to the origins of matter in the early universe. In situations where fundamental particles interact weakly with one another (for example, in the interactions of light with charged particles) there exists a set of techniques in QFT that rely on perturbation theory and have been extraordinarily successful at explaining observed physical phenomena. On the other hand, many fundamental phenomena, including phase transitions and nuclear interactions, are described by strongly interacting systems for which perturbative techniques are insufficient and a rigorous, predictive theoretical formulation is lacking.
There are reasons to expect that a satisfactory reformulation of QFT which is valid beyond weak coupling requires developing a novel understanding of its fundamental degrees of freedom. This is the overarching goal of my project. In particular, there is evidence that the dynamics of strongly interacting systems cannot be understood solely in terms of the interactions between fundamental particles. Rather, extended degrees of freedom like strings or membranes also play an essential role. A prototypical example of this are flux tubes which stretch between quarks and bind them within the nucleus.
I aim to obtain a precise description of the extended degrees of freedom in a variety of QFTs and use this knowledge to improve our understanding of strongly coupled QFTs beyond what can be achieved by more conventional means. A major component of this is to identify the algebraic and geometric structures which encode the physics of extended degrees of freedom. Achieving this enables us to tackle a number of fundamental questions, including understanding how these mathematical structures manifest themselves in the physics of strongly coupled QFTs, how they can be used to perform exact computations beyond perturbation theory, and how they constrain the space of possible QFTs. In the course of the project we have found applications beyond what was expected at the outset: in particular, we have discovered a novel way to employ extended degrees of freedom to study QFTs that interact with gravity.
