Phases of quantum field theories: symmetries and vacua

Quantum field theory (QFT) is a framework used in theoretical physics to explain various physical phenomena, ranging from the smallest subatomic scales to the larger scales of our universe. While QFT has been successful in many ways, there is still much we do not understand about it. To address pressing open questions, we focus on a simplified version called supersymmetric QFTs. This allows us to study specific QFT aspects in a controlled manner. In a sense, this is a laboratory environment for quantum field theories. Two key features of supersymmetric QFTs are their vacua (ground states) and symmetries. Vacua play a crucial role as they determine the phase of the system by defining what particles can exist. Symmetries, on the other hand, define the rules for motion and interaction of the particles. In my recent research, I made some exciting discoveries that challenge our current understanding of vacua. Surprisingly, I found that some vacua exhibit unexpected quantum behaviours. To analyse these phenomena, I developed a specialised program that provides a powerful algorithm and exact computational techniques to explore these quantum vacua. Using this program, I can determine the exact spectrum (i.e. what particles exist), identify the symmetries involved, and even create phase diagrams. This project has three main objectives. Firstly, together with my START-team, I aim to classify all the quantum vacua in six-dimensional supersymmetric QFTs, which hold a pivotal role in the realm of supersymmetric quantum field theories. This will lay the foundation for a systematic study of QFTs across different dimensions, not just in six space-time dimensions. Secondly, we will analyse the underlying mathematical structure of these new quantum vacua, which sheds light on the new rules governing the quantum behaviour. Lastly, we will explore a novel direction: studying four-dimensional QFTs by utilising the lessons learnt from the quantum vacua of the six-dimensional supersymmetric QFTs. The outcomes of this project will advance both physics and mathematics. In physics, the results will provide unprecedented insights into the quantum vacua of QFTs in various dimensions. In mathematics, the intuitive QFT techniques offer a fresh perspective on the underlying geometric structures.