Generalized Symmetries in Quantum Field Theories

This PhD project will explore the mathematical and physical implications of higher-form symmetries in Quantum Field Theories (QFTs), and extensions thereof includ-ing higher-group structures and other categorical symmetries. Higher-form sym-metries have charged objects that are higher-dimensional, defect operators in the QFT, e.g. line operators in Yang-Mills theory. These are charged under topological operators, which form a so-called higher-form symmetry group.
The higher-group structure arises whenever higher-form symmetries of a given theory do not simply form a product group, but a "generalized extension".
Such symmetries have only recently been uncovered in Mathematical Physics. Complementing this, their mathematical structure has recently seen a lot of exciting developments in algebraic topology/category theory.
This project aims to synergize these developments and to explore the physical im-plications of such symmetries upon 4d and lower dimensional gauge theories: specifically, understanding the role of higher-groups in constraining the dynamics and vacuum structure of QFTs, and in particular N=1 supersymmetric gauge theo-ries.
The goal is to obtain a comprehensive understanding of all higher-group and gen-eralized symmetry structures in 4d.
Concretely the objectives are:

1. Determining the global flavor symmetry group, 1-form symmetries and the 2-group structures or anomalies that these higher-form symmetries form, for 4d gauge theories with matter as well as quiver gauge theories.
2. Determining the implications on the vacuum structure of such QFTs, in par-ticular in view of confinement.
3. Matching these generalized symmetries across QFT dualities (e.g. Seiberg-like dualities).
4. Developing methods to realize these generalized symmetries in string theo-ry constructions of QFTs.
5. Generalizing to lower-dimensional QFTs (with applications in condensed matter) as well as non-supersymmetric theories.


The methodology is founded in Mathematical Physics, with a very strong compo-nent in algebraic topology and category theory.
The emergence of categorical symmetries is very recent and the project intends to combine these with modern tools in QFT such as dualities, geometric realization in string theory and applications to lower dimensional theories, such as condensed matter.

This aligns with the EPSRC research strategy for Mathematical Physics,
algebraic topology, and, on the Physics side, condensed matter. This project stands out as an interdisciplinary research program, which connects to the vibrant re-search environment in the UK in said subjects.