SYMSPEC · Symmetries and Spectra

Symmetry is ubiquitous in nature. It is a fundamental concept and organising principle in quantum field theory, the common language of modern theoretical physics. The past few years have witnessed a radical evolution in the notion of symmetry that transcends the paradigm of groups and representation theory and marches into the realm of higher mathematics. This is known as categorical symmetry. The field has experienced a period of rapid discovery and exploration but the underlying mathematical structure remains poorly understood and this hinders progress on the most pressing open problems. Now is the time to take stock of recent developments and establish new mathematical foundations and computational tools in order to take advantage of this unprecedented opportunity to explore novel symmetries and gain new insights into physical systems described by quantum field theory.The purpose of this research programme is to establish a coherent mathematical framework for categorical symmetry. The research programme introduces and explores the notion of ``categorical spectrum"", which compiles topological quantum field theories of all dimensions into a mathematical structure that opens up powerful conceptual and computational advances. This is applied to the exploration of categorical symmetries, 't Hooft anomalies and gapped phases of matter, catalysing a step change in the discovery of new examples and applications.""