Higher Structures from Symmetries in Quantum Field Theory

Abstract

Symmetries in the form of actions of groups and Lie algebras have been an important tool for the analysis of physical systems for a very long time. It has been realized only relatively recently that quantum field theories admit symmetries that are described by more general mathematical structures: fusion categories and their higher-categorical generalizations.

The understanding of such higher categories and the construction of concrete examples raise many mathematical challenges which are at present addressed with a wide variety of mathematical tools, including homotopy theory, the theory of $C^*$-algebras, factorization algebras and operads.

An improved mathematical understanding of higher categories has an immediate impact on the classification and characterization of quantum systems which can realize these categorical structures as symmetries. Particularly note-worthy applications of these symmetries include integrable systems with fusion category symmetries, defects in topological and conformal field theories, the characterization of three-dimensional topological order and systems in quantum information theory realizing quantum codes.

The workshop "Higher Structures from Symmetries in Quantum Field Theory", brought a wide range of mathematical researchers together, working on a broad range of topics, from category theory to condensed matter physics, and provided an extremely fruitful interaction across disciplines. For some leading researchers from neighbouring fields, this was their first visit to Oberwolfach.