at the

American Institute of Mathematics, San Jose, California

organized by

Jacob Bridgeman, Tobias Osborne, David Penneys, and Julia Plavnik

While the symmetries of a classical mathematical object form a group, we have seen the emergence of quantum mathematical objects such as topological quantum field theories (TQFTs) and non-commutative spaces whose quantum symmetries are better described by tensor categories. Of particular importance are fusion categories, which simultaneously generalize groups and their representation categories. In turn, fusion categories yield TQFTs and associated quantum invariants for links and 3-manifolds. In quantum many body physics, unitary modular tensor categories have recently risen to prominence as a means to describe the topological order of (2+1)D topological phases of matter, whose ground state manifolds are described by TQFTs.

During the past decade, we have witnessed extraordinary progress in understanding complex quantum many body systems. For example, in 2016, the Nobel prize was awarded for the study of topological phases of matter. In particular, variational methods exploiting tensor networks, emerging at the interface between quantum information theory and condensed matter physics, have led to unparalleled progress in understanding strongly correlated quantum many body systems. A growing realisation arising here is that unitary fusion categories label and classify the low energy physics of complex quantum phases of matter. This has led to the development of tensor network techniques which exploit this fusion category structure to efficiently describe these quantum many body systems.

Both the study of fusion categories in mathematics and tensor networks in physics have witnessed tremendous recent progress, leading to many challenging and fascinating problems. Intriguingly, many commonalities and complementary ideas between these two fields have surfaced, promising new collaborative progress on difficult problems. A key objective, of central importance in condensed matter physics, is to understand unitary fusion categories, i.e., fusion categories with a positive dagger structure. Conversely, physical intuition can be used to guide our understanding of fusion categories, their classification, or additional structure that can be imposed.

Specific focus topics for the workshop include:

- tensor-network algorithms for fusion categories and G-crossed braided fusion categories
- higher algebra/category approaches to topological phases and their boundaries
- physical realisations of unitary fusion categories and unitary modular categories

The workshop schedule.

A report on the workshop activities.