Quantum Symmetries: Subfactors, Topological Phases, and Higher Categories

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Symmetry arises in many places in mathematics and the physical sciences. Typically, symmetries of a system are described by the mathematical notion of a group, which is a set with a unit and multiplication, where every operation has an inverse. Groups act on objects by structure-preserving maps. In recent decades, we have seen the emergence of quantum mathematical objects, like von Neumann algebras and topological phases of matter, which naturally live in "higher categories," and so their symmetries are better described by tensor categories. This project aims to study the higher categories arising in the study of von Neumann algebras and topological phases of matter to better understand these quantum systems and their higher quantum symmetries. The project provides research training opportunities for undergraduate and graduate students.This project has three main focuses. First, it continues the investigation of small index subfactors and unitary fusion categories to search for new examples of exotic quantum symmetries. Second, it investigates a nets of operator algebras approach to (2+1)D topological phases of matter, analogous to the conformal net description of conformal field theory. Third, it investigate unitarity for higher categories and its relationship to boundaries and phase transitions for (2+1)D topological phases of matter.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.