EAPSI: Classification of Fusion Categories with one Non-Invertible Object

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Fusion categories are mathematical structures which are used to encode invariants of a wide variety of mathematical systems. A mathematical invariant is a quantity or other property of a mathematical object that remains unchanged under transformations of that object. For example, the symmetries of a given geometric object may be expressed as the data of a fusion category; certain properties of these symmetries must remain the same under deformations of the object, and this is reflected in the invariant theory of fusion categories. One primary goal of this project is to unify several different approaches to the study of fusion categories coming from both mathematics and physics. Kyoto University Professor Masaki Izumi, the host scientist for this project, is a leading international researcher in fusion category theory from the physics-influenced point of view. This collaboration will further the interaction between the two communities and will result in new mathematical questions arising from established physical theories.This project aims to classify fusion categories having one object that is non-invertible under the tensor product: these are called the near-group fusion categories. Izumi has developed a technique utilizing the theory of algebras of operators on Hilbert spaces to make this classification; specifically, he realizes their objects as endomorphisms on Cuntz C* algebras and classifies their possible images. Together Izumi and the principal investigator (PI) will realize the near-groups in this way and provide classification parameters for the possible equivalence classes of fusion categories having the near-group fusion rule. Several different important families of fusion category invariants will also be computed, including the modular data for the Drinfel'd centers and the Frobenius-Schur indicators.This award under the East Asia and Pacific Summer Institutes program supports summer research by a U.S. graduate student and is jointly funded by NSF and the Japan Society for the Promotion of Science.