Categorical Symmetries of Operator Algebras

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Symmetries are a fundamental part of the way we mathematically model the physical world. They provide a paradigm for characterizing physical theories across the spectrum, from high energy to condensed matter physics. Traditionally, symmetries are described by mathematical objects called groups. However, in recent decades, interesting quantum theories have emerged whose fundamental symmetries are not reversible, requiring an extension of our classical ideas of symmetry beyond groups. Quantum theories can be described in the language of operator algebras, and these new kinds of symmetry can be realized by mathematical objects called tensor categories acting on operator algebras. The goal of this project is to provide classification results for categorical symmetry of operator algebras with an emphasis on situations relevant to quantum spin systems. These results will, in particular, help provide a rigorous understanding of topologically ordered phases of matter. This project will incorporate research opportunities for graduate and undergraduate students at North Carolina State University, with an emphasis on the recruitment of students from underrepresented groups.This project has two main components. In the first, the principal investigator will provide a classification of approximately finite dimensional actions of amenable tensor categories on approximately finite dimensional C*-algebras in terms of K-theoretic invariants. Amenable tensor categories simultaneously generalize discrete amenable groups and the representation categories of compact groups, while approximately finite dimensional actions provide categorical generalizations of global symmetries of 1D lattice spin systems. The principal investigator will extend their previous results in this direction from the case of fusion categories to the infinite amenable setting and apply these results to obtain new classifications for topologically ordered spin systems. In the second component, the principal investigator will generalize their previously developed categorical chi invariant for von Neumann algebras to an invariant for group actions on von Neumann algebras. The principal investigator will use this invariant to distinguish group actions on McDuff factors.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.