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1. Unitary braided-enriched monoidal categories

   https://doi.org/10.4171/QT/226  

   Dell, Zachary; Huston, Peter; Penneys, David (October 2024, Quantum Topology)

   Braided-enriched monoidal categories were introduced in the work of Morrison–Penneys, where they were characterized using braided central functors. The recent work of Kong–Yuan–Zhang–Zheng and Dell extended this characterization to an equivalence of 2-categories. Since their introduction, braided-enriched fusion categories have been used to describe certain phenomena in topologically ordered systems in theoretical condensed matter physics. While these systems are unitary, there was previously no general notion of unitarity for enriched categories in the literature. We supply the notion of unitarity for enriched categories and braided-enriched monoidal categories and extend the above 2-equivalence to the unitary setting. 

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2. Classification of $$\mathbb{Z}/2\mathbb{Z}$$-quadratic unitary fusion categories (with an appendix by Ryan Johnson, Siu-Hung Ng, David Penneys, Jolie Roat, Matthew Titsworth, and Henry Tucker)

   https://doi.org/10.4171/QT/201  

   Edie-Michell, Cain; Izumi, Masaki; Penneys, David (February 2024, Quantum Topology)

   A unitary fusion category is called $$\mathbb{Z}/2\mathbb{Z}$$-quadratic if it has a $$\mathbb{Z}/2\mathbb{Z}$$ group of invertible objects and one other orbit of simple objects under the action of this group. We give a complete classification of $$\mathbb{Z}/2\mathbb{Z}$$-quadratic unitary fusion categories. The main tools for this classification are skein theory, a generalization of Ostrik's results on formal codegrees to analyze the induction of the group elements to the center, and a computation similar to Larson's rank-finiteness bound for $$\mathbb{Z}/3\mathbb{Z}$$-near group pseudounitary fusion categories. This last computation is contained in an appendix coauthored with attendees from the 2014 AMS MRC on Mathematics of Quantum Phases of Matter and Quantum Information. 

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3. On classification of super-modular categories of rank 8

   https://doi.org/10.1142/S021949882140017X  

   Bruillard, Paul; Plavnik, Julia; Rowell, Eric C.; Zhang, Qing (January 2021, Journal of Algebra and Its Applications)

   null (Ed.)

   We develop categorical and number-theoretical tools for the classification of super-modular categories. We apply these tools to obtain a partial classification of super-modular categories of rank [Formula: see text]. In particular we find three distinct families of prime categories in rank [Formula: see text] in contrast to the lower rank cases for which there is only one such family. 

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4. Unitary Anchored Planar Algebras

   https://doi.org/10.1007/s00220-024-04985-w  

   Henriques, André; Penneys, David; Tener, James (May 2024, Communications in Mathematical Physics)

   Abstract In our previous article (http://arxiv.org/abs/1607.06041), we established an equivalence between pointed pivotal module tensor categories and anchored planar algebras. This article introduces the notion of unitarity for both module tensor categories and anchored planar algebras, and establishes the unitary analog of the above equivalence. Our constructions use Baez’s 2-Hilbert spaces (i.e., semisimple$$\textrm{C}^*$$ ${\text{C}}^{\ast }$-categories equipped with unitary traces), the unitary Yoneda embedding, and the notion of unitary adjunction for dagger functors between 2-Hilbert spaces. 

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5. Integral metaplectic modular categories

   https://doi.org/10.1142/S0218216520500327  

   Deaton, Adam; Gustafson, Paul; Mavrakis, Leslie; Rowell, Eric C.; Poltoratski, Sasha; Timmerman, Sydney; Warren, Benjamin; Zhang, Qing (April 2020, Journal of Knot Theory and Its Ramifications)

   A braided fusion category is said to have Property F if the associated braid group representations factor through a finite group. We verify integral metaplectic modular categories have property F by showing these categories are group-theoretical. For the special case of integral categories [Formula: see text] with the fusion rules of [Formula: see text] we determine the finite group [Formula: see text] for which [Formula: see text] is braided equivalent to [Formula: see text]. In addition, we determine the associated classical link invariant, an evaluation of the 2-variable Kauffman polynomial at a point. 

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