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1. On dualizability of braided tensor categories

   https://doi.org/10.1112/S0010437X20007630  

   Brochier, Adrien; Jordan, David; Snyder, Noah (March 2021, Compositio Mathematica)

   null (Ed.)

   We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $$q$$ . 

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2. Support varieties without the tensor product property

   https://doi.org/10.1112/blms.13048  

   Bergh, Petter Andreas; Plavnik, Julia Yael; Witherspoon, Sarah (June 2024, Bulletin of the London Mathematical Society)

   Abstract We show that over a perfect field, every non‐semisimple finite tensor category with finitely generated cohomology embeds into a larger such category where the tensor product property does not hold for support varieties. 

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3. Stratification and the comparison between homological and tensor triangular support

   https://doi.org/10.1093/qmath/haac040  

   Barthel, Tobias; Heard, Drew; Sanders, Beren (December 2022, The Quarterly Journal of Mathematics)

   Abstract We compare the homological support and tensor triangular support for ‘big’ objects in a rigidly-compactly generated tensor triangulated category. We prove that the comparison map from the homological spectrum to the tensor triangular spectrum is a bijection and that the two notions of support coincide whenever the category is stratified, extending the work of Balmer. Moreover, we clarify the relations between salient properties of support functions and exhibit counter-examples highlighting the differences between homological and tensor triangular support. 

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4. Vertex operator superalgebras and the 16-fold way

   https://doi.org/10.1090/tran/8454  

   Dong, Chongying; Ng, Siu-Hung; Ren, Li (July 2021, Transactions of the American Mathematical Society)

   Let V V be a vertex operator superalgebra with the natural order 2 automorphism σ \sigma . Under suitable conditions on V V , the σ \sigma -fixed subspace V 0 ¯ V_{\bar 0} is a vertex operator algebra and the V 0 ¯ V_{\bar 0} -module category C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a modular tensor category. In this paper, we prove that C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a fermionic modular tensor category and the Müger centralizer C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 of the fermion in C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is generated by the irreducible V 0 ¯ V_{\bar 0} -submodules of the V V -modules. In particular, C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 is a super-modular tensor category and C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a minimal modular extension of C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 . We provide a construction of a vertex operator superalgebra V l V^l for each positive integer l l such that C ( V l ) 0 ¯ \mathcal {C}_{{(V^l)_{\bar 0}}} is a minimal modular extension of C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 . We prove that these modular tensor categories C ( V l ) 0 ¯ \mathcal {C}_{{(V^l)_{\bar 0}}} are uniquely determined, up to equivalence, by the congruence class of l l modulo 16. 

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5. Tensor category describing anyons in the quantum Hall effect and quantization of conductance

   https://doi.org/10.1142/S0129055X24610075  

   Bachmann, Sven; Corbelli, Matthew; Fraas, Martin; Ogata, Yoshiko (May 2025, Reviews in Mathematical Physics)

   In this study, we examine the quantization of Hall conductance in an infinite plane geometry. We consider a microscopic charge-conserving system with a pure, gapped infinite-volume ground state. While Hall conductance is well-defined in this scenario, existing proofs of its quantization have relied on assumptions of either weak interactions, or properties of finite volume ground state spaces, or invertibility. Here, we assume that the conditions necessary to construct the braided [Formula: see text]-tensor category which describes anyonic excitations are satisfied, and we demonstrate that the Hall conductance is rational if the tensor category is finite. 

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