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We develop the rewriting theory for monoidal supercategories and2-supercategories. This extends the theory of higher-dimensional rewriting established for (linear) 2-categories to the super setting, providing a suite of tools for constructing bases and normal forms for2-supercategories given by generators and relations. We then employ this newly developed theory to prove the non-degeneracy conjecture for the odd categorification of quantum\mathfrak{sl}(2)from A. Ellis and A. Lauda [Quantum Topol. 7 (2016), 329–433] and J. Brundan and A. Ellis [Proc. Lond. Math. Soc. (3) 115 (2017), 925–973] As a corollary, this gives a classification of dg-structures on the odd2-category conjectured by A. Lauda and I. Egilmez [Quantum Topol. 11 (2020), 227–294].
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Abstract We relate the Fukaya category of the symmetric power of a genus zero surface to deformed category$$\mathcal {O}$$ of a cyclic hypertoric variety by establishing an isomorphism between algebras defined by Ozsváth–Szabó in Heegaard–Floer theory and Braden–Licata–Proudfoot–Webster in hypertoric geometry. The proof extends work of Karp–Williams on sign variation and the combinatorics of the$$m=1$$ amplituhedron. We then use the algebras associated to cyclic arrangements to construct categorical actions of$$\mathfrak {gl}(1|1)$$ , and generalize our isomorphism to give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement.more » « less
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Abstract We develop the categorical context for defining Hermitian non‐semisimple topological quantum field theories (TQFTs). We prove that relative Hermitian modular categories give rise to modified Hermitian Witten–Reshetikhin–Turaev TQFTs and provide numerous examples of these structures coming from the representation theory of quantum groups and quantum superalgebras. The Hermitian theory developed here for the modified Turaev–Viro TQFT is applied to define new pseudo‐Hermitian topological phases that can be considered as non‐semisimple analogs of Levin–Wen models.more » « less
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Abstract Rickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology,Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y -ification of Khovanov–Rozansky homology,preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer,Knot homology via derived categories of coherent sheaves IV, colored links,Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).more » « less
