Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups I

Lentner, Simon; Mierach, Svea Nora; Schweigert, Christoph; Sommerhäuser, Yorck

doi:10.1007/978-981-19-4645-5

Cited by 1 publication

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“…Generalizations of part of the theory and applications of modular fusion categories to lowdimensional topology have been obtained for non-semisimple (i.e., not necessarily semisimple) modular categories. These constructions include equivalent characterizations of modularity conditions [42], mapping class group actions and modular functors [19,29,41], and partially defined non-semisimple TFTs [10,25]. In general, it is still open whether the non-semisimple braided categories of representations of a logarithmic conformal field theories are modular [24,30].…”

Section: Introduction 1motivationmentioning

confidence: 99%

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Frobenius Monoidal Functors of Dijkgraaf-Witten Categories and Rigid Frobenius Algebras

Hannah,

Laugwitz,

Ros Camacho

2023

SIGMA

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We construct a separable Frobenius monoidal functor from $\mathcal{Z}\big(\mathsf{Vect}_H^{\omega|_H}\big)$ to $\mathcal{Z}\big(\mathsf{Vect}_G^\omega\big)$ for any subgroup $H$ of $G$ which preserves braiding and ribbon structure. As an application, we classify rigid Frobenius algebras in $\mathcal{Z}\big(\mathsf{Vect}_G^\omega\big)$, recovering the classification of étale algebras in these categories by Davydov-Simmons [J. Algebra 471 (2017), 149-175, arXiv:1603.04650] and generalizing their classification to algebraically closed fields of arbitrary characteristic. Categories of local modules over such algebras are modular tensor categories by results of Kirillov-Ostrik [Adv. Math. 171 (2002), 183-227, arXiv:math.QA/0101219] in the semisimple case and Laugwitz-Walton [Comm. Math. Phys., to appear, arXiv:2202.08644] in the general case.

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