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1. Rank-finiteness for modular categories

   https://doi.org/10.1090/jams/842  

   Bruillard, Paul; Ng, Siu-Hung; Rowell, Eric; Wang, Zhenghan (July 2016, Journal of the American Mathematical Society)

   null (Ed.)

   We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the context of spherical fusion categories. For a modular category C \mathcal {C} with N = ord ( T ) N= \textrm {ord}(T) , the order of the modular T T -matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimension D 2 D^2 in the Dedekind domain Z [ e 2 π i N ] \mathbb {Z}[e^{\frac {2\pi i}{N}}] is identical to that of N N . 

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2. The Cohomological Brauer Group of a Torsion $${\mathbb{G}}_{m}$$-Gerbe

   https://doi.org/10.1093/imrn/rnz235  

   Shin, Minseon (November 2019, International Mathematics Research Notices)

   Abstract Let $$S$$ be a scheme and let $$\pi : \mathcal{G} \to S$$ be a $${\mathbb{G}}_{m,S}$$-gerbe corresponding to a torsion class $$[\mathcal{G}]$$ in the cohomological Brauer group $${\operatorname{Br}}^{\prime}(S)$$ of $$S$$. We show that the cohomological Brauer group $${\operatorname{Br}}^{\prime}(\mathcal{G})$$ of $$\mathcal{G}$$ is isomorphic to the quotient of $${\operatorname{Br}}^{\prime}(S)$$ by the subgroup generated by the class $$[\mathcal{G}]$$. This is analogous to a theorem proved by Gabber for Brauer–Severi schemes. 

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3. Picard sheaves, local Brauer groups, and topological modular forms

   https://doi.org/10.1112/topo.12333  

   Antieau, Benjamin; Meier, Lennart; Stojanoska, Vesna (May 2024, Journal of Topology)

   Abstract We develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real ‐theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2‐torsion group. 

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4. Monodromy of the Casimir connection of a symmetrisable Kac–Moody algebra

   https://doi.org/10.1007/s00222-024-01242-8  

   Appel, Andrea; Toledano Laredo, Valerio (May 2024, Inventiones mathematicae)

   Let g be a symmetrisable Kac-Moody algebra and V an integrable g-module in category O. We show that the monodromy of the (normally ordered) rational Casimir connection on V can be made equivariant with respect to the Weyl group W of g, and therefore defines an action of the braid group BW on V. We then prove that this action is canonically equivalent to the quantum Weyl group action of BW on a quantum deformation of V, that is an integrable, category O module V over the quantum group Uhg such that V/hV is isomorphic to V. This extends a result of the second author which is valid for g semisimple. 

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5. Fukaya categories of surfaces, spherical objects and mapping class groups

   https://doi.org/10.1017/fms.2021.21  

   Auroux, Denis; Smith, Ivan (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least $$2$$ whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank $$1$$ local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included. 

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