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1. The Jordan–Chevalley decomposition for 𝐺-bundles on elliptic curves

   https://doi.org/10.1090/ert/631  

   Frăţilă, Dragoş; Gunningham, Sam; Li, Penghui (December 2022, Representation Theory of the American Mathematical Society)

   We study the moduli stack of degree 0 0 semistable G G -bundles on an irreducible curve E E of arithmetic genus 1 1 , where G G is a connected reductive group in arbitrary characteristic. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups H H of G G (the E E -pseudo-Levi subgroups), where each stratum is computed in terms of H H -bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan–Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where E E has a single cusp (respectively, node), this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra g \mathfrak {g} (respectively, algebraic group G G ). We also provide a Tannakian description of these moduli stacks and use it to show that if E E is not a supersingular elliptic curve, the moduli of framed unipotent bundles on E E are equivariantly isomorphic to the unipotent cone in G G . Finally, we classify the E E -pseudo-Levi subgroups using the Borel–de Siebenthal algorithm, and compute some explicit examples. 

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2. Categorical Equivalence and the Renormalization Group: LMS/EPSRC Durham Symposium on Higher Structures in M‐Theory

   https://doi.org/10.1002/prop.201910019  

   Sharpe, Eric (May 2019, Fortschritte der Physik)

   Abstract In this article we review how categorical equivalences are realized by renormalization group flow in physical realizations of stacks, derived categories, and derived schemes. We begin by reviewing the physical realization of sigma models on stacks, as (universality classes of) gauged sigma models, and look in particular at properties of sigma models on gerbes (equivalently, sigma models with restrictions on nonperturbative sectors), and ‘decomposition,’ in which two‐dimensional sigma models on gerbes decompose into disjoint unions of ordinary theories. We also discuss stack structures on examples of moduli spaces of SCFTs, focusing on elliptic curves, and implications of subtleties there for string dualities in other dimensions. In the second part of this article, we review the physical realization of derived categories in terms of renormalization group flow (time evolution) of combinations of D‐branes, antibranes, and tachyons. In the third part of this article, we review how Landau–Ginzburg models provide a physical realization of derived schemes, and also outline an example of a derived structure on a moduli spaces of SCFTs. 

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3. Proof of the elliptic expansion moonshine conjecture of Căldăraru, He, and Huang

   https://doi.org/10.1090/proc/16032  

   Hong, Letong; Mertens, Michael; Ono, Ken; Zhang, Shengtong (January 2022, Proceedings of the American Mathematical Society)

   Using predictions in mirror symmetry, Căldăraru, He, and Huang recently formulated a “Moonshine Conjecture at Landau-Ginzburg points” [arXiv:2107.12405, 2021] for Klein’s modular j j -function at j = 0 j=0 and j = 1728. j=1728. The conjecture asserts that the j j -function, when specialized at specific flat coordinates on the moduli spaces of versal deformations of the corresponding CM elliptic curves, yields simple rational functions. We prove this conjecture, and show that these rational functions arise from classical 2 F 1 _2F_1 -hypergeometric inversion formulae for the j j -function. 

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4. 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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5. Wall crossing for K‐moduli spaces of plane curves

   https://doi.org/10.1112/plms.12615  

   Ascher, Kenneth; DeVleming, Kristin; Liu, Yuchen (June 2024, Proceedings of the London Mathematical Society)

   Abstract We construct proper good moduli spaces parametrizing K‐polystable ‐Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anticanonical divisor. We then establish a wall‐crossing framework of these K‐moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient is small, the K‐moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K‐moduli spaces are weighted blow‐ups of Kirwan type. We also describe all wall crossings for degree 4,5,6 and relate the final K‐moduli spaces to Hacking's compactification and the moduli of K3 surfaces. 

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