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1. On the moduli spaces of 4d N=3 SCFTs I: triple special Kähler structure

   Argyres, Philip; Bouget, Antoine; Martone, Mario (December 2019, ArXivorg)

   We initiate a systematic analysis of moduli spaces of vacua of four dimensional =3 SCFTs. Our analysis is based on the one hand on the properties of =3 chiral rings --- which we review in detail and contrast with chiral rings of theories with less supersymmetry --- and on the other hand on constraints coming from low-energy supersymmetry. This leads us to introduce a new type of geometric structure, which characterizes =3 SCFT moduli spaces, and that we call triple special Kähler (TSK). A rank-n TSK moduli space has complex dimension 3n, and is singular at complex co-dimension 3 subspaces where charged states become massless. The structure of singularities defines a stratification of the TSK space in terms of lower-dimensional TSK manifolds. 

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2. On the tensor rank of the 3 x 3 permanent and determinant

   https://doi.org/10.13001/ela.2021.5107  

   Krishna, Siddharth; Makam, Visu (January 2021, The Electronic Journal of Linear Algebra)

   null (Ed.)

   The tensor rank and border rank of the $$3 \times 3$$ determinant tensor are known to be $$5$$ if the characteristic is not two. In characteristic two, the existing proofs of both the upper and lower bounds fail. In this paper, we show that the tensor rank remains $$5$$ for fields of characteristic two as well. 

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3. Tate’s thesis in the de Rham setting

   https://doi.org/10.1090/jams/1010  

   Hilburn, Justin; Raskin, Sam (July 2023, Journal of the American Mathematical Society)

   We calculate the category of D D -modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham local systems with a flat section. Our result establishes a conjecture coming out of the 3 d 3d mirror symmetry program, which obtains new compatibilities for the geometric Langlands program from rich dualities of QFTs that are themselves obtained from string theory conjectures. 

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4. Compactifications of Moduli of Points and Lines in the Projective Plane

   https://doi.org/10.1093/imrn/rnab200  

   Schaffler, Luca; Tevelev, Jenia (August 2021, International Mathematics Research Notices)

   Abstract Projective duality identifies the moduli spaces $$\textbf{B}_n$$ and $$\textbf{X}(3,n)$$ parametrizing linearly general configurations of $$n$$ points in $$\mathbb{P}^2$$ and $$n$$ lines in the dual $$\mathbb{P}^2$$, respectively. The space $$\textbf{X}(3,n)$$ admits Kapranov’s Chow quotient compactification $$\overline{\textbf{X}}(3,n)$$, studied also by Lafforgue, Hacking, Keel, Tevelev, and Alexeev, which gives an example of a KSBA moduli space of stable surfaces: it carries a family of certain reducible degenerations of $$\mathbb{P}^2$$ with $$n$$ “broken lines”. Gerritzen and Piwek proposed a dual perspective, a compact moduli space parametrizing certain reducible degenerations of $$\mathbb{P}^2$$ with $$n$$ smooth points. We investigate the relation between these approaches, answering a question of Kapranov from 2003. 

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5. K-stability and birational models of moduli of quartic K3 surfaces

   https://doi.org/10.1007/s00222-022-01170-5  

   Ascher, Kenneth; DeVleming, Kristin; Liu, Yuchen (May 2023, Inventiones mathematicae)

   Abstract We show that the K-moduli spaces of log Fano pairs $$({\mathbb {P}}^3, cS)$$ ( P 3 , c S ) where S is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily–Borel compactification of moduli of quartic K3 surfaces as c varies in the interval (0, 1). We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza–O’Grady’s prediction on the Hassett–Keel–Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of $${\mathbb {P}}^3$$ P 3 . 

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