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1. Functional Transcendence of Periods and the Geometric André–Grothendieck Period Conjecture

   https://doi.org/10.1017/fms.2025.10036  

   Bakker, Benjamin; Tsimerman, Jacob (January 2025, Forum of Mathematics, Sigma)

   Abstract We prove a functional transcendence theorem for the integrals of algebraic forms in families of algebraic varieties. This allows us to prove a geometric version of André’s generalization of the Grothendieck period conjecture, which we state using the formalism of Nori motives. More precisely, we prove a version of the Ax–Schanuel conjecture for the comparison between the flat and algebraic coordinates of an arbitrary admissible graded polarizable variation of integral mixed Hodge structures. This can be seen as a generalization of the recent Ax–Schanuel theorems of [13, 18] for mixed period maps. 

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2. Classification of Classical Spin Liquids: Typology and Resulting Landscape

   Yan, Han; Benton, Owen; Moessner, Roderich; Nevidomskyy, Andriy H. (April 2023, arXivorg)

   Classical spin liquids (CSL) lack long-range magnetic order and are characterized by an extensive ground state degeneracy. We propose a classification scheme of CSLs based on the structure of the flat bands of their Hamiltonians. Depending on absence or presence of the gap from the flat band, the CSL are classified as algebraic or fragile topological, respectively. Each category is further classified: the algebraic case by the nature of the emergent Gauss's law at the gap-closing point(s), and the fragile topological case by the homotopy of the eigenvector winding around the Brillouin zone. Previously identified instances of CSLs fit snugly into our scheme, which finds a landscape where algebraic CSLs are located at transitions between fragile topological ones. It also allows us to present a new, simple family of models illustrating that landscape, which hosts both fragile topological and algebraic CSLs, as well as transitions between them 

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3. Classification of Classical Spin Liquids: Detailed Formalism and Suite of Examples

   Yan, Han; Benton, Owen; Nevidomskyy, Andriy H.; Moessner, Roderich (May 2023, arXivorg)

   The hallmark of highly frustrated systems is the presence of many states close in energy to the ground state. Fluctuations between these states can preclude the emergence of any form of order and lead to the appearance of spin liquids. Even on the classical level, spin liquids are not all alike: they may have algebraic or exponential correlation decay, and various forms of long wavelength description, including vector or tensor gauge theories. Here, we introduce a classification scheme, allowing us to fit the diversity of classical spin liquids (CSLs) into a general framework as well as predict and construct new kinds. CSLs with either algebraic or exponential correlation-decay can be classified via the properties of the bottom flat band(s) in their soft-spin Hamiltonians. The classification of the former is based on the algebraic structures of gapless points in the spectra, which relate directly to the emergent generalized Gauss's laws that control the low temperature physics. The second category of CSLs, meanwhile, are classified by the fragile topology of the gapped bottom band(s). Utilizing the classification scheme we construct new models realizing exotic CSLs, including one with anisotropic generalized Gauss's laws and charges with subdimensional mobility, one with a network of pinch-line singularities in its correlation functions, and a series of fragile topological CSLs connected by zero-temperature transitions. 

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4. Subspace foliations and collapse of closed flat manifolds

   https://doi.org/10.1002/mana.202000156  

   Bettiol, Renato G.; Derdzinski, Andrzej; Mossa, Roberto; Piccione, Paolo (October 2022, Mathematische Nachrichten)

   Abstract We study relations between certain totally geodesic foliations of a closed flat manifold and its collapsed Gromov–Hausdorff limits. Our main results explicitly identify such collapsed limits as flat orbifolds, and provide algebraic and geometric criteria to determine whether they are singular. 

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5. A Hitchin connection on nonabelian theta functions for parabolic 𝐺-bundles

   https://doi.org/10.1515/crelle-2023-0049  

   Biswas, Indranil; Mukhopadhyay, Swarnava; Wentworth, Richard (September 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract For a simple, simply connected complex affine algebraic group 𝐺, we prove the existence of a flat projective connection on the bundle of nonabelian theta functions on the moduli spaces of semistable parabolic 𝐺-bundles for families of smooth projective curves with marked points. 

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