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1. Spectral Data for Spin Higgs Bundles

   https://doi.org/10.1093/imrn/rny296  

   Mukhopadhyay, Swarnava; Wentworth, Richard (January 2019, International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper we determine the spectral data parametrizing Higgs bundles in a generic fiber of the Hitchin map for the case where the structure group is the special Clifford group with fixed Clifford norm. These are spin and “twisted” spin Higgs bundles. The method used relates variations in spectral data with respect to the Hecke transformations for orthogonal bundles introduced by Abe. The explicit description also recovers a result from the geometric Langlands program, which states that the fibers of the Hitchin map are the dual abelian varieties to the corresponding fibers of the moduli spaces of projective orthogonal Higgs bundles (in the even case) and projective symplectic Higgs bundles (in the odd case). 

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2. Elliptic bihamiltonian structures from relative shifted Poisson structures

   https://doi.org/10.1112/topo.12315  

   Hua, Zheng; Polishchuk, Alexander (December 2023, Journal of Topology)

   Abstract In this paper, generalizing our previous construction, we equip the relative moduli stack of complexes over a Calabi–Yau fibration (possibly with singular fibers) with a shifted Poisson structure. Applying this construction to the anticanonical linear systems on surfaces, we get examples of compatible Poisson brackets on projective spaces extending Feigin–Odesskii Poisson brackets. Computing explicitly the corresponding compatible brackets coming from Hirzebruch surfaces, we recover the brackets defined by Odesskii–Wolf. 

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3. Integral equivariant cohomology of affine Grassmannians

   https://doi.org/10.4153/S0008439524000122  

   Anderson, David (February 2024, Canadian Mathematical Bulletin)

   Abstract We give explicit presentations of the integral equivariant cohomology of the affine Grassmannians and flag varieties in type A, arising from their natural embeddings in the corresponding infinite (Sato) Grassmannian and flag variety. These presentations are compared with results obtained by Lam and Shimozono, for rational equivariant cohomology of the affine Grassmannian, and by Larson, for the integral cohomology of the moduli stack of vector bundles on. 

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4. Heights on stacks and a generalized Batyrev–Manin–Malle conjecture

   https://doi.org/10.1017/fms.2023.5  

   Ellenberg, Jordan S.; Satriano, Matthew; Zureick-Brown, David (January 2023, Forum of Mathematics, Sigma)

   Abstract We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack $$\mathcal {X}$$ , which specializes to the Batyrev–Manin conjecture when $$\mathcal {X}$$ is a scheme and to Malle’s conjecture when $$\mathcal {X}$$ is the classifying stack of a finite group. 

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5. Vertex algebras of CohFT-type

   Chiara Damiolini, Angela Gibney (January 2022, Cambridge Univ. Press, Cambridge, 2022)

   Sam Payne, et al (Ed.)

   Representations of certain vertex algebras, here called of CohFT-type, can be used to construct vector bundles of coinvariants and conformal blocks on moduli spaces of stable curves [DGT2]. We show that such bundles define semisimple cohomological field theories. As an application, we give an expression for their total Chern character in terms of the fusion rules, following the approach and computation in [MOP+2] for bundles given by integrable modules over ane Lie alge- bras. It follows that the Chern classes are tautological. Examples and open problems are discussed. 

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