关于库仑分支的几何
On the geometry of Coulomb branches
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中文总结 AI 辅助
本文对三维\(N = 4\)规范理论的库仑分支进行几何描述，通过环面紧致化爆破统一处理不同情形。利用局部描述分类辛叶并描述横向切片，还讨论了相关主题，如改进定理假设及阐明与横向希尔伯特概型关系。
中文摘要 AI 辅助
由Braverman - Finkelberg - Nakajima定义的三维\(N = 4\)规范理论的库仑分支，构成了一类庞大且有趣的辛奇点。本文依据环面紧致化的爆破，对这些簇进行几何描述，涵盖有理、\(K\)-理论和椭圆情形。利用所得局部描述对库仑分支的辛叶进行分类，并依据某些较小库仑分支的零维叶分类描述其横向切片。还讨论了相关主题，如去除Gannon和作者关于库仑分支函子性定理中不必要假设，以及通过改进Bielawski和Foscolo方法阐明与横向希尔伯特概型的关系。
英文摘要
Coulomb branches of 3-dimensional $N=4$ gauge theories, as defined by Braverman--Finkelberg--Nakajima, form a large and interesting class of symplectic singularities. In this paper, we give a geometric description of these varieties in terms of blowups of toric compactifications, uniformly across the rational, $K$-theoretic, and elliptic settings, which in physics are often associated with 3-, 4-, and 5-dimensional versions of these theories. We use the resulting local description to classify the symplectic leaves of Coulomb branches and describe their transverse slices in terms of the classification of zero-dimensional leaves of certain smaller Coulomb branches. We also discuss related topics, including removing an unnecessary hypothesis from a theorem of Gannon and the author on functoriality for Coulomb branches, and clarifying the relationship with transverse Hilbert schemes by refining the approach of Bielawski and Foscolo.
