论文标题
广义仿射施普林格理论和希尔伯特计划在平面曲线上
Generalized affine Springer theory and Hilbert schemes on planar curves
论文作者
论文摘要
我们表明，如Goresky-Kottwitz-Macpherson所定义的，平面曲线奇异性及其抛物线变体的希尔伯特方案可以解释为某些$ gl_n $的弹簧弹簧纤维。利用Aggine Springer理论的概括为Braverman-Finkelberg-Nakajima的库仑分支代数，我们在希尔伯特计划的同源性方面构建了一个理性的Cherednik代数作用，并在示例中对其进行了计算。在此过程中，我们概括了帕哈里奇（Parahoric）的近期建造希尔本·卡姆尼泽（Hilburn-Kamnitzer）周，这可能引起了独立的兴趣。在球形案例中，我们通过用于库仑分支的新的一般定位公式来明确计算。通过Hogancamp-Mellit的结果，我们还显示了理性的Cherednik代数作用于圆环结的homfly同源性。这项工作的启发是部分是由三维$ \ Mathcal {n} = 4 $量学理论的构造。
We show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for $GL_n$, as defined by Goresky-Kottwitz-MacPherson. Using a generalization of affine Springer theory for Braverman-Finkelberg-Nakajima's Coulomb branch algebras, we construct a rational Cherednik algebra action on the homology of the Hilbert schemes, and compute it in examples. Along the way, we generalize to the parahoric setting the recent construction of Hilburn-Kamnitzer-Weekes, which may be of independent interest. In the spherical case, we make our computations explicit through a new general localization formula for Coulomb branches. Via results of Hogancamp-Mellit, we also show the rational Cherednik algebra acts on the HOMFLY homologies of torus knots. This work was inspired in part by a construction in three-dimensional $\mathcal{N}=4$ gauge theory.
