发表机构
Department of Mathematics, UNC Chapel Hill（北卡罗来纳大学教堂山分校数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究任意秩平面曲线奇点的动机超多项式与瞬子切片超多项式等的联系，通过平面曲线奇点适当族超多项式的归纳极限得瞬子和新公式，有望影响相关领域，还对特定双曲结超多项式有动机解释及相关猜想。
AI 中文摘要
关键定理是任意秩平面曲线奇点的动机超多项式与相应瞬子切片的超多项式、带导体的涅克拉索夫型瞬子和之间的联系。瞬子切片与紧化雅可比有关，但形成更广泛的类，通常与平面曲线奇点无关，可针对任何孤立曲面奇点定义，本文聚焦于\(\mathbb{A}^2\)中的\((0,0)\)。这有望影响仿射斯普林格纤维理论等相关领域。例如，得到了双曲结\(K12n242\)和\(K12n725\)等超多项式的动机解释。利用平面曲线奇点适当族的超多项式的归纳极限得到任意秩瞬子和的新公式。本文一个重要方向是导体为单项式的情况，猜想相应的瞬子超多项式若超对偶，与新型DAHA超多项式一致，有望与相应考克斯特结的约化霍万诺夫 - 罗赞斯基多项式一致，上述双曲结是最简单的非代数例子。
英文摘要
The key theorem is a connection between motivic superpolynomials of plane curve singularities in any ranks with superpolynomials of the corresponding instanton slices, Nekrasov-type instanton sums with conductors. In this case, instanton slices are related to compactified Jacobians, but they form a much wider class and, generally, have no connection to plane curve singularities. This development is expected to impact theory of affine Springer fibers (at least, in type $A$), and related fields. For instance, we obtain a motivic interpretation (counting $\mathbb{F}_q$-points of some stacks) of superpolynomials for hyperbolic knots K12n242, K12n725, among many others. New formulas for instanton sums in any ranks are obtained using that they are inductive limits of superpolynomials of proper families of plane curve singularities. The main conjecture states that instanton superpolynomials for arbitrary Young diagrams (almost all are not from plane curve singularities) can be transformed to the corresponding DAHA superpolynomials if and only if the latter are positive or (equivalently) the former are superdual. Our DAHA superpolynomials are parallel to Galashin-Lam EHA ones, but we used the nonsymmetric approach. The connection with reduced Khovanov-Rozansky polynomials of the corresponding Coxeter knots is expected. Among applications, formulas for Nekrasov's instanton sums with an impressive connection to nonsymmetric Macdonald polynomials, topological invariance of unibranch motivic superpolynomials for valuation semigroups with 3-4 generators, some mixed-characteristic conjecture, and an upgrade of Weak Riemann Hypothesis.
CommentsSignificant revision: stronger main connection conjecture, enhanced instanton formulas via nonsymmetric Macdonald polynomials, topological invariance of certain families of motivic superpolynomials, and some mixed characteristic conjecture. update of v2, 106 pages
