{"title":"Quantum Differential and Difference Equations for $$\\textrm{Hilb}^{n}(\\mathbb {C}^2)$$","authors":"Andrey Smirnov","doi":"10.1007/s00220-024-05056-w","DOIUrl":null,"url":null,"abstract":"<p>We consider the quantum <i>difference</i> equation of the Hilbert scheme of points in <span>\\({{\\mathbb {C}}}^2\\)</span>. This equation is the K-theoretic generalization of the quantum differential equation discovered by A. Okounkov and R. Pandharipande in [27]. We obtain two explicit descriptions for the monodromy of these equations - representation-theoretic and algebro-geometric. In the representation-theoretic description, the monodromy acts via certain explicit elements in the quantum toroidal algebra . In the algebro-geometric description, the monodromy features as transition matrices between the stable envelope bases in equivariant K-theory and elliptic cohomology. Using the second approach we identify the monodromy matrices for the differential equation with the K-theoretic <i>R</i>-matrices of cyclic quiver varieties, which appear as subvarieties in the 3<i>D</i>-mirror Hilbert scheme. Most of the results in the paper are illustrated by explicit examples for cases <span>\\(n=2\\)</span> and <span>\\(n=3\\)</span> in the Appendix.\n</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s00220-024-05056-w","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the quantum difference equation of the Hilbert scheme of points in \({{\mathbb {C}}}^2\). This equation is the K-theoretic generalization of the quantum differential equation discovered by A. Okounkov and R. Pandharipande in [27]. We obtain two explicit descriptions for the monodromy of these equations - representation-theoretic and algebro-geometric. In the representation-theoretic description, the monodromy acts via certain explicit elements in the quantum toroidal algebra . In the algebro-geometric description, the monodromy features as transition matrices between the stable envelope bases in equivariant K-theory and elliptic cohomology. Using the second approach we identify the monodromy matrices for the differential equation with the K-theoretic R-matrices of cyclic quiver varieties, which appear as subvarieties in the 3D-mirror Hilbert scheme. Most of the results in the paper are illustrated by explicit examples for cases \(n=2\) and \(n=3\) in the Appendix.
我们考虑的是({{\mathbb {C}}}^2\ )中点的希尔伯特方案的量子差分方程。这个方程是 A. Okounkov 和 R. Pandharipande 在 [27] 中发现的量子微分方程的 K 理论广义化。我们对这些方程的单调性有两种明确的描述--表示理论描述和几何代数描述。在表征理论描述中,单反通过量子环代数中的某些明确元素起作用。在椭圆几何描述中,单旋转是等变 K 理论和椭圆同调中稳定包络基之间的过渡矩阵。利用第二种方法,我们将微分方程的单旋转矩阵与循环簇变体的 K 理论 R 矩阵相识别,后者作为子变体出现在三维镜像希尔伯特方案中。论文中的大部分结果都在附录中以 \(n=2\) 和 \(n=3\) 两种情况的明确例子加以说明。
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.