论文标题
A型A型kleinian奇异性的扭曲痕迹和正形式
Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A
论文作者
论文摘要
以下[Beem C.，Peelaers W.，Rastelli L.，Comm。数学。物理。 354（2017），345-392，Arxiv：1601.05378]和[Etingof P.，Stryker D.，Sigma 16（2020），014，28页，Arxiv：1909.13588]，我们对kleinian symilimilial signimolinemlimilial symilimilitial symilimilitial symilinelimilition $a__________________________________________________________________________________________________________________________。特别是，我们为这些痕迹提供了明确的积分公式，并使用它们来确定痕迹何时在相应的代数上定义了阳性遗传形式。这导致了此类量化的单一短星形成的分类，这是Beem，Peelaers和Rastelli与3维超符合场理论相关的问题。特别是，我们确认他们的猜想是，对于$ n \ le 4 $，统一的短星级产物是唯一的，并将其参数计算为量化参数的函数，从而提供了Beem，eemaers和peelaers和rastelli的数值函数的精确公式。如果$ n = 2 $，则尤其是，以$ {\ mathfrak {sl}} _ 2 $恢复统一的球形harish-chandra bimodules的理论。因此，本文的结果可能被视为对统一的Harish-Chandra Bimodules概括的概括的起点，而不是围绕还原的代数代数代数[Vogan Jr. D.A.，《数学研究年鉴》，第1卷。 118，普林斯顿大学出版社，新泽西州普林斯顿，1987年]到更通用的量子代数。最后，我们得出复发以计算与扭曲痕迹相对应的短星形产品的系数，这些痕迹是离散painlevé系统的概括。
Following [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392, arXiv:1601.05378] and [Etingof P., Stryker D., SIGMA 16 (2020), 014, 28 pages, arXiv:1909.13588], we undertake a detailed study of twisted traces on quantizations of Kleinian singularities of type $A_{n-1}$. In particular, we give explicit integral formulas for these traces and use them to determine when a trace defines a positive Hermitian form on the corresponding algebra. This leads to a classification of unitary short star-products for such quantizations, a problem posed by Beem, Peelaers and Rastelli in connection with 3-dimensional superconformal field theory. In particular, we confirm their conjecture that for $n\le 4$ a unitary short star-product is unique and compute its parameter as a function of the quantization parameters, giving exact formulas for the numerical functions by Beem, Peelaers and Rastelli. If $n=2$, this, in particular, recovers the theory of unitary spherical Harish-Chandra bimodules for ${\mathfrak{sl}}_2$. Thus the results of this paper may be viewed as a starting point for a generalization of the theory of unitary Harish-Chandra bimodules over enveloping algebras of reductive Lie algebras [Vogan Jr. D.A., Annals of Mathematics Studies, Vol. 118, Princeton University Press, Princeton, NJ, 1987] to more general quantum algebras. Finally, we derive recurrences to compute the coefficients of short star-products corresponding to twisted traces, which are generalizations of discrete Painlevé systems.
