Mathematical Sciences: Holonomic Fields

数学科学：完整域

• 批准号: 9401594
• 负责人: John Palmer
• 金额: $ 6.52万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401594&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Holonomic; Fields

9401594 Palmer Holonomic fields are quantum field theories with an intimate connection to elliptic operators with isolated singularities. The correlation function or tau-function of such fields have appeared in a wide variety of contexts including, the scaling limit of the two dimensional Ising model, the Riemann-Hilbert problem on the complex sphere, level spacing distributions for random matrices, the KdV hierarchy and its relatives, the asymptotics of Toeplitz determinants, and recently intersection theory on the moduli space of Riemann surfaces. A reformulation of the original Sato, Miwa and Jimbo theory that has been worked out by Professor Palmer makes it possible to consider a variety of generalizations. The proposal here is to complete the analysis of the Dirac operators on the hyperbolic disk and to make the field theory connection by developing an analytic version of the theory of vertex operators. These vertex operators should also prove useful in simplifying (and generalizing) some of the algebraic analysis that goes into the theory of vertex operator algebras. Quantum field theories are mathematical models for quantum mechanics. In addition the use in mathematical physics in these theories often represent the beginning of new developments in mathematics. The proposal here involves explicit description of invariants of earlier quantum field theories. These descriptions lead to surprising connections with important recent mathematical developments in algebra and geometry. ***

小行星9401594 完整场是量子场论，与孤立奇点的椭圆算子有着密切的联系。这种场的相关函数或τ函数出现在各种各样的背景下，包括二维伊辛模型的标度极限，复球面上的黎曼-希尔伯特问题，随机矩阵的水平间距分布，KdV层次及其相关，Toeplitz行列式的渐近性，以及最近黎曼曲面模空间上的相交理论。帕默教授重新表述了原来的佐藤、三和金宝理论，使人们有可能考虑各种推广。这里的建议是完成双曲圆盘上狄拉克算子的分析，并通过发展顶点算子理论的解析版本来建立场论的联系。这些顶点算子在简化（和推广）顶点算子代数理论中的一些代数分析时也应该是有用的。量子场论是量子力学的数学模型。此外，在数学物理中使用这些理论往往代表着数学新发展的开始。这里的提议涉及到对早期量子场论不变量的明确描述。这些描述导致了令人惊讶的联系与重要的最近数学发展代数和几何。 ***