Mathematical Sciences: Holonomic Fields

[401594] Palmer完整场是与具有孤立奇点的椭圆算子密切相关的量子场论。这些场的相关函数或tau函数已经出现在各种各样的背景下，包括二维Ising模型的缩放极限，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交理论。帕尔默教授对原来的佐藤、三和和金宝理论进行了重新表述，使得考虑各种概括成为可能。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联系。这些顶点运算符在简化（和推广）一些进入顶点运算符代数理论的代数分析方面也应该证明是有用的。量子场论是量子力学的数学模型。此外，这些理论在数学物理中的应用往往代表了数学新发展的开始。这里的建议涉及对早期量子场论的不变量的明确描述。这些描述与近代代数和几何的重要数学发展有着惊人的联系。＊＊＊

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. ***