Mathematical Sciences: Holonomic Fields

9401594帕尔默完整体场是与具有孤立奇点的椭圆算符有密切联系的量子场论。这些场的关联函数或tau函数已经出现在各种各样的背景中，包括二维Ising模型的标度极限，复球面上的Riemann-Hilbert问题，随机矩阵的能级间距分布，KdV族及其近亲，Toeplitz行列式的渐近性，以及最近关于Riemann曲面的模空间的相交理论。帕尔默教授对最初的佐藤、三和和金博理论进行了重新表述，使我们有可能考虑各种概括。这里的建议是通过发展顶点算符理论的解析版本来完成对双曲圆盘上的狄拉克算符的分析，并将场论联系起来。这些顶点算子在简化(和推广)一些进入顶点算子代数理论的代数分析时也应该被证明是有用的。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着数学新发展的开始。这里的提议涉及到对早期量子场理论中不变量的明确描述。这些描述导致了与代数和几何领域最近重要的数学发展之间令人惊讶的联系。***

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. ***