Resurgence理论及其应用

Resurgence theory and its applications

本文综述了法国数学家Jeanécalle发展的复现理论及其在数学和物理中的若干重要应用。系统介绍简单复现函数的基本理论及其模运算,并提及更一般奇性的处理。给出该理论基本构造的同时,以欧拉级数、斯特林级数、艾里函数以及量子双对数函数作为基本例子来阐释这一理论的精妙之处。并提及其在量子力学(精密WKB理论、半经典迹公式)、规范场理论(特别是复陈-赛蒙斯理论)以及拓扑弦理论中的若干应用。

This paper gives an introduction to the resurgence theory developed by French mathematician JeanÉcalle and its important applications in mathematics and physics.The theory of simple resurgent functions and mould calculus are systematically presented with briefly mentioning general singularities.Euler series,Stirling series,Airy function and quantum dilogarithm are taken as examples to explain the subtlety of the theory.Applications in quantum mechanics(exact WKB theory,semi-classical trace formula),gauge theory(especially complex Chern-Simons theory)and topological string theory are also mentioned.