Equivariant Localization of Path Integrals
Equivariant Localization of Path Integrals
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发表时间：
1996-08
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通讯作者：
R. Szabo
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作者：
R. Szabo
我们回顾了计算费曼路径积分的等变定位技术。我们发展了系统的几何方法来研究动力系统相空间路径积分的半经典性质，强调了与可积和拓扑量子场论的关系。从详细回顾相关的数学背景——等变上同调和Duistermaat-Heckman定理开始，我们展示了局部化思想如何与经典可积性相关，以及如何使用BRST量化技术将它们正式扩展到在特殊情况下导出路径积分的显式局部化公式。在量子可积性和拓扑场论中，提出了各种环空间局部化的概念。我们强调了这些可局部化模型通常具有的一般对称性，并利用这些对称性讨论了局部化公式的适用范围。在基本量子力学、摩尔斯理论、指数定理、半简单李群的特征公式、自旋系统的量子化、矩阵模型中的酉积分、黎曼曲面的模不变量、超对称量子场论、二维杨-米尔斯理论、共形场论、上同调场论和量子场论中的环展开等方面提出了一些物理和数学应用。本文还讨论了路径积分量化的一些现代技术，如相干态方法。给出了等变局域化与拓扑场论中的其他思想，如Batalin-Fradkin-Vilkovisky和Mathai-Quillen形式的关系。
We review equivariant localization techniques for the evaluation of Feynman path integrals. We develop systematic geometric methods for studying the semi-classical properties of phase space path integrals for dynamical systems, emphasizing the relations with integrable and topological quantum field theories. Beginning with a detailed review of the relevant mathematical background -- equivariant cohomology and the Duistermaat-Heckman theorem, we demonstrate how the localization ideas are related to classical integrability and how they can be formally extended to derive explicit localization formulas for path integrals in special instances using BRST quantization techniques. Various loop space localizations are presented and related to notions in quantum integrability and topological field theory. We emphasize the common symmetries that such localizable models always possess and use these symmetries to discuss the range of applicability of the localization formulas. A number of physical and mathematical applications are presented in connection with elementary quantum mechanics, Morse theory, index theorems, character formulas for semi-simple Lie groups, quantization of spin systems, unitary integrations in matrix models, modular invariants of Riemann surfaces, supersymmetric quantum field theories, two-dimensional Yang-Mills theory, conformal field theory, cohomological field theories and the loop expansion in quantum field theory. Some modern techniques of path integral quantization, such as coherent state methods, are also discussed. The relations between equivariant localization and other ideas in topological field theory, such as the Batalin-Fradkin-Vilkovisky and Mathai-Quillen formalisms, are presented.