Moment maps and diffeomorphisms
Moment maps and diffeomorphisms
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DOI：
10.4310/ajm.1999.v3.n1.a1
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
S. Donaldson
中科院分区：
文献类型：
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作者：
S. Donaldson
Atiyah 和 Bott 在[1]中指出，表面上的束上的连接曲率可以被视为与规范组的作用相对应的“动量”。这一观察以及各种扩展激发了大量工作，并提供了理解杨-米尔斯理论中许多现象的概念框架。我们本文的目的是在微分同胚群的框架中探索一些类似的想法。我们首先在相当一般的环境中识别矩图，然后看看这些想法在一些更具体的情况下如何发挥作用。我们希望表明，矩图观点是有用的，无论是在理解某些既定结果方面，还是在提出几何和分析中的新问题方面。虽然这些分析问题是这项工作的主要动机，但我们在这里将集中于形式方面，不会对分析做出任何认真的进展。
Atiyah and Bott pointed out, in [1], that the curvature of a connection on a bundle over a surface can be viewed as the “momentum” corresponding to the action of the gauge group. This observation, together with various extensions, has stimulated a great deal of work and provides a conceptual framework to understand many phenomena in Yang-Mills theory. Our purpose in this paper is to explore some similar ideas in the framework of diffeomorphism groups. We begin by identifying a moment map in a rather general setting, and then see how the ideas work in some more specific situations. We hope to show that the moment map point of view is useful, both in understanding certain established results and also in suggesting new problems in geometry and analysis. While these analytical questions are the main motivation for the work, we will concentrate here on the formal aspects and will not make any serious inroads on the analysis.