Actions of loop groups on harmonic maps
Actions of loop groups on harmonic maps
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DOI：
10.1090/s0002-9947-1991-1062870-5
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发表时间：
1991-02
影响因子：
1.3
通讯作者：
M. Bergvelt;M. Guest
中科院分区：
文献类型：
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作者：
M. Bergvelt;M. Guest
我们描述了一个一般的框架，其中循环群阿金（：）的子群作用在从S2到Gln（：）的调和映射空间上。这代表了Zakharov-Mikhailov-Shabat [ZM，ZS]所考虑的作用的简化，因为我们将黎曼-希尔伯特问题的轮廓取为圆的并集;然而，它将基本成分简化为阿金（S）的著名Birkhoff分解，这便于严格处理。我们给各种具体的例子的行动，并使用这些调查建议乌伦贝克[呃]，一个限制版本的这样一个行动（“完成”）引起了她的基本过程“增加一个uniton”。事实证明，这种情况不会发生，因为完备化保持了调和映射的能量。然而，在特殊情况下的调和映射从S2到复杂的射影空间，我们描述了一个修改的完成过程确实重现“添加一个uniton”。微分方程对称群理论的一个方面是“解的增殖”的思想。这意味着，从一个明显的解决方案开始，对称性的应用可以导致进一步的（也许不那么明显的）解决方案。本文将Zakharov等人[ZM，ZS]所描述的一般框架应用于调和映射方程（数学物理的主要手征或sigma模型）。为了把这一点纳入更广泛的背景下，我们回顾说，一个基本的发现，在研究的Korteweg-de弗里斯和相关的方程是，空间的解决方案承认一定的无限维对称群。历史上，这一点的第一个表现（通过诺特定理）是无穷多个守恒定律的出现。后来这导致了一个优雅的代数描述的解决方案的方程作为点的轨道表示的一个无限维组。(See例如，[Wi]对其中一些想法进行了简要描述。）其他方程存在以前未被注意到的对称性（“隐藏的对称性”）成为一种有趣的可能性，人们试图挖掘它们（到目前为止，比KdV方程的成功要少得多）。一个这样的例子是调和映射方程的映射从表面D到一个紧李群G（或齐次空间G/H）。基本的观察[宝，呃，ZM，ZS]，其中介绍了一个无限维群是调和映射从D到G对应于某些收到的编辑1990年2月9日。1980年数学学科分类（1985年修订）。第58集E20（r）1991年美国数学学会0002-9947/91每页1.00美元+0.25美元
We describe a general framework in which subgroups of the loop group AGIn(: act on the space of harmonic maps from S2 to Gln(: . This represents a simplification of the action considered by Zakharov-Mikhailov-Shabat [ZM, ZS] in that we take the contour for the Riemann-Hilbert problem to be a union of circles; however, it reduces the basic ingredient to the well-known Birkhoff decomposition of AGIn(S, and this facilitates a rigorous treatment. We give various concrete examples of the action, and use these to investigate a suggestion of Uhlenbeck [Uh] that a limiting version of such an action ("completion") gives rise to her fundamental process of "adding a uniton". It turns out that this does not occur, because completion preserves the energy of harmonic maps. However, in the special case of harmonic maps from S2 to complex projective space, we describe a modification of this completion procedure which does indeed reproduce "adding a uniton". One aspect of the theory of symmetry groups for diffierential equations is the idea of "proliferation of solutions". By this is meant that, starting with an obvious solution, application of symsmetries can lead to further (perhaps less obvious) solutions. In this paper we shall apply the general framework described by Zakharov et al. [ZM, ZS] to the harmonic map equation (the principal chiral or sigma model of mathematical physics). To put this into a wider context, we recall that a fundamental discovery in the study of the Korteweg-de Vries and related equations was that the space of solutions admits a certain infinite-dimensional symmetry group. Historically, the first manifestation of this (via Noether's theorem) was the appearance of infinitely many conservation laws. Later this led to an elegant algebraic description of solutions of the equation as the points of an orbit of a representation of an infinite-dimensional group. (See, for example, [Wi] for a brief description of some of these ideas.) The existence of previously unnoticed symmetries ("hidden symmetries") for other equations became an interesting possibility, and attempts have been made (so far with much less success than in the case of the KdV equation) to unearth them. One such example is the harmonic map equation for maps from a surface D into a compact Lie group G (or homogeneous space G/H ). The basic observation [Po, Uh, ZM, ZS] which introduces an infinite-dimensional group is that harmonic maps from D to G correspond to certain Received by the editors February 9, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 58E20. (r)1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page