Instantons and Harmonic Maps
Instantons and Harmonic Maps
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瞬子和谐波图
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发表时间：
1985
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通讯作者：
M. Guest
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作者：
M. Guest
在物理学中出现的微分方程经常表现为映射空间上某个泛函的欧拉-拉格朗日方程;在这次讲座中，我们将关注两个最近的例子，即杨-米尔斯理论和σ-模型理论。这里有两个显著的特征：（a）存在称为瞬子的特殊解;（B）瞬子的参数空间或模空间与定义该泛函的空间之间可能存在拓扑关系。杨-米尔斯泛函的技术问题促使人们将其与已有结果的一般情况进行比较，即能量泛函的研究
$$ E：f \mapsto \frac{1}{2}\int\limits_M {{\left| {df} \right|}^2}} $$
定义在（紧）黎曼流形M，N之间的光滑映射Map（M，N）空间上。根据定义，E的临界点是调和映射。这实际上包括通常的σ模型示例[10，11，20]，其中M = CP 1且N = CPn。最近的工作M。F. Atiyah和S. K.然而，唐纳森（参见[2]）指出，这不仅仅是一个有用的类比：如果M = CP 1，并且N被G上由环组成的无限维李群ΩG所取代，则S4上G-丛的杨-米尔斯瞬子可以被识别为“σ-模型瞬子”。
Differential equations arising in Physics often appear as the Euler-Lagrange equations for a certain functional on a space of maps; in this lecture we shall be concerned with two recent examples, namely Yang-Mills theory and the theory of σ-models. Two striking features here are (a) the existence of special solutions known as instantons, and (b) the possibility of a topological relation between the parameter space or moduli space of instantons and the space on which the functional is defined. Technical problems with the Yang-Mills functional have prompted comparison with a general situation where some results already exist, i.e. the study of the energy functional
$$ E:f \mapsto \frac{1}{2}\int\limits_M {{{\left| {df} \right|}^2}} $$
defined on the space of smooth maps Map(M,N) between (compact) Riemannian manifolds M,N. The critical points for E are, by definition, harmonic maps. This actually includes the usual σ-model example [10,11,20] where M = CP1 and N = CPn. Recent work of M. F. Atiyah and S. K. Donaldson (see [2]) indicates that this is much more than a useful analogy, however: Yang-Mills instantons for a G-bundle over S4 may be identified with “σ-model instantons” if M = CP1 and N is replaced by the infinite dimensional Lie group ΩG consisting of loops on G.