On the space-time monopole equation. (English) Zbl 1157.53016

Dai, Bo; Terng, Chuu-Lian; Uhlenbeck, Karen

Yau, Shing Tung (ed.), Essays in geometry in memory of S. S. Chern. Somerville, MA: International Press (ISBN 978-1-57146-116-2/hbk). Surveys in Differential Geometry 10, 1-30 (2006).

In the Riemannian case, via the dimension reduction, translation invariant instantons in \(\mathbb{R}^4\) correspond to Euclidean monopoles in \(\mathbb{R}^3\). This program was initiated by Atiyah and Hitchin and has been an important research area in geometry by many people. In the case \(\mathbb{R}^{2,2}\) of a different signature, the (anti) self dual equations are perhaps not very interesting to study. However their dimension reduction to the space-time monopole equations in \(\mathbb{R}^{2,1}\) yields an extremely interesting and rich system of nonlinear wave equations. In this article the authors present a nice introduction and survey of the literature on these equations. As in the Riemannian case, the space-time monopole equations come with Lax pairs. On the other hand, with a mild restriction and gauge fixing, the equations lead to Ward’s equations for maps into Lie groups. There have been intensive studies of Ward’s equations by various people. The alternative explanations of these works are included in the paper. A new Hamiltonian formulation for the Ward equations are introduced. By using Bäcklund transformations and the inverse scattering theory, the authors discover a large class of space-time monopoles which are global in time.
For the entire collection see [Zbl 1117.53003].

Reviewer: Shuguang Wang (Columbia)

MSC:

53C07	Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C80	Applications of global differential geometry to the sciences
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K35	Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems

Keywords:

space-time monopole; Ward equations; Lax pairs; Bäcklund transformation