The higher flows of harmonic maps and an application to the Virasoro action

Abstract

One characteristic of soliton equations is their infinite number of conservation laws. These conservation laws give rise to a hierarchy of partial differential equations. One example is the AKNS hierarchy. Terng and Uhlenbeck, Poisson Actions and Scattering Theory for Integrable Systems, Surveys in Differential Geometry: Integrable Systems (A supplement to J. Diff. Geom.), 4 (1998) described the conservation laws that give rise to the AKNS hierarchy. This hierarchy can also be realized in a loop group setting, where the equations are called the positive flows. The name arises from the power of λ which generates each flow. Terng (Loop Groups and Integrable Systems, Preliminary notes) also showed that in some sense, the harmonic map equation arises as a −1-flow in this hierarchy. The harmonic map equation also has a hierarchy of negative flows. However, these are not differential equations like the positive flows. It remained to be asked how the positive and negative flows relate and if they commute. The answer to the latter question is in the positive. As for the first question, the two hierarchies are part of one larger hierarchy. This was solved by using a more complex loop group and explicitly determining the loop algebra splitting. This splitting has applications to physics and is applied to an example of Schwarz involving the Virasoro action in Sec. VIII.