Karen Uhlenbeck and Mihaela B. Vajiac

Abstract:

The Virasoro algebra is the formal algebra which arises as the infinitesimal algebra of the diffeomorphism of the line. It has been known for a long time that a half Virasoro algebra acts as an infinitesimal symmetry on the KdV equations and the higher order general Gelfand-Dickey equations (KdV-r). This action occurs in many integrable systems, and is viewed as an important ingredient in quantum cohomology. Since harmonic maps from a two-dimensional domain into a Lie group target have many of the properties of integrable systems, it is not surprising that these half-Virasoro actions occur in the context of harmonic maps. In this paper, we elaborate on a construction of John Schwarz for Virasoro actions on harmonic maps from R

For related information, see the abstract of my Ritt lectures.

References

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[2] M. Guest "Harmonic Maps, Loop Groups and Integrable Systems" London Math. Society Student Texts 38, Cambridge University Press (l997).

[3] J. Schwarz, Classical Duality Symmetries in Two Dimensions, Nuclear Phys. B 447 (l995) 137-182.

[4] K. Uhlenbeck, Harmonic Maps into Lie Groups, J. Diff. Geo. 30 (l989), 1-50.

[5] van Moerbeke, Integrable Foundations of String Theory, in Lectures on Integrable Systems, 163-269, World Sci. Publishing (l994).