Virasoro Actions on Harmonic Maps (after J. Schwarz)
Karen Uhlenbeck   and Mihaela B. Vajiac
 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(1,1) into a Lie group. We give a general explanation of how such actions arise, and construct the Euclidean analogues.

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

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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).