Virasoro Actions on Harmonic Maps (after J. Schwarz)
                         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^(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 slides from Uhlenbeck's Ritt lectures.

References
[1] E. Getzler, The Virasoro Conjecture for Gromov-Witten Invariants, Algebraic Geometry Hirzebruch 70 (Warsaw l998), Cont Math 241, Amer. Math. Soc. 147-176.
[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).