Virasoro Actions on Harmonic Maps (after J. Schwarz)
Karen Uhlenbeck and Mihaela B.
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
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
integrable systems, it is not surprising that these half-Virasoro
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
 E. Getzler, The Virasoro Conjecture for Gromov-Witten Invariants,
Algebraic Geometry Hirzebruch 70 (Warsaw l998), Cont Math 241, Amer.
Math. Soc. 147-176.
 M. Guest "Harmonic Maps, Loop Groups and Integrable Systems" London
Math. Society Student Texts 38, Cambridge University Press (l997).
 J. Schwarz, Classical Duality Symmetries in Two Dimensions, Nuclear
B 447 (l995) 137-182.
 K. Uhlenbeck, Harmonic Maps into Lie Groups, J. Diff. Geo. 30
 van Moerbeke, Integrable Foundations of String Theory, in Lectures
on Integrable Systems, 163-269, World Sci. Publishing (l994).