THE ORIGIN OF GAUGE SYMMETRIES IN INTEGRABLE SYSTEMS OF THE KdV TYPE
Abstract
Generalized systems of integrable nonlinear differential equations of the KdV type are considered from the point of view of self-dual Yang-Mills theory in space-times with signature (2, 2). We present a systematic method for embedding the rth flows of the SL(N) KdV hierarchy with N≥2 and r<N in the dimensionally reduced self-dual system using SL(N) as gauge group. We also find that for r>N the corresponding equations can be described in a similar fashion, provided that (in general) the rank of the gauge group increases accordingly. Certain connections of this formalism with WN algebras are also discussed. Finally, we obtain a new class of nonlinear systems in two dimensions by introducing self-dual Ansätze associated with the algebras of Bershadsky and Polyakov.
This work was supported in part by the National Science Foundation under grants PHY-87–17155 and PHY-88–16001, and by a graduate fellowship from the Physics Department (to D.A.D.).
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