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MiS Preprint
90/2007
Virasoro Action on Pseudo-differential Symbols and (Noncommutative) Supersymmetric Peakon Type Integrable Systems
Partha Guha
Abstract
Using Grozman's formalism of invariant differential operators we demonstrate the derivation of $N = 2$ Camassa-Holm equation from the action of $Vect(S^{1|2})$ on the space of pseudo-differential symbols. We also use generalized logarithmic $2$-cocycles to derive $N = 2$ super KdV equations.
We show this method is equally effective to derive Camassa-Holm family of equations and these system of equations can also be interpreted as geodesic flows on the Bott-Virasoro group with respect to right invariant $H^1$- metric. In the second half of the paper we focus on the derivations of the fermionic extension of a new peakon type systems. This new one-parameter family of $N = 1$ super peakon type equations, known as $N = 1$ super $b$- field equations, are derived from the action of $Vect(S^{1|1})$ on tensor densities of arbitrary weights. Finally, using the formal Moyal deformed action of $Vect(S^{1|1})$ on the space of Pseudo-differential symbols to derive the noncommutative analogues of $N = 1$ super $b$- field equations.
