Virasoro action on pseudo-differential symbols and (noncommutative) supersymmetric peakon type integrable systems
From the author's abstract: ``Using Grozman's formalism of invariant differential operators we demonstrate the derivation of the \(N=2\) Camassa-Holm equation from the action of \(\text{Vect}(S^{1|2})\) 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 \(\text{Vect}(S^{1|1})\) on tensor densities of arbitrary weights. Finally, using the formal Moyal deformed action of \(\text{Vect}(S^{1|1})\) on the space of pseudo-differential symbols to derive the noncommutative analogues of \(N=1\) super \(b\)-field equations.''
scientific article; zbMATH DE number 5633618
scientific article