2024/07/29 by Carpentier, Sylvain, Suh, UhiRinn
#17B69 #17B80 #35Q53 #37K10 #37K30 #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2407.19717
In this paper we realize the supersymmetric classical W-algebras W(\mathfrakgl(n+1|n)) and W(\mathfrakgl(n|n)) as differential algebras generated by the coefficients of a monic superdifferential operator L. In the case of W(\mathfrakgl(n|n)) (resp. W(\mathfrakgl(n+1|n))) this operator is even (resp. odd). We show that the supersymmetric Poisson vertex algebra bracket on these supersymmetric W-algebras is the supersymmetric analogue of the quadratic Gelfand-Dickey bracket associated to the operator L. Finally, we construct integrable hierarchies of evolutionary Hamiltonian PDEs on both W-algebras. A key observation is that to construct these hierarchies on the algebra W(\mathfrakgl(n+1|n)) one needs to introduce a new concept of even supersymmetric Poisson vertex algebras.