Continuum limit of the N=(1|1) supersymmetric Toda lattice hierarchy

Abstract

Generalizing the graded commutator in superalgebras, we propose a new bracket operation on the space of graded operators with an involution. We show that the Lax representation of the two-dimensional N =(1|1) supersymmetric Toda lattice hierarchy can be realized via the generalized bracket operation; this is important in constructing the semiclassical (continuum) limit of this hierarchy. We construct the continuum limit of the N =(1|1) Toda lattice hierarchy, the dispersionless N =(1|1) Toda hierarchy. In this limit, we obtain the Lax representation, with the generalized graded bracket becoming the corresponding Poisson bracket on the graded phase superspace.