Fermionic flows and tau function of the N = ( 1 | 1 ) superconformal Toda lattice hierarchy

An infinite class of fermionic flows of the N=(1|1) superconformal Toda lattice hierarchy is constructed and their algebraic structure is studied. We completely solve the semi-infinite N=(1|1) Toda lattice and chain hierarchies and derive their tau functions, which may be relevant for building supersymmetric matrix models. Their bosonic limit is also discussed. 1. Introduction. Recently the N=(1|1) supersymmetric generalization of the Darboux transformation was proposed, and an infinite class of bosonic solutions of its symmetry equation was constructed in [1]. These solutions generate bosonic flows of the N=(1|1) supersymmetric Toda lattice hierarchy in the same way as their bosonic counterparts – solutions of the symmetry equation of the Darboux transformation [2] – produce the flows of the bosonic Toda lattice hierarchy. However, due to the supersymmetry it is obvious that besides bosonic flows the supersymmetric hierarchy also possesses fermionic ones. A natural question arises about finding solutions of the symmetry equation which are responsible for the fermionic flows.