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

Abstract

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.

Generalized fermionic discrete Toda hierarchy

“…In order to derive the second representation, we introduce a new notation for the fields at odd and even values of the lattice coordinate j: 15) and rewrite (2.3), (2.5), and (2.6) in the following form: 18) where e j andē j are the composite fields…”

“…The operator e l∂ (l ∈ Z) acts on these fields as the discrete lattice shift 15) and the subscript +(−) in (3.12) means the part of the corresponding operators which includes the operators e l∂ at l ≥ 0 (l < 0). The explicit form for the functionals u (m) k, j and v (m) k, j can be obtained through the representation of the composite Lax operators (L ± ) m * in (3.12) and (3.14) in terms of the Lax operators L ± in (3.12).…”

“…In the canonical basis (4.10), the bosonic Hamiltonians (4.9) become 15) and they generate, via the first (4.12) and second (4.13) Hamiltonian structures, the following equations [3]:…”

“…They are the N = (2|2) [6,8,9,11,15,16] and N = (0|2) [11] supersymmetric TL hierarchies. Actually, besides a different number of supersymmetries, they have different bosonic limits which are decoupled systems of two infinite bosonic TL hierarchies and single infinite bosonic TL hierarchy, respectively.…”

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

“…The first property is that the new composite fields defined in accordance with the rules 5) are regular in the semiclassical limit. From rules (3.2-3.5) and the obvious identities…”