The N = 2 supersymmetric Toda lattice hierarchy

Abstract

We propose a new integrable N = 2 supersymmetric Toda lattice hierarchy which may be relevant for constructing a supersymmetric one-matrix model. We define its first two Hamiltonian structures, the recursion operator and Lax-pair representation. We provide partial evidence for the existence of an infinite-dimensional N = 2 superalgebra of its flows. We study its bosonic limit and introduce new Lax-pair representations for the bosonic Toda lattice hierarchy. Finally we discuss the relevance this approach for constructing N = 2 supersymmetric generalized Toda lattice hierarchies.

Generalized fermionic discrete Toda hierarchy

“…Our further purpose is to construct a bi-Hamiltonian structure of the generalized fermionic TL equations (2.3) (consequently, originating from them (2.13) and (2.16)) in 1D space when all the fields depend on only one bosonic coordinate z = z 1 + z 2 . This task was solved in [3] for the 1D N = 2 TL hierarchy obtained by reduction (2.21) of the 1D generalized fermionic TL hierarchy. Here we solve this task for the original 1D generalized fermionic TL hierarchy.…”

“…A more complicated problem is to obtain the next fermionic integral s + 2 (u). First, using (3.12), one can find the explicit form of the functional u (3) 3 ,…”

“…The 1D N = 2 TL hierarchy was proposed and studied in detail in [3]. In this section, we reproduce its biHamiltonian description [3] reducing the bi-Hamiltonian structure (2.28) and (2.29) of the 1D generalized fermionic TL hierarchy by reduction constraint (2.21). Our starting point is (2.16) with boundary conditions (2.4a) and (2.4c) in 1D space (2.22).…”

“…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]:…”

A Note on the Appearance of Self-Dual Yang-Mills Fields in Integrable Hierarchies

“…Therefore, we conclude that the complex twisted N = 2 supersymmetric Toda chain hierarchy with the complex conjugations (2.4)-(2.6) possesses twisted real N = 2 supersymmetry. For this reason we like to call it the "twisted N = 2 supersymmetric Toda chain hierarchy" (for the supersymmetric Toda chain hierarchy possessing untwisted N = 2 supersymmetry see [17] and references therein). Let us remark that a combination of the two involutions (2.6) and (2.5) generates the infinite-dimensional group of discrete Darboux transformations [10] (v, u)…”