Toda lattice hierarchy and conservation laws

Time evolutions of the Toda lattice hierarchies of Ueno and Takasaki are induced by Hamiltonians which are conservation laws for the original (one and two dimensional) Toda lattice obtained by Olive and Turok. Moreover these Hamiltonians for two dimensional Toda lattice hierarchy are also conserved quantities of the two component KP hierarchy in which that system is embedded. The one dimensional Toda lattice hierarchy is characterized by the bilinear relations, and a new version of the one dimensional Toda lattice hierarchy is constructed. Generalized Toda lattice hierarchies associated to all affine Lie algebras are presented.     

The N = 2 supersymmetric Toda lattice hierarchy

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.     

Toda lattice hierarchy and generalized string equations

String equations of thep ^th generalized Kontsevich model and the compactifiedc=1 string theory are re-examined in the language of the Toda lattice hierarchy. As opposed to a hypothesis postulated in the literature, the generalized Kontsevich model atp=–1 does not coincide with thec=1 string theory at selfdual radius. A broader family of solutions of the Toda lattice hierarchy including these models is constructed, and shown to satisfy generalized string equations. The status of a variety ofc1 string models is discussed in this new framework.     

On tau-functions for the Toda lattice hierarchy

Letters in Mathematical Physics - We extend a recent result of Dubrovin et al. in On tau-functions for the KdV hierarchy, arXiv:1812.08488 to the Toda lattice hierarchy. Namely, for an arbitrary...     

A supersymmetric extension of the Toda lattice hierarchy

A supersymmetric extension of the Toda lattice (STL) hierarchy is introduced. Explicit representation of solutions of the STL hierarchy is given by means of the Riemann-Hilbert decomposition. The STL hierarchy connected with the infinite-dimensional Lie super algebra osp (/) is studied.     

Integrable boundary problems for 2D Toda lattice

Spatially two-dimensional Toda lattice is examined in the aspect of correct formulation of boundary problems that can be solved within the scheme of the Inverse Scattering Method. It is shown that there exists a large set of integrable boundary problems and various curves can be chosen as boundaries for those problems. Explicit solutions are presented for problems on closed and unclosed curves taken as boundary contours.     

The Toda hierarchy and the KdV hierarchy

We show that spin generalization of elliptic Calogero-Moser system, elliptic extension of Gaudin model and their cousins are the degenerations of Hitchin systems. Applications to the constructions of integrals of motion, angle-action variables and quantum systems are discussed. The constructions of classical systems are motivated by Conformal Field Theory, and their quantum counterparts can be thought of as being the degenerations of the critical level Knizhnik-Zamolodchikov-Bernard equations.     

On the CP topological sigma model and the Toda lattice hierarchy

We propose, in bihamiltonian formalism, a version of the Toda lattice hierarchy that is satisfied by the two point correlation functions of the CP¹ topological sigma model at genus one approximation, and we also show that this bihamiltonian hierarchy is compatible with the Virasoro constraints of Eguchi–Hori–Xiong up to genus two approximation.     

A hierarchy of Hamiltonian lattice equations associated with the relativistic Toda type system

By considering a discrete iso-spectral problem, a hierarchy of bi-Hamiltonian relativistic Toda type lattice equations are revisited. After introducing a semi-direct sum Lie algebras of four by four matrices, integrable coupling system associated with the relativistic Toda type lattice are derived. It is shown that the resulting lattice soliton hierarchy possesses Hamiltonian structures and infinitely many common commuting symmetries as well infinitely many conserved functions. The Liouville integrability of the resulting system is then demonstrated.     

Two-dimensional generalized Toda lattice

The zero curvature representation is obtained for the two-dimensional generalized Toda lattices connected with semisimple Lie algebras. The reduction group and conservation laws are found and the mass spectrum is calculated.     

The relation between the Toda hierarchy and the KdV hierarchy

Under three relations connecting the field variables of Toda flows and that of KdV flows, we present three new sequences of combinations of the equations in the Toda hierarchy which have the KdV hierarchy as a continuous limit. The relation between the Poisson structures of the KdV hierarchy and the Toda hierarchy in the continuous limit is also studied.     

Supersymmetric two-dimensional Toda lattice

The two-dimensional Toda lattice connected with contragradient Lie superalgebras is studied. The systems of linear equations associated with the models for which the inverse scattering method is applicable are written down. The reduction group is calculated.     

Representation theoretical meaning of the initial value problem for the Toda lattice hierarchy: I

The Toda lattice hierarchy is shown to have the Bruhat decomposition of the A group as its parameter space instead of the Grassmann manifold for the KP hierarchy. Takasaki's work on the initial value problem for the Toda lattice hierarchy is reinterpreted from this point of view.     

Dispersionless Toda hierarchy and two-dimensional string theory

The dispersionless Toda hierarchy turns out to lie in the heart of a recently proposed Landau-Ginzburg formulation of two-dimensional string theory at self-dual compactification radius. The dynamics of massless tachyons with discrete momenta is shown to be encoded into the structure of a special solution of this integrable hierarchy. This solution is obtained by solving a Riemann-Hilbert problem. Equivalence to the tachyon dynamics is proven by deriving recursion relations of tachyon correlation functions in the machinery of the dispersionless Toda hierarchy. Fundamental ingredients of the Landau-Ginzburg formulation, such as Landau-Ginzburg potentials and tachyon Landau-Ginzburg fields, are translated into the language of the Lax formalism. Furthermore, a wedge algebra is pointed out to exist behind the Riemann-Hilbert problem, and speculations on its possible role as generators of extra states and fields are presented.     

The generalized non-abelian Toda lattice

We construct a Backlund transformation and inverse scattering form for the generalized nonabelian Toda lattice. We investigate the continuum limit and we show that our model, in this limit, is equivalent to the self-dual sector of the three dimensionalSU(N) Yand-Mills field theory.     

From the Toda lattice to the Volterra lattice and back

We discuss the relationship between the multiple Hamiltonian structures of the generalized Toda lattices and that of the generalized Volterra lattices.     

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.     

Quasi-classical limit of Toda hierarchy andW-infinity symmetries

Previous results on quasi-classical limit of the KP hierarchy and itsW-infinity symmetries are extended to the Toda hierarchy. The Planck constant now emerges as the spacing unit of difference operators in the Lax formalism. Basic notions, such as dressing operators, Baker-Akhiezer functions, and tau function, are redefined.W _{1 +} symmetries of the Toda hierarchy are realized by suitable rescaling of the Date-Jimbo-Kashiara-Miwa vertex operators. These symmetries are contracted tow _{1 +} symmetries of the dispersionless hierarchy through their action on the tau function.