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A. H. Zimerman

Publications and source records attributed to A. H. Zimerman.

103 records · Page 6Linked to original sources
arXiv1994

Two-Matrix String Model as Constrained (2+1)-Dimensional Integrable System

We show that the 2-matrix string model corresponds to a coupled system of $2+1$-dimensional KP and modified KP ($\KPm$) integrable equations subject to a specific ``symmetry'' constraint. The latter together with the Miura-Konopelchenko map for $\KPm$ are the continuum incarnation of the matrix string equation. The $\KPm$ Miura and Bäcklund transformations are natural consequences of the underlying lattice structure. The constrained $\KPm$ system is equivalent to a $1+1$-dimensional generalized KP-KdV hierarchy related to graded ${\bf SL(3,1)}$. We provide an explicit representation of this hierarchy, including the associated ${\bf W(2,1)}$-algebra of the second Hamiltonian structure, in terms of free currents.

hep-th↗
arXiv1993

On Discrete Symmetries of the Multi-Boson KP Hierarchies

We show that the multi-boson KP hierarchies possess a class of discrete symmetries linking them to the discrete Toda systems. These discrete symmetries are generated by the similarity transformation of the corresponding Lax operator. This establishes a canonical nature of the discrete transformations. The spectral equation, which defines both the lattice system and the corresponding Lax operator, plays a key role in determining pertinent symmetry structure. We also introduce a concept of the square-root lattice leading to a family of new pseudo-differential operators with covariance under additional Bäcklund transformations.

hep-th↗
arXiv1993

Toda and Volterra Lattice Equations from Discrete Symmetries of KP Hierarchies

The discrete models of the Toda and Volterra chains are being constructed out of the continuum two-boson KP hierarchies. The main tool is the discrete symmetry preserving the Hamiltonian structure of the continuum models. The two-boson currents of KP hierarchy are being associated with sites of the corresponding chain by successive actions of discrete symmetry.

hep-th↗
arXiv1993

On ${\bf W_{\infty}}$ Algebras, Gauge Equivalence of KP Hierarchies, Two-Boson Realizations and their KdV Reductions

The gauge equivalence between basic KP hierarchies is discussed. The first two Hamiltonian structures for KP hierarchies leading to the linear and non-linear $\Winf$ algebras are derived. The realization of the corresponding generators in terms of two boson currents is presented and it is shown to be related to many integrable models which are bi-Hamiltonian. We can also realize those generators by adding extra currents, coupled in a particular way, allowing for instance a description of multi-layered Benney equations or multi-component non-linear Schroedinger equation. In this case we can have a second Hamiltonian bracket structure which violates Jacobi identity. We consider the reduction to one-boson systems leading to KdV and mKdV hierarchies. A Miura transformation relating these two hierarchies is obtained by restricting gauge transformation between corresponding two-boson hierarchies. Connection to Drinfeld-Sokolov approach is also discussed in the $SL(2,\IR)$ gauge theory. (Lectures presented at the VII J.A. Swieca Summer School, Section: Particles and Fields, Campos do Jordão - Brasil - January/93)

hep-th↗
arXiv1993

Construction of Affine and Conformal Affine Toda Solitons by Hirota's Method

In this talk we report some results about the construction of soliton solutions for the Affine and Conformal Affine Toda models using the Hirota's method. We obtain new classes of solitons connected to the degeneracies of the Cartan matrix eigenvalues as well as to some particular features of the recursive scheme developed here. We obtain an universal mass formula for all those solitons. The examples of $SU(6)$ and $Sp(3)$ are discussed in some detail. ( Talk presented at the VII J.A. Swieca Summer School, Section: Particles and Fields, Campos do Jordão - Brasil - January/93)

hep-th↗
arXiv1992

Hirota's Solitons in the Affine and the Conformal Affine Toda Models

We use Hirota's method formulated as a recursive scheme to construct complete set of soliton solutions for the affine Toda field theory based on an arbitrary Lie algebra. Our solutions include a new class of solitons connected with two different type of degeneracies encountered in the Hirota's perturbation approach. We also derive an universal mass formula for all Hirota's solutions to the Affine Toda model valid for all underlying Lie groups. Embedding of the Affine Toda model in the Conformal Affine Toda model plays a crucial role in this analysis.

hep-th↗
arXiv1992

Connection between the Affine and Conformal Affine Toda Models and their Hirota's Solution

It is shown that the Affine Toda models (AT) constitute a ``gauge fixed'' version of the Conformal Affine Toda model (CAT). This result enables one to map every solution of the AT models into an infinite number of solutions of the corresponding CAT models, each one associated to a point of the orbit of the conformal group. The Hirota's $τ$-function are introduced and soliton solutions for the AT and CAT models associated to $\hat {SL}(r+1)$ and $\hat {SP}(r)$ are constructed.

hep-th↗
arXiv1992

On Two-Current Realization of KP Hierarchy

A simple description of the KP hierarchy and its multi-hamiltonian structure is given in terms of two Bose currents. A deformation scheme connecting various W-infinity algebras and relation between two fundamental nonlinear structures are discussed. Properties of Faá di Bruno polynomials are extensively explored in this construction. Applications of our method are given for the Conformal Affine Toda model, WZNW models and discrete KP approach to Toda lattice chain.

hep-th↗
arXiv1992

Supersymmetric Construction of W-Algebras from Super Toda and Wznw Theories

A systematic construction of super W-algebras in terms of the WZNW model based on a super Lie algebra is presented. These are shown to be the symmetry structure of the super Toda models, which can be obtained from the WZNW theory by Hamiltonian reduction. A classification, according to the conformal spin defined by an improved energy-momentum tensor, is dicussed in general terms for all super Lie algebras whose simple roots are fermionic . A detailed discussion employing the Dirac bracket structure and an explicit construction of W-algebras for the cases of $OSP(1,2)$, $OSP(2,2)$ , $OSP(3,2)$ and $D(2,1 \mid α)$ are given. The $N=1$ and $N=2$ super conformal algebras are discussed in the pertinent cases.

hep-th↗
arXiv1992

A New Deformation of W-Infinity and Applications to the Two-loop WZNW and Conformal Affine Toda Models

We construct a centerless W-infinity type of algebra in terms of a generator of a centerless Virasoro algebra and an abelian spin-1 current. This algebra conventionally emerges in the study of pseudo-differential operators on a circle or alternatively within KP hierarchy with Watanabe's bracket. Construction used here is based on a special deformation of the algebra $w_{\infty}$ of area preserving diffeomorphisms of a 2-manifold. We show that this deformation technique applies to the two-loop WZNW and conformal affine Toda models, establishing henceforth $W_{\infty}$ invariance of these models.

hep-th↗
arXiv1991

Comments on Two-Loop Kac-Moody Algebras

It is shown that the two-loop Kac-Moody algebra is equivalent to a two variable loop algebra and a decoupled $β$-$γ$ system. Similarly WZNW and CSW models having as algebraic structure the Kac-Moody algebra are equivalent to an infinity of versions of the corresponding ordinary models and decoupled abelian fields.

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