329 research outputs found
Tau-Functions and Generalized Integrable Hierarchies
The tau-function formalism for a class of generalized ``zero-curvature''
integrable hierarchies of partial differential equations, is constructed. The
class includes the Drinfel'd-Sokolov hierarchies. A direct relation between the
variables of the zero-curvature formalism and the tau-functions is established.
The formalism also clarifies the connection between the zero-curvature
hierarchies and the Hirota-type hierarchies of Kac and Wakimoto.Comment: 23 page
Universality and Non-Perturbative Definitions of 2D Quantum Gravity from Matrix Models
The universality of the non-perturbative definition of Hermitian one-matrix
models following the quantum, stochastic, or -like stabilization is
discussed in comparison with other procedures. We also present another
alternative definition, which illustrates the need of new physical input for
matrix models to make contact with 2D quantum gravity at the
non-perturbative level.Comment: 20 page
Non-local conservation laws and flow equations for supersymmetric integrable hierarchies
An infinite series of Grassmann-odd and Grassmann-even flow equations is
defined for a class of supersymmetric integrable hierarchies associated with
loop superalgebras. All these flows commute with the mutually commuting bosonic
ones originally considered to define these hierarchies and, hence, provide
extra fermionic and bosonic symmetries that include the built-in N=1
supersymmetry transformation. The corresponding non-local conserved quantities
are also constructed. As an example, the particular case of the principal
supersymmetric hierarchies associated with the affine superalgebras with a
fermionic simple root system is discussed in detail.Comment: 36 pages, LaTeX fil
Pohlmeyer reduction revisited
A systematic group theoretical formulation of the Pohlmeyer reduction is
presented. It provides a map between the equations of motion of sigma models
with target-space a symmetric space M=F/G and a class of integrable
multi-component generalizations of the sine-Gordon equation. When M is of
definite signature their solutions describe classical bosonic string
configurations on the curved space-time R_t\times M. In contrast, if M is of
indefinite signature the solutions to those equations can describe bosonic
string configurations on R_t\times M, M\times S^1_\vartheta or simply M. The
conditions required to enable the Lagrangian formulation of the resulting
equations in terms of gauged WZW actions with a potential term are clarified,
and it is shown that the corresponding Lagrangian action is not unique in
general. The Pohlmeyer reductions of sigma models on CP^n and AdS_n are
discussed as particular examples of symmetric spaces of definite and indefinite
signature, respectively.Comment: 45 pages, LaTeX, more references added, accepted for publication in
JHE
q-Deformation of the AdS5 x S5 Superstring S-matrix and its Relativistic Limit
A set of four factorizable non-relativistic S-matrices for a multiplet of
fundamental particles are defined based on the R-matrix of the quantum group
deformation of the centrally extended superalgebra su(2|2). The S-matrices are
a function of two independent couplings g and q=exp(i\pi/k). The main result is
to find the scalar factor, or dressing phase, which ensures that the unitarity
and crossing equations are satisfied. For generic (g,k), the S-matrices are
branched functions on a product of rapidity tori. In the limit k->infinity, one
of them is identified with the S-matrix describing the magnon excitations on
the string world sheet in AdS5 x S5, while another is the mirror S-matrix that
is needed for the TBA. In the g->infinity limit, the rapidity torus
degenerates, the branch points disappear and the S-matrices become meromorphic
functions, as required by relativistic S-matrix theory. However, it is only the
mirror S-matrix which satisfies the correct relativistic crossing equation. The
mirror S-matrix in the relativistic limit is then closely related to that of
the semi-symmetric space sine-Gordon theory obtained from the string theory by
the Pohlmeyer reduction, but has anti-symmetric rather than symmetric bound
states. The interpolating S-matrix realizes at the quantum level the fact that
at the classical level the two theories correspond to different limits of a
one-parameter family of symplectic structures of the same integrable system.Comment: 41 pages, late
The Hamiltonian Structure of Soliton Equations and Deformed W-Algebras
The Poisson bracket algebra corresponding to the second Hamiltonian structure
of a large class of generalized KdV and mKdV integrable hierarchies is
carefully analysed. These algebras are known to have conformal properties, and
their relation to -algebras has been previously investigated in some
particular cases. The class of equations that is considered includes
practically all the generalizations of the Drinfel'd-Sokolov hierarchies
constructed in the literature. In particular, it has been recently shown that
it includes matrix generalizations of the Gelfand-Dickey and the constrained KP
hierarchies. Therefore, our results provide a unified description of the
relation between the Hamiltonian structure of soliton equations and -algebras, and it comprises almost all the results formerly obtained by other
authors. The main result of this paper is an explicit general equation showing
that the second Poisson bracket algebra is a deformation of the Dirac bracket
algebra corresponding to the -algebras obtained through Hamiltonian
reduction.Comment: 41 pages, plain TeX, no figures. New introduction and references
added. Version to be published in Annals of Physics (N.Y.
Semi-classical spectrum of the Homogeneous sine-Gordon theories
The semi-classical spectrum of the Homogeneous sine-Gordon theories
associated with an arbitrary compact simple Lie group G is obtained and shown
to be entirely given by solitons. These theories describe quantum integrable
massive perturbations of Gepner's G-parafermions whose classical
equations-of-motion are non-abelian affine Toda equations. One-soliton
solutions are constructed by embeddings of the SU(2) complex sine-Gordon
soliton in the regular SU(2) subgroups of G. The resulting spectrum exhibits
both stable and unstable particles, which is a peculiar feature shared with the
spectrum of monopoles and dyons in N=2 and N=4 supersymmetric gauge theories.Comment: 28 pages, plain TeX, no figure
Tau-functions and Dressing Transformations for Zero-Curvature Affine Integrable Equations
The solutions of a large class of hierarchies of zero-curvature equations
that includes Toda and KdV type hierarchies are investigated. All these
hierarchies are constructed from affine (twisted or untwisted) Kac-Moody
algebras~. Their common feature is that they have some special ``vacuum
solutions'' corresponding to Lax operators lying in some abelian (up to the
central term) subalgebra of~; in some interesting cases such subalgebras
are of the Heisenberg type. Using the dressing transformation method, the
solutions in the orbit of those vacuum solutions are constructed in a uniform
way. Then, the generalized tau-functions for those hierarchies are defined as
an alternative set of variables corresponding to certain matrix elements
evaluated in the integrable highest-weight representations of~. Such
definition of tau-functions applies for any level of the representation, and it
is independent of its realization (vertex operator or not). The particular
important cases of generalized mKdV and KdV hierarchies as well as the abelian
and non abelian affine Toda theories are discussed in detail.Comment: 27 pages, plain Te
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