74 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
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
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
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
Integrable deformations of strings on symmetric spaces
A general class of deformations of integrable sigma-models with symmetric
space F/G target-spaces are found. These deformations involve defining the
non-abelian T dual of the sigma-model and then replacing the coupling of the
Lagrange multiplier imposing flatness with a gauged F/F WZW model. The original
sigma-model is obtained in the limit of large level. The resulting deformed
theories are shown to preserve both integrability and the equations-of-motion,
but involve a deformation of the symplectic structure. It is shown that this
deformed symplectic structure involves a linear combination of the original
Poisson bracket and a generalization of the Faddeev-Reshetikhin Poisson bracket
which we show can be re-expressed as two decoupled F current algebras. It is
then shown that the deformation can be incorporated into the classical model of
strings on R x F/G via a generalization of the Pohlmeyer reduction. In this
case, in the limit of large sigma-model coupling it is shown that the theory
becomes the relativistic symmetric space sine-Gordon theory. These results
point to the existence of a deformation of this kind for the full Green-Schwarz
superstring on AdS5 x S5.Comment: 41 pages, typos corrected, references adde
Symplectic deformations of integrable field theories and AdS/CFT
Relativistic integrable field theories like the sine-Gordon equation have an
infinite set of conserved charges. In a light-front formalism these conserved
charges are closely related to the integrable modified KdV hierarchy at the
classical level. The latter hierarchy admits a family of symplectic structures
which we argue can be viewed as deformations of the relativistic sine-Gordon
symplectic structure. These deformed theories are integrable but no longer
relativistic and the basic excitations of the theory, the solitons, have an
interesting non-relativistic dispersion relation that in a certain limit
becomes the dispersion relation of dyonic giant magnons of string theory in the
AdS/CFT correspondence. We argue that the deformed classical theories can be
lifted to quantum theories when the sine-Gordon theory is embedded in a larger
theory that describes the string world-sheet sigma model in AdS(5)xS(5).Comment: 4 page
A New and Elementary CP^n Dyonic Magnon
We show that the dressing transformation method produces a new type of dyonic
CP^n magnon in terms of which all the other known solutions are either
composites or arise as special limits. In particular, this includes the
embedding of Dorey's dyonic magnon via an RP^3 subspace of CP^n. We also show
how to generate Dorey's dyonic magnon directly in the S^n sigma model via the
dressing method without resorting to the isomorphism with the SU(2) principle
chiral model when n=3. The new dyon is shown to be either a charged dyon or
topological kink of the related symmetric-space sine-Gordon theories associated
to CP^n and in this sense is a direct generalization of the soliton of the
complex sine-Gordon theory.Comment: 21 pages, JHEP3, typos correcte
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