120,421 research outputs found
Supersymmetry Flows, Semi-Symmetric Space Sine-Gordon Models And The Pohlmeyer Reduction
We study the extended supersymmetric integrable hierarchy underlying the
Pohlmeyer reduction of superstring sigma models on semi-symmetric superspaces
F/G. This integrable hierarchy is constructed by coupling two copies of the
homogeneous integrable hierarchy associated to the loop Lie superalgebra
extension f of the Lie superalgebra f of F and this is done by means of the
algebraic dressing technique and a Riemann-Hilbert factorization problem. By
using the Drinfeld-Sokolov procedure we construct explicitly, a set of 2D spin
\pm1/2 conserved supercharges generating supersymmetry flows in the phase space
of the reduced model. We introduce the bi-Hamiltonian structure of the extended
homogeneous hierarchy and show that the two brackets are of the
Kostant-Kirillov type on the co-adjoint orbits defined by the light-cone Lax
operators L_\pm. By using the second symplectic structure, we show that these
supersymmetries are Hamiltonian flows, we compute part of the supercharge
algebra and find the supersymmetric field variations they induce. We also show
that this second Poisson structure coincides with the canonical
Lorentz-Invariant symplectic structure of the WZNW model involved in the
Lagrangian formulation of the extended integrable hierarchy, namely, the
semi-symmetric space sine-Gordon model (SSSSG), which is the Pohlmeyer reduced
action functional for the transverse degrees of freedom of superstring sigma
models on the cosets F/G. We work out in some detail the Pohlmeyer reduction of
the AdS_2xS^2 and the AdS_3xS^3 superstrings and show that the new conserved
supercharges can be related to the supercharges extracted from 2D superspace.
In particular, for the AdS_2xS^2 example, they are formally the same.Comment: V2: Two references added, V3: Modifications in section 2.6, V4:
Published versio
Integrable hierarchy underlying topological Landau-Ginzburg models of D-type
A universal integrable hierarchy underlying topological Landau-Ginzburg
models of D-tye is presented. Like the dispersionless Toda hierarchy, the new
hierarchy has two distinct (``positive" and ``negative") set of flows. Special
solutions corresponding to topological Landau-Ginzburg models of D-type are
characterized by a Riemann-Hilbert problem, which can be converted into a
generalized hodograph transformation. This construction gives an embedding of
the finite dimensional small phase space of these models into the full space of
flows of this hierarchy. One of flat coordinates in the small phase space turns
out to be identical to the first ``negative" time variable of the hierarchy,
whereas the others belong to the ``positive" flows.Comment: 14 pages, Kyoto University KUCP-0061/9
Beyond pressureless gas dynamics: Quadrature-based velocity moment models
Following the seminal work of F. Bouchut on zero pressure gas dynamics which
has been extensively used for gas particle-flows, the present contribution
investigates quadrature-based velocity moments models for kinetic equations in
the framework of the infinite Knudsen number limit, that is, for dilute clouds
of small particles where the collision or coalescence probability
asymptotically approaches zero. Such models define a hierarchy based on the
number of moments and associated quadrature nodes, the first level of which
leads to pressureless gas dynamics. We focus in particular on the four moment
model where the flux closure is provided by a two-node quadrature in the
velocity phase space and provide the right framework for studying both smooth
and singular solutions. The link with both the kinetic underlying equation as
well as with zero pressure gas dynamics is provided and we define the notion of
measure solutions as well as the mathematical structure of the resulting system
of four PDEs. We exhibit a family of entropies and entropy fluxes and define
the notion of entropic solution. We study the Riemann problem and provide a
series of entropic solutions in particular cases. This leads to a rigorous link
with the possibility of the system of macroscopic PDEs to allow particle
trajectory crossing (PTC) in the framework of smooth solutions. Generalized
-choc solutions resulting from Riemann problem are also investigated.
Finally, using a kinetic scheme proposed in the literature without mathematical
background in several areas, we validate such a numerical approach in the
framework of both smooth and singular solutions.Comment: Submitted to Communication in Mathematical Science
Matrix models without scaling limit
In the context of hermitean one--matrix models we show that the emergence of
the NLS hierarchy and of its reduction, the KdV hierarchy, is an exact result
of the lattice characterizing the matrix model. Said otherwise, we are not
obliged to take a continuum limit to find these hierarchies. We interpret this
result as an indication of the topological nature of them. We discuss the
topological field theories associated with both and discuss the connection with
topological field theories coupled to topological gravity already studied in
the literature.Comment: Latex, SISSA-ISAS 161/92/E
Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces
A moving frame formulation of non-stretching geometric curve flows in
Euclidean space is used to derive a 1+1 dimensional hierarchy of integrable
SO(3)-invariant vector models containing the Heisenberg ferromagnetic spin
model as well as a model given by a spin-vector version of the mKdV equation.
These models describe a geometric realization of the NLS hierarchy of soliton
equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet
equations of the moving frame. This derivation yields an explicit
bi-Hamiltonian structure, recursion operator, and constants of motion for each
model in the hierarchy. A generalization of these results to geometric surface
flows is presented, where the surfaces are non-stretching in one direction
while stretching in all transverse directions. Through the Frenet equations of
a moving frame, such surface flows are shown to encode a hierarchy of 2+1
dimensional integrable SO(3)-invariant vector models, along with their
bi-Hamiltonian structure, recursion operator, and constants of motion,
describing a geometric realization of 2+1 dimensional bi-Hamiltonian NLS and
mKdV soliton equations. Based on the well-known equivalence between the
Heisenberg model and the Schrodinger map equation in 1+1 dimensions, a
geometrical formulation of these hierarchies of 1+1 and 2+1 vector models is
given in terms of dynamical maps into the 2-sphere. In particular, this
formulation yields a new integrable generalization of the Schrodinger map
equation in 2+1 dimensions as well as a mKdV analog of this map equation
corresponding to the mKdV spin model in 1+1 and 2+1 dimensions.Comment: Published version with typos corrected. Significantly expanded
version of a talk given by the first author at the 2008 BIRS workshop on
"Geometric Flows in Mathematics and Physics
Topological conformal field theory with a rational W potential and the dispersionless KP hierarchy
We present a new class of topological conformal field theories (TCFT)
characterized by a rational potential, which includes the minimal models of
A and D types as its subclasses. An explicit form of the potential is found
by solving the underlying dispersionless KP hierarchy in a particular small
phase space. We discuss also the dispersionless KP hierarchy in large phase
spaces by reformulating the hierarchy, and show that the potential takes a
universal form, which does not depend on a specific form of the solution in a
large space.Comment: 14 pages, plain TEX, KUL-TF-94/
Integrability vs Supersymmetry: Poisson Structures of The Pohlmeyer Reduction
We construct recursively an infinite number of Poisson structures for the
supersymmetric integrable hierarchy governing the Pohlmeyer reduction of
superstring sigma models on the target spaces AdS_{n}\times S^n, n=2,3,5. These
Poisson structures are all non-local and not relativistic except one, which is
the canonical Poisson structure of the semi-symmetric space sine-Gordon model
(SSSSG). We verify that the superposition of the first three Poisson structures
corresponds to the canonical Poisson structure of the reduced sigma model.
Using the recursion relations we construct commuting charges on the reduced
sigma model out of those of the SSSSG model and in the process we explain the
integrable origin of the Zukhovsky map and the twisted inner product used in
the sigma model side. Then, we compute the complete Poisson superalgebra for
the conserved Drinfeld-Sokolov supercharges associated to an exotic kind of
extended non-local rigid 2d supersymmetry recently introduced in the SSSSG
context. The superalgebra has a kink central charge which turns out to be a
generalization to the SSSSG models of the well-known central extensions of the
N=1 sine-Gordon and N=2 complex sine-Gordon model Poisson superalgebras
computed from 2d superspace. The computation is done in two different ways
concluding the proof of the existence of 2d supersymmetry in the reduced sigma
model phase space under the boost invariant SSSSG Poisson structure.Comment: 33 pages, Published versio
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