1,616 research outputs found
Thermodynamic limit of random partitions and dispersionless Toda hierarchy
We study the thermodynamic limit of random partition models for the instanton
sum of 4D and 5D supersymmetric U(1) gauge theories deformed by some physical
observables. The physical observables correspond to external potentials in the
statistical model. The partition function is reformulated in terms of the
density function of Maya diagrams. The thermodynamic limit is governed by a
limit shape of Young diagrams associated with dominant terms in the partition
function. The limit shape is characterized by a variational problem, which is
further converted to a scalar-valued Riemann-Hilbert problem. This
Riemann-Hilbert problem is solved with the aid of a complex curve, which may be
thought of as the Seiberg-Witten curve of the deformed U(1) gauge theory. This
solution of the Riemann-Hilbert problem is identified with a special solution
of the dispersionless Toda hierarchy that satisfies a pair of generalized
string equations. The generalized string equations for the 5D gauge theory are
shown to be related to hidden symmetries of the statistical model. The
prepotential and the Seiberg-Witten differential are also considered.Comment: latex2e using amsmath,amssymb,amsthm packages, 55 pages, no figure;
(v2) typos correcte
Dispersionless integrable equations as coisotropic deformations. Extensions and reductions
Interpretation of dispersionless integrable hierarchies as equations of
coisotropic deformations for certain algebras and other algebraic structures
like Jordan triple systInterpretation of dispersionless integrable hierarchies
as equations of coisotropic deformations for certain algebras and other
algebraic structures like Jordan triple systems is discussed. Several
generalizations are considered. Stationary reductions of the dispersionless
integrable equations are shown to be connected with the dynamical systems on
the plane completely integrable on a fixed energy level. ems is discussed.
Several generalizations are considered. Stationary reductions of the
dispersionless integrable equations are shown to be connected with the
dynamical systems on the plane completely integrable on a fixed energy level.Comment: 21 pages, misprints correcte
SDiff(2) Toda equation -- hierarchy, function, and symmetries
A continuum limit of the Toda lattice field theory, called the SDiff(2) Toda
equation, is shown to have a Lax formalism and an infinite hierarchy of higher
flows. The Lax formalism is very similar to the case of the self-dual vacuum
Einstein equation and its hyper-K\"ahler version, however now based upon a
symplectic structure and the group SDiff(2) of area preserving diffeomorphisms
on a cylinder . An analogue of the Toda lattice tau function is
introduced. The existence of hidden SDiff(2) symmetries are derived from a
Riemann-Hilbert problem in the SDiff(2) group. Symmetries of the tau function
turn out to have commutator anomalies, hence give a representation of a central
extension of the SDiff(2) algebra.Comment: 16 pages (``vanilla.sty" is attatched to the end of this file after
``\bye" command
Multi-Hamiltonian structures for r-matrix systems
For the rational, elliptic and trigonometric r-matrices, we exhibit the links
between three "levels" of Poisson spaces: (a) Some finite-dimensional spaces of
matrix-valued holomorphic functions on the complex line; (b) Spaces of spectral
curves and sheaves supported on them; (c) Symmetric products of a surface. We
have, at each level, a linear space of compatible Poisson structures, and the
maps relating the levels are Poisson. This leads in a natural way to Nijenhuis
coordinates for these spaces. At level (b), there are Hamiltonian systems on
these spaces which are integrable for each Poisson structure in the family, and
which are such that the Lagrangian leaves are the intersections of the
symplective leaves over the Poisson structures in the family. Specific examples
include many of the well-known integrable systems.Comment: 26 pages, Plain Te
Elliptic Calogero-Moser Systems and Isomonodromic Deformations
We show that various models of the elliptic Calogero-Moser systems are
accompanied with an isomonodromic system on a torus. The isomonodromic partner
is a non-autonomous Hamiltonian system defined by the same Hamiltonian. The
role of the time variable is played by the modulus of the base torus. A
suitably chosen Lax pair (with an elliptic spectral parameter) of the elliptic
Calogero-Moser system turns out to give a Lax representation of the
non-autonomous system as well. This Lax representation ensures that the
non-autonomous system describes isomonodromic deformations of a linear ordinary
differential equation on the torus on which the spectral parameter of the Lax
pair is defined. A particularly interesting example is the ``extended twisted
model'' recently introduced along with some other models by Bordner
and Sasaki, who remarked that this system is equivalent to Inozemtsev's
generalized elliptic Calogero-Moser system. We use the ``root type'' Lax pair
developed by Bordner et al. to formulate the associated isomonodromic system on
the torus.Comment: latex2e using amsfonts package, 50pages; (v2) typos corrected; (v3)
typos in (3.35), (3.46), (3.48) and (B.26) corrected; (v4) errors in
(1.7),(1.12),(3.46),(3.47) and (3.48) corrected; (v5) final version for
publication, errors in (2.31),(2.35),(3.12),(3.30),(3.45),(4.16) and (4.37)
correcte
A family of linearizable recurrences with the Laurent property
We consider a family of non-linear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of Laurent phenomenon algebras, as introduced recently by Lam and Pylyavskyy. Furthermore, each member of this family is shown to be linearizable in two different ways, in the sense that its iterates satisfy both a linear relation with constant coefficients and a linear relation with periodic coefficients. Associated monodromy matrices and first integrals are constructed, and the connection with the dressing chain for Schrödinger operators is also explained
Open/closed duality for FZZT branes in c=1
We describe how the matrix integral of Imbimbo and Mukhi arises from a limit
of the FZZT partition function in the double-scaled c=1 matrix model. We show a
similar result for 0A and comment on subtleties in 0B.Comment: 26 pages, 2 figure
Isolation and properties of lysophospholipases from the venom of an Australian elapid snake, Pseudechis australis
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