536 research outputs found
A note on fermionic flows of the N=(1|1) supersymmetric Toda lattice hierarchy
We extend the Sato equations of the N=(1|1) supersymmetric Toda lattice
hierarchy by two new infinite series of fermionic flows and demonstrate that
the algebra of the flows of the extended hierarchy is the Borel subalgebra of
the N=(2|2) loop superalgebra.Comment: 4 pages LaTe
-analogue of modified KP hierarchy and its quasi-classical limit
A -analogue of the tau function of the modified KP hierarchy is defined by
a change of independent variables. This tau function satisfies a system of
bilinear -difference equations. These bilinear equations are translated to
the language of wave functions, which turn out to satisfy a system of linear
-difference equations. These linear -difference equations are used to
formulate the Lax formalism and the description of quasi-classical limit. These
results can be generalized to a -analogue of the Toda hierarchy. The results
on the -analogue of the Toda hierarchy might have an application to the
random partition calculus in gauge theories and topological strings.Comment: latex2e, a4 paper 15 pages, no figure; (v2) a few references are
adde
Toda Lattice Hierarchy and Generalized String Equations
String equations of the -th generalized Kontsevich model and the
compactified string theory are re-examined in the language of the Toda
lattice hierarchy. As opposed to a hypothesis postulated in the literature, the
generalized Kontsevich model at does not coincide with the
string theory at self-dual radius. A broader family of solutions of the Toda
lattice hierarchy including these models are constructed, and shown to satisfy
generalized string equations. The status of a variety of string
models is discussed in this new framework.Comment: 35pages, LaTeX Errors are corrected in Eqs. (2.21), (2.36), (2.33),
(3.3), (5.10), (6.1), sentences after (3.19) and theorem 5. A few references
are update
Quantum and Classical Aspects of Deformed Strings.
The quantum and classical aspects of a deformed matrix model proposed
by Jevicki and Yoneya are studied. String equations are formulated in the
framework of Toda lattice hierarchy. The Whittaker functions now play the role
of generalized Airy functions in strings. This matrix model has two
distinct parameters. Identification of the string coupling constant is thereby
not unique, and leads to several different perturbative interpretations of this
model as a string theory. Two such possible interpretations are examined. In
both cases, the classical limit of the string equations, which turns out to
give a formal solution of Polchinski's scattering equations, shows that the
classical scattering amplitudes of massless tachyons are insensitive to
deformations of the parameters in the matrix model.Comment: 52 pages, Latex
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
Matrix biorthogonal polynomials on the unit circle and non-Abelian Ablowitz-Ladik hierarchy
Adler and van Moerbeke \cite{AVM} described a reduction of 2D-Toda hierarchy
called Toeplitz lattice. This hierarchy turns out to be equivalent to the one
originally described by Ablowitz and Ladik \cite{AL} using semidiscrete
zero-curvature equations. In this paper we obtain the original semidiscrete
zero-curvature equations starting directly from the Toeplitz lattice and we
generalize these computations to the matrix case. This generalization lead us
to the semidiscrete zero-curvature equations for the non-abelian (or
multicomponent) version of Ablowitz-Ladik equations \cite{GI}. In this way we
extend the link between biorthogonal polynomials on the unit circle and
Ablowitz-Ladik hierarchy to the matrix case.Comment: 23 pages, accepted on publication on J. Phys. A., electronic link:
http://stacks.iop.org/1751-8121/42/36521
String Equation for String Theory on a Circle
We derive a constraint (string equation) which together with the Toda Lattice
hierarchy determines completely the integrable structure of the compactified 2D
string theory. The form of the constraint depends on a continuous parameter,
the compactification radius R. We show how to use the string equation to
calculate the free energy and the correlation functions in the dispersionless
limit. We sketch the phase diagram and the flow structure and point out that
there are two UV critical points, one of which (the sine-Liouville string
theory) describes infinitely strong vortex or tachyon perturbation.Comment: 20 pages, 1 figure, slight changes in the text and many typos
correcte
Second and Third Order Observables of the Two-Matrix Model
In this paper we complement our recent result on the explicit formula for the
planar limit of the free energy of the two-matrix model by computing the second
and third order observables of the model in terms of canonical structures of
the underlying genus g spectral curve. In particular we provide explicit
formulas for any three-loop correlator of the model. Some explicit examples are
worked out.Comment: 22 pages, v2 with added references and minor correction
Integrable Structure of Supersymmetric Yang-Mills and Melting Crystal
We study loop operators of SYM in background.
For the case of U(1) theory, the generating function of correlation functions
of the loop operators reproduces the partition function of melting crystal
model with external potential. We argue the common integrable structure of
SYM and melting crystal model.Comment: 12 pages, 1 figure, based on an invited talk presented at the
international workshop "Progress of String Theory and Quantum Field Theory"
(Osaka City University, December 7-10, 2007), to be published in the
proceeding
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
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