2,202 research outputs found
Dispersionless Toda hierarchy and two-dimensional string theory
The dispersionless Toda hierarchy turns out to lie in the heart of a recently
proposed Landau-Ginzburg formulation of two-dimensional string theory at
self-dual compactification radius. The dynamics of massless tachyons with
discrete momenta is shown to be encoded into the structure of a special
solution of this integrable hierarchy. This solution is obtained by solving a
Riemann-Hilbert problem. Equivalence to the tachyon dynamics is proven by
deriving recursion relations of tachyon correlation functions in the machinery
of the dispersionless Toda hierarchy. Fundamental ingredients of the
Landau-Ginzburg formulation, such as Landau-Ginzburg potentials and tachyon
Landau-Ginzburg fields, are translated into the language of the Lax formalism.
Furthermore, a wedge algebra is pointed out to exist behind the Riemann-Hilbert
problem, and speculations on its possible role as generators of ``extra''
states and fields are presented.Comment: LaTeX 21 pages, KUCP-0067 (typos are corrected and a brief note is
added
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
Explicit solutions of the classical Calogero & Sutherland systems for any root system
Explicit solutions of the classical Calogero (rational with/without harmonic
confining potential) and Sutherland (trigonometric potential) systems is
obtained by diagonalisation of certain matrices of simple time evolution. The
method works for Calogero & Sutherland systems based on any root system. It
generalises the well-known results by Olshanetsky and Perelomov for the A type
root systems. Explicit solutions of the (rational and trigonometric) higher
Hamiltonian flows of the integrable hierarchy can be readily obtained in a
similar way for those based on the classical root systems.Comment: 18 pages, LaTeX, no figur
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
Toda Tau Functions with Quantum Torus Symmetries
The quantum torus algebra plays an important role in a special class of solutions of the Toda hierarchy. Typical examples are the solutions related to the melting crystal model of topological strings and 5D SUSY gauge theories. The quantum torus algebra is realized by a 2D complex free fermion system that underlies the Toda hierarchy, and exhibits mysterious “shift symmetries”. This article is based on collaboration with Toshio Nakatsu
An hbar-expansion of the Toda hierarchy: a recursive construction of solutions
A construction of general solutions of the \hbar-dependent Toda hierarchy is
presented. The construction is based on a Riemann-Hilbert problem for the pairs
(L,M) and (\bar L,\bar M) of Lax and Orlov-Schulman operators. This
Riemann-Hilbert problem is translated to the language of the dressing operators
W and \bar W. The dressing operators are set in an exponential form as W =
e^{X/\hbar} and \bar W = e^{\phi/\hbar}e^{\bar X/\hbar}, and the auxiliary
operators X,\bar X and the function \phi are assumed to have \hbar-expansions X
= X_0 + \hbar X_1 + ..., \bar X = \bar X_0 + \hbar\bar X_1 + ... and \phi =
\phi_0 + \hbar\phi_1 + .... The coefficients of these expansions turn out to
satisfy a set of recursion relations. X,\bar X and \phi are recursively
determined by these relations. Moreover, the associated wave functions are
shown to have the WKB form \Psi = e^{S/\hbar} and \bar\Psi = e^{\bar S/\hbar},
which leads to an \hbar-expansion of the logarithm of the tau function.Comment: 37 pages, no figures. arXiv admin note: substantial text overlap with
arXiv:0912.486
Kernel Formula Approach to the Universal Whitham Hierarchy
We derive the dispersionless Hirota equations of the universal Whitham
hierarchy from the kernel formula approach proposed by Carroll and Kodama.
Besides, we also verify the associativity equations in this hierarchy from the
dispersionless Hirota equations and give a realization of the associative
algebra with structure constants expressed in terms of the residue formulas.Comment: 18 page
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