18 research outputs found
Conformal Mappings and Dispersionless Toda hierarchy II: General String Equations
In this article, we classify the solutions of the dispersionless Toda
hierarchy into degenerate and non-degenerate cases. We show that every
non-degenerate solution is determined by a function of
two variables. We interpret these non-degenerate solutions as defining
evolutions on the space of pairs of conformal mappings ,
where is a univalent function on the exterior of the unit disc, is a
univalent function on the unit disc, normalized such that ,
and . For each solution, we show how to define the
natural time variables , as complex coordinates on the space
. We also find explicit formulas for the tau function of the
dispersionless Toda hierarchy in terms of . Imposing
some conditions on the function , we show that the
dispersionless Toda flows can be naturally restricted to the subspace
of defined by . This recovers
the result of Zabrodin.Comment: 25 page
Conformal Mappings and Dispersionless Toda hierarchy
Let be the space consists of pairs , where is a
univalent function on the unit disc with , is a univalent function
on the exterior of the unit disc with and
. In this article, we define the time variables , on which are holomorphic with respect to the natural
complex structure on and can serve as local complex coordinates
for . We show that the evolutions of the pair with
respect to these time coordinates are governed by the dispersionless Toda
hierarchy flows. An explicit tau function is constructed for the dispersionless
Toda hierarchy. By restricting to the subspace consists
of pairs where , we obtain the integrable hierarchy
of conformal mappings considered by Wiegmann and Zabrodin \cite{WZ}. Since
every homeomorphism of the unit circle corresponds uniquely to
an element of under the conformal welding
, the space can be naturally
identified as a subspace of characterized by . We
show that we can naturally define complexified vector fields \pa_n, n\in \Z
on so that the evolutions of on
with respect to \pa_n satisfy the dispersionless Toda
hierarchy. Finally, we show that there is a similar integrable structure for
the Riemann mappings . Moreover, in the latter case, the time
variables are Fourier coefficients of and .Comment: 23 pages. This is to replace the previous preprint arXiv:0808.072
The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type
We obtain the necessary and sufficient conditions for a two-component
(2+1)-dimensional system of hydrodynamic type to possess infinitely many
hydrodynamic reductions. These conditions are in involution, implying that the
systems in question are locally parametrized by 15 arbitrary constants. It is
proved that all such systems possess three conservation laws of hydrodynamic
type and, therefore, are symmetrizable in Godunov's sense. Moreover, all such
systems are proved to possess a scalar pseudopotential which plays the role of
the `dispersionless Lax pair'. We demonstrate that the class of two-component
systems possessing a scalar pseudopotential is in fact identical with the class
of systems possessing infinitely many hydrodynamic reductions, thus
establishing the equivalence of the two possible definitions of the
integrability. Explicit linearly degenerate examples are constructed.Comment: 15 page
Spectral Duality Between Heisenberg Chain and Gaudin Model
In our recent paper we described relationships between integrable systems
inspired by the AGT conjecture. On the gauge theory side an integrable spin
chain naturally emerges while on the conformal field theory side one obtains
some special reduced Gaudin model. Two types of integrable systems were shown
to be related by the spectral duality. In this paper we extend the spectral
duality to the case of higher spin chains. It is proved that the N-site GL(k)
Heisenberg chain is dual to the special reduced k+2-points gl(N) Gaudin model.
Moreover, we construct an explicit Poisson map between the models at the
classical level by performing the Dirac reduction procedure and applying the
AHH duality transformation.Comment: 36 page
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
Generic critical points of normal matrix ensembles
The evolution of the degenerate complex curve associated with the ensemble at
a generic critical point is related to the finite time singularities of
Laplacian Growth. It is shown that the scaling behavior at a critical point of
singular geometry is described by the first Painlev\'e
transcendent. The regularization of the curve resulting from discretization is
discussed.Comment: Based on a talk given at the conference on Random Matrices, Random
Processes and Integrable Systems, CRM Montreal, June 200