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Non-Point Invertible Transformations and Integrability of Partial Difference Equations
This article is devoted to the partial difference quad-graph equations that
can be represented in the form , where the map
is injective. The transformation relates any
of such equations to a quad-graph equation. It is proved that this
transformation maps Darboux integrable equations of the above form into Darboux
integrable equations again and decreases the orders of the transformed
integrals by one in the -direction. As an application of this fact, the
Darboux integrable equations possessing integrals of the second order in the
-direction are described under an additional assumption. The transformation
also maps symmetries of the original equations into symmetries of the
transformed equations (i.e. preserves the integrability in the sense of the
symmetry approach) and acts as a difference substitution for symmetries of a
special form. The latter fact allows us to derive necessary conditions of
Darboux integrability for the equations defined in the first sentence of the
abstract
Geometric transitions and integrable systems
We consider {\bf B}-model large duality for a new class of noncompact
Calabi-Yau spaces modeled on the neighborhood of a ruled surface in a
Calabi-Yau threefold. The closed string side of the transition is governed at
genus zero by an Hitchin integrable system on a genus Riemann surface
. The open string side is described by a holomorphic Chern-Simons
theory which reduces to a generalized matrix model in which the eigenvalues lie
on the compact Riemann surface . We show that the large planar
limit of the generalized matrix model is governed by the same Hitchin
system therefore proving genus zero large duality for this class of
transitions.Comment: 70 pages, 1 figure; version two: minor change
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