9,099 research outputs found

    Non-Point Invertible Transformations and Integrability of Partial Difference Equations

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    This article is devoted to the partial difference quad-graph equations that can be represented in the form φ(u(i+1,j),u(i+1,j+1))=ψ(u(i,j),u(i,j+1))\varphi (u(i+1,j),u(i+1,j+1))=\psi (u(i,j),u(i,j+1)), where the map (w,z)(φ(w,z),ψ(w,z))(w,z) \rightarrow (\varphi(w,z),\psi(w,z)) is injective. The transformation v(i,j)=φ(u(i,j),u(i,j+1))v(i,j)=\varphi (u(i,j),u(i,j+1)) 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 jj-direction. As an application of this fact, the Darboux integrable equations possessing integrals of the second order in the jj-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

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    We consider {\bf B}-model large NN 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 A1A_1 Hitchin integrable system on a genus gg Riemann surface Σ\Sigma. 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 Σ\Sigma. We show that the large NN planar limit of the generalized matrix model is governed by the same A1A_1 Hitchin system therefore proving genus zero large NN duality for this class of transitions.Comment: 70 pages, 1 figure; version two: minor change
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