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Register a new userBacklund-Transformation-Related Recursion Operators: Application to the Self-Dual Yang-Mills Equation
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Added by
Costas Papachristou
Publication date
2009
fields
Physics
and research's language is
English
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By using the self-dual Yang-Mills (SDYM) equation as an example, we study a method for relating symmetries and recursion operators of two partial differential equations connected to each other by a non-auto-Backlund transformation. We prove the Lie-algebra isomorphism between the symmetries of the SDYM equation and those of the potential SDYM (PSDYM) equation, and we describe the construction of the recursion operators for these two systems. Using certain known aspects of the PSDYM symmetry algebra, we draw conclusions regarding the Lie algebraic structure of the potential symmetries of the SDYM equation.
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Starting from a self-dual $SU(infty)$ Yang-Mills theory in $(2+2)$ dimensions, the Plebanski second heavenly equation is obtained after a suitable dimensional reduction. The self-dual gravitational background is the cotangent space of the internal two-dimensional Riemannian surface required in the formulation of $SU(infty)$ Yang-Mills theory. A subsequent dimensional reduction leads to the KP equation in $(1+2)$ dimensions after the relationship from the Plebanski second heavenly function, $Omega$, to the KP function, $u$, is obtained. Also a complexified KP equation is found when a different dimensional reduction scheme is performed . Such relationship between $Omega$ and $u$ is based on the correspondence between the $SL(2,R)$ self-duality conditions in $(3+3)$ dimensions of Das, Khviengia, Sezgin (DKS) and the ones of $SU(infty)$ in $(2+2)$ dimensions . The generalization to the Supersymmetric KP equation should be straightforward by extending the construction of the bosonic case to the previous Super-Plebanski equation, found by us in [1], yielding self-dual supergravity backgrounds in terms of the light-cone chiral superfield, $Theta$, which is the supersymmetric analog of $Omega$. The most important consequence of this Plebanski-KP correspondence is that $W$ gravity can be seen as the gauge theory of $phi$-diffeomorphisms in the space of dimensionally-reduced $D=2+2,~SU^*(infty)$ Yang-Mills instantons. These $phi$ diffeomorphisms preserve a volume-three-form and are, precisely, the ones which provide the Plebanski-KP correspondence.
We demonstrate how the Moutard transformation of two-dimensional Schrodinger operators acts on the Faddeev eigenfunctions on the zero energy level and present some explicitly computed examples of such eigenfunctions for smooth fast decaying potentials of operators with non-trivial kernel and for deformed potentials which correspond to blowing up solutions of the Novikov-Veselov equation.
We prove the Makeenko-Migdal equation for two-dimensional Euclidean Yang-Mills theory on an arbitrary compact surface, possibly with boundary. In particular, we show that two of the proofs given by the first, third, and fourth authors for the plane case extend essentially without change to compact surfaces.
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