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  • 1
    ISSN: 0550-3213
    Source: Elsevier Journal Backfiles on ScienceDirect 1907 - 2002
    Topics: Physics
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Amsterdam : Elsevier
    Physics Letters B 76 (1978), S. 589-592 
    ISSN: 0370-2693
    Source: Elsevier Journal Backfiles on ScienceDirect 1907 - 2002
    Topics: Physics
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The quotient $${{\widetilde{LG}} \mathord{\left/ {\vphantom {{\widetilde{LG}} G}} \right. \kern-\nulldelimiterspace} G}$$ of a super loop group $$\widetilde{LG}$$ by the subgroup of constant loops is given a supersymplectic structure and identified through a moment map embedding with a coadjoint orbit of the centrally extended super loop algebra . The algebra $$\widetilde{diff}^c S^1 $$ of super-conformal vector fields on the circle is shown to have a natural representation as Hamiltonian vector fields on $${{\widetilde{LG}} \mathord{\left/ {\vphantom {{\widetilde{LG}} G}} \right. \kern-\nulldelimiterspace} G}$$ generated by an equivariant moment map . This map is obtained by composition of315-8 with a super Poisson map defining a supersymmetric extension of the classical Sugawara formula. Upon quantization, this yields the corresponding formula of Kac and Todorov on unitary highest weight representations. For any homomorphism ρ:u(1)→G, an associated “twisted” moment map is also derived, generating a super Poisson bracket realization of a super Virasoro subalgebra $$\widetilde{Vir}$$ of the semi-direct sum . The corresponding super Poisson map is interpreted as a nonabelian generalization of the super Miura map and applied to two super KdV hierarchies to derive corresponding integrable generalized super MKdV hierarchies in .
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  • 4
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We give a complete proof of the equivalence between constraint equations and field equations for thed=10,N=1 supersymmetric Yang-Mills theory, a result proposed and partially proved recently by Witten [1]. Our approach explicitly reconstructs the superconnection satisfying the constraints from the on shell component fields. A key ingredient of the method is the choice of a suitable family of gauges, effectively eliminating all gauge dependence on anti-commuting co-ordinates. As a corollary, obtained by dimensional reduction, we also deduce the equivalence of constraints and field equations for thed=4,N=4 theory, as well as ford=6,N=2.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Communications in mathematical physics 166 (1994), S. 337-365 
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The Hamiltonian structure of the monodromy preserving deformation equations of Jimboet al [JMMS] is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to “dual” pairs of matrix differential operators whose monodromy is preserved under the same family of deformations. As illustrative examples, involving discrete and continuous reductions, a higher rank generalization of the Hamiltonian equations governing the correlation functions for an impenetrable Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painlevé transcendentsP V and VI .
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  • 6
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra $$\widetilde{\mathfrak{g}\mathfrak{l}}^{ + *} (2,\mathbb{R})$$ are integrated by separation of variables in the Hamilton-Jacobi equation in hyperellipsoidal coordinates. The canonically quantized systems are then shown to also be completely integrable and separable within the same coordinates. Pairs of second class constraints defining reduced phase spaces are implemented in the quantized systems by choosing one constraint as an invariant, and interpreting the other as determining a quotient (i.e. by treating one as a first class constraint and the other as a gauge condition). Completely integrable, separable systems on spheres and ellipsoids result, but those on ellipsoids require a further modification of order $$\mathcal{O}(\hbar ^2 )$$ in the commuting invariants in order to assure self-adjointness and to recover the Laplacian for the case of free motion. for each case — in the ambient space ℝ n , the sphere and the ellipsoid — the Schrödinger equations are completely separated in hyperellipsoidal coordinates, giving equations of generalized Lamé type.
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  • 7
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract This work deals with Bäcklund transformations for the principal SL(n, ℂ) sigma model together with all reduced models with values in Riemannian symmetric spaces. First, the dressing method of Zakharov, Mikhailov, and Shabat is shown, for the case of a meromorphic dressing matrix, to be equivalent to a Bäcklund transformation for an associated, linearly extended system. Comparison of this multi-Bäcklund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem. A new method of solution for such multi-Bäcklund transformations (MBT) is developed, by the introduction of a “soliton correlation matrix” which satisfies a Riccati system equivalent to the MBT. Using the geometric structure of this system, a linearization is achieved, leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single Bäcklund transformation through purely linear algebraic relations. A systematic study of all reductions of the system by involutive automorphisms is made, thereby defining the multi-Bäcklund transformations and their solution for all Riemannian symmetric spaces.
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  • 8
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract A moment map $$\tilde J_r :M_A \to (\widetilde{gl(r)}^ + )^*$$ is constructed from the Poisson manifold ℳA of rank-r perturbations of a fixedN×N matrixA to the dual $$(\widetilde{gl(r)}^ + )^*$$ of the positive part of the formal loop algebra $$\widetilde{gl(r)}$$ =gl(r)⊗ℂ[[λ, λ−1]]. The Adler-Kostant-Symes theorem is used to give hamiltonians which generate commutative isospectral flows on $$(\widetilde{gl(r)}^ + )^*$$ . The pull-back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in ℳA. The latter may be identified with flows on finite dimensional coadjoint orbits in $$(\widetilde{gl(r)}^ + )^*$$ and linearized on the Jacobi variety of an invariant spectral curveX r which, generically, is anr-sheeted Riemann surface. Reductions of ℳA are derived, corresponding to subalgebras ofgl(r, ℂ) andsl(r, ℂ), determined as the fixed point set of automorphism groupes generated by involutions (i.e., all the classical algebras), as well as reductions to twisted subalgebras of $$\widetilde{sl(r,\mathbb{C}})$$ . The theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of PDE's.
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  • 9
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local reductions of Hamiltonian flows generated by monodromy invariants on the dual of a loop algebra. Following earlier work of De Groot et al., reductions based upon graded regular elements of arbitrary Heisenberg subalgebras are considered. We show that, in the case of the nontwisted loop algebra ℓ(gl n ), graded regular elements exist only in those Heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn=pr or to equal numbers plus onen=pr+1. We prove that the reduction belonging to the grade 1 regular elements in the casen=pr yields thep×p matrix version of the Gelfand-Dickeyr-KdV hierarchy, generalizing the scalar casep=1 considered by DS. The methods of DS are utilized throughout the analysis, but formulating the reduction entirely within the Hamiltonian framework provided by the classical r-matrix approach leads to some simplifications even forp=1.
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  • 10
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part $$\tilde{\mathfrak{g}}^+$$ of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouvile-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. As illustrative examples, the caseg =sl(2), together with its real forms, is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, with the Liouville generating function expressed in hyperellipsoidal coordinates. Forg =sl(3), the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schrödinger equation, a case which requires further symplectic constraints in order to deal with singularities in the spectral data at ∞.
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