Manifestly supersymmetric Lax integrable hierarchies

Aratyn, H.; Rasinariu, Constantin

doi:10.1016/s0370-2693(96)01463-3

Cited by 19 publications

(20 citation statements)

References 25 publications

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“…Now we are ready to show that (22) is, in fact, equivalent to the modified hierarchy equation (21). From (26) and (24) we have…”

mentioning

confidence: 92%

“…In other words, the fundamental Poisson brackets of the V i can be calculated from the supersymmetric GD bracket defined by Λ n . However, motivated by the recent studies on super pseudo-differential operators with reductions and truncations [23][24][25] it would be interesting to consider the supersymmetric GD bracket defined by the Lax operator L = AB −1 where both A and B are superdifferential operators; in fact, in view of (8), this situation is reduced to consider the case L = B −1 . More precisely, if B = B i D i be an invertible superdifferential Lax operator such that the Poisson brackets among the B i be given by (7).…”

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confidence: 99%

“…Thus, from the Lax formalism point of view, we can identify the one-step permutation Ω : L Ω i−1 → L Ω i as an elementary Bäcklund transformation due to the fact that the form of the Lax operator and the hierarchy equation are preserved under such transformation. Moreover, from (24) such one-step permutation can be viewed as a gauge transformation triggered by the gauge…”

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confidence: 99%

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On the Miura and Bäcklund Transformations Associated With the Supersymmetric Gelfand–dickey Bracket

Shaw

1998

Mod. Phys. Lett. A

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The supersymmetric version of the Miura and Bäcklund transformations associated with the supersymmetric Gelfand-Dickey bracket are investigated from the point of view of the Kupershmidt-Wilson theorem.Soon after the work of Zamolodchikov, 1 much attention has been paid to the Walgebras in two-dimensional conformal field theory and integrable systems. 2 Now it is well-known that the classical version of the W n -algebras arises naturally as the second Gelfand-Dickey (GD) bracket of the nth generalized Korteweg de Vries (KdV) hierarchy 3,4 defined by the Lax operator L = ∂ n +u n−1 ∂ n−1 +· · ·+u 0 . In such connection, the Kupershmidt-Wilson (KW) theorem 5,6 plays an important role in such a way that under the factorization L = (∂−φ n )(∂−φ n−1 ) · · · (∂−φ 1 ), the second GD bracket is transformed into a much simple one in terms of the Miura fields φ i . Thus the factorization not only provides the Miura transformation which maps the nth KdV hierarchy to the corresponding modified hierarchies but also gives a free-field realization of the associated classical W n -algebras. 7 On the other hand, as observed by Adler in Ref. 8, the cyclic permutation of the Miura fields φ i in the factorized Lax operator generates the Bäcklund transformation of the nth KdV hierarchy. Hence from the point of view of the KW theorem, one has a unified description of the Miura and Bäcklund transformations associated with the second GD structure for the nth KdV hierarchy. In general, the above scheme is encoded in a particular form of the Lax operator and its associated Poisson structure. Several integrable systems have been studied based on this scheme. For example, the KW theorem and Adler's work have been generalized to the constrained KP hierarchy 9-14 *

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“…Now we are ready to show that (22) is, in fact, equivalent to the modified hierarchy equation (21). From (26) and (24) we have…”

mentioning

confidence: 92%

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confidence: 99%

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confidence: 99%

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On the Miura and Bäcklund Transformations Associated With the Supersymmetric Gelfand–dickey Bracket

Shaw

1998

Mod. Phys. Lett. A

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“…2 An alternative Lax representation of the N = 2 supersymmetric (n, m)-GNLS hierarchy was proposed in [3]. Its relation to our Lax representation is not completely clear to us.…”

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confidence: 92%

“…In [1] a large class of such hierarchies was introduced: the N = 2 supersymmetric (n, m) Generalized Nonlinear Schrödinger (GNLS) hierarchies. They were subsequently studied in a number of other papers, [2,3,4,5,6].The goal of the present letter is to fill a gap in our knowledge of these hierarchies by analyzing their Hamiltonian structures. In particular we produce here the first two Hamiltonian structures and the relevant recursion operators, as well as related local and non-local conserved charges, in two different superfield representations which possess local flow equations.…”

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confidence: 97%

The Hamiltonian structure of the N = 2 supersymmetric GNLS hierarchy

Bonora

Сорин²

1997

Physics Letters B

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The first two Hamiltonian structures and the recursion operator connecting all evolution systems and Hamiltonian structures of the N = 2 supersymmetric (n, m)-GNLS hierarchy are constructed in terms of N = 2 superfields in two different superfield bases with local evolution equations. Their bosonic limits are studied in detail. New local and nonlocal bosonic and fermionic integrals both for the N = 2 supersymmetric (n, m)-GNLS hierarchy and its bosonic counterparts are derived. As an example, in the n=1, m=1 case, the algebra and the symmetry transformations for some of them are worked out and a rich N=4 supersymmetry structure is uncovered.Recently there has been an intense research activity on N=2 supersymmetric integrable hierarchy. Basically this stems from the interest spurred by the ordinary integrable hierarchies and their relation to the 2D gravity and N=2 supersymmetric models representing superstring vacua. Although an analysis of this connection for the N=2 supersymmetric integrable hierarchies is still lacking, it is nevertheless a fact that the latter hierarchies have extremely rich and interesting structures. In [1] a large class of such hierarchies was introduced: the N = 2 supersymmetric (n, m) Generalized Nonlinear Schrödinger (GNLS) hierarchies. They were subsequently studied in a number of other papers, [2,3,4,5,6].The goal of the present letter is to fill a gap in our knowledge of these hierarchies by analyzing their Hamiltonian structures. In particular we produce here the first two Hamiltonian structures and the relevant recursion operators, as well as related local and non-local conserved charges, in two different superfield representations which possess local flow equations.Let us start with a short summary of the main facts concerning the N = 2 supersymmetric (n, m)-GNLS hierarchy [1, 6] which will be useful in what follows.The Lax operator of the N = 2 supersymmetric (n, m)-GNLS hierarchy has the following form

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Super Nonlinear Schrödinger Equation

Lozano¹,

Spallucci²,

Henkel³

et al. 2004

Concise Encyclopedia of Supersymmetry

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S S-DUALITYA nonperturbative symmetry that relates the weak and strong coupling regimes of a quantum theory. In a gauge theory the inversion of the coupling is accompanied by the interchange of the electric and magnetic degrees of freedom [1]. When a h-term is included in the Lagrangian there is an additional symmetry under h ! h þ 2p, and the theory is then invariant under s ! À1=s and s ! s þ 1, withand g the coupling constant. These two transformations generate the SLð2; Z Þ group, or S-duality group [2]. Examples in which it is conjectured to be a symmetry: Four dimensional N¼4 supersymmetric Yang Mills theory, where it is referred as Montonen Olive duality [3]. In string theory: symmetry of the ten dimensional Type IIB superstring theory [4], it relates the Type I and heterotic SO(32) theories [5,6,7], in four dimensions it is a symmetry of the heterotic compactified on a 6-torus [8]. Let G be a real simple Lie algebra with Cartan decomposition G ¼ T È ? P with T a maximal compact subalgebra of G and P the orthogonal complement of T with respect to the Killing form which is negative definite when restricted to T . A linear Cartan involution h is defined by h j P ¼ À1 and h j T ¼ 1. For the split form, the Cartan Chevalley involution h is given by hðh a Þ ¼ Àh a , hðe a Þ ¼ Àe Àa , then, the maximal compact subalgebra T is generated over R by fe a À e Àa g and P by fh a g and fe a þ e Àa g. Let A be a maximal abelian subalgebra of P, and let g be (a maximally non compact h-stable) Cartan subalgebra containing A (A Ã is its dual). The dimension of A corresponds to the real rank of G. For the split form, A is generated over R by the Cartan generators fh a g and the real rank coincides with the rank of its complexification. The conjugation r of G C , the complexification of G, with respect to G defines an involutive automorphism and all the fixed elements of G C by r are the real form G. Thus, we have rðXÞ ¼ X and rðiXÞ ¼ ÀiX 8X 2 G. In the same way, the conjugation s ¼ h:r of K G C is defined with respect to the compact real form G ¼ T È iP. The Satake diagram ðG; AÞ permits to obtain easily the conjugation r and then to find the real form associated to it and consists of: 4. The Dynkin diagram of the root system: R ¼ fa a : a 2 D 1 g.

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Manifestly supersymmetric Lax integrable hierarchies

Cited by 19 publications

References 25 publications

On the Miura and Bäcklund Transformations Associated With the Supersymmetric Gelfand–dickey Bracket

On the Miura and Bäcklund Transformations Associated With the Supersymmetric Gelfand–dickey Bracket

The Hamiltonian structure of the N = 2 supersymmetric GNLS hierarchy

Super Nonlinear Schrödinger Equation

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