Exotic nonlinear supersymmetry and integrable systems
aa r X i v : . [ h e p - t h ] J a n Exotic nonlinear supersymmetry andintegrable systems
Mikhail S. Plyushchay
Departamento de F´ısica, Universidad de Santiago de Chile, Casilla 307, Santiago, Chile
E-mail: [email protected]
Abstract
Peculiar properties of many classical and quantum systems can be related to, orderived from those of a free particle. In this way we explain the appearance andpeculiarities of the exotic nonlinear Poincar´e supersymmetry in reflectionless and finite-gap quantum systems related to the Korteweg-de Vries equation. The same approachis used to explain the origin and the nature of nonlinear symmetries in the perfectlyinvisible PT -regularized conformal and superconformal mechanics systems. Peculiar properties of many classical and quantum systems can be related to, or derivedfrom those of a free particle. The peculiarity of the simplest 1D free particle system is thatit has a local integral of motion p = − i ddx , which separates the left- and right-moving planewaves e ± ikx of the same energy, and distinguishes the unique non-degenerate state ψ = 1 ofzero energy at the very edge of the continuous spectrum by annihilating it. This simplestsystem also is characterized by the sl (2 , R ) conformal symmetry that expands up to theSchr¨odinger symmetry due to the presence of the integral p [1].The interesting class of the quantum systems intimately related to the simplest caseof the one-dimensional free particle corresponds to reflectionless systems [2]. Applying ageneralized Darboux transformation [3] of order n ≥ H ,a quantum reflectionless system H n is generated. Each energy level in the continuous part ofthe spectrum of H n corresponds to a deformed plane wave Ψ ± k ( x ) = A − e ± ikx propagatingto the left or to the right. The generator of the Darboux transformation A − is a differentialoperator of order n based here on an appropriately chosen set of the seed states ψ sj ( x ), j = 1 , . . . , n , which are formal (non-physical) eigenstates of H , ker A − = span { ψ s , . . . , ψ sn } .Reflectionless system H n has then n bound states Ψ j ( x ) = A − f ψ sj , j = 1 , . . . , n , generatedfrom linearly independent formal eigenstates f ψ sj ( x ) = ψ sj ( x ) R x dξ/ ( ψ sj ( ξ )) of H of the sameeigenvalues as the states ψ sj [4]. A non-degenerate state Ψ ( x ) = A − ψ = 1of the free particle. All these spectral peculiarities of reflectionless system H n are detectedby a nontrivial integral of motion P = A − p A + , A + = ( A − ) † , which is a differential operatorof the odd order 2 n + 1 being a Darboux-dressed momentum operator of the free particle.The operator P annihilates all the n bound states as well as the lowest state Ψ ( x ) in thecontinuos part of the spectrum of H n , and separates the left- and right-moving deformedplane waves Ψ ± k ( x ) of equal energy being eigenstates of the P of opposite eigenvalues [5].1otentials of the quantum reflectionless systems can be promoted to multi-soliton so-lutions of the Korteweg-de Vries (KdV) equation by exploiting the covariance of its Laxrepresentation with respect to the Darboux transformations [3, 6]. In this way, potential ofany reflectionless system with n bound states represents a snapshot of an n -soliton solutionto the KdV equation, whose temporal evolution corresponds to an isospectral deformationof the reflectionless system. Operator P in such a picture is a Lax-Novikov integral of the n -th stationary equation of the KdV hierarchy [5].By periodization of reflectionless systems, some finite-gap quantum systems can be ob-tained, whose potentials are solutions of the stationary equations of the KdV hierarchy. Lax-Novikov integral of an n -gap quantum system, being differential operator of order 2 n + 1,separates the left- and right-moving Bloch states of the same energy inside the valence andconduction bands, and annihilates all the 2 n + 1 periodic and anti-periodic edge states at theedges of the bands, on which two irreducible non-unitary finite-dimensional representationsof the conformal sl (2 , R ) algebra are realized [7]. The Darboux covariance of the Lax repre-sentation allows to promote the potentials of finite-gap quantum systems to the cnoidal-typesolutions of the KdV equation [6].Darboux transformations also can be applied to the finite-gap systems of the most generalform to produce some finite-gap systems completely isospectral to the initial ones, or togenerate finite-gap systems with the added arbitrary number of the bound states inside theprohibited zones or at their edges. In the latter case the generated potentials and related tothem super-potentials are promoted to solutions of the KdV and the modified KdV equationsin the form of the soliton defects propagating in a finite-gap background [6].With the pairs of the quantum systems produced starting from the quantum 1D freeparticle or a finite-gap system, an exotic nonlinear N = 4 supersymmetry can be associated.The emergence of the exotic supersymmetry in such systems is rooted in existence of themomentum integral in the free particle, or the Lax-Novikov integral in a finite-gap system.In the simplest case of a reflectionless system, due to the presence of p in the structure ofthe Lax-Novikov integral P , the latter can be factorized into the product of two non-singularoperators, P = A − ( p A + ). In correspondence with this, the given reflectionless system H n can be generated from and intertwined with the free particle H not only by the order n differential operators A − and A + , but also by the operators A − p and p A + of differentialorder n + 1. As a consequence, the extended system H composed from H and H n hasnot only a pair of supercharges Q a , a = 1 ,
2, constructed from the operators A − and A + ,but also possesses a pair of supercharges S a of differential order n + 1 constructed from theintertwining operators A − p and p A + . The anti-commutators of the supercharges Q a and Q b produce a polynomial of order n in H , while S a and S b anti-commute for a polynomialof order n + 1 in H . The anti-commutator of Q a and S b gives rise to an additional evengenerator L of the superalgebra composed from pH n and Lax-Novikov integral P . As a result,instead of the N = 2 (nonlinear in the case of n >
1) Poincar´e supersymmetry generatedby two supercharges and Hamiltonian H , we obtain an exotic nonlinear N = 4 Poincar´esupersymmetry which includes an additional bosonic integral L . Analogous exotic nonlinear N = 4 Poincar´e supersymmetric structure describes extended systems H composed fromisospectral, or almost isospectral pairs of reflectionless systems with multi-soliton potentials u n ( x, . . . ) and u n ′ ( x, . . . ), where the ellipsis corresponds to the sets of 2 n and 2 n ′ parameterscharacterizing the amplitudes and phases of the n - and n ′ -soliton solutions of the KdVequation. The concrete form of the superalgebra depends on the choice of those parameters,and its supercharges undergo some restructuring associated with lowering their differential2rders each time when some sets of the amplitude parameters in H n coincide with thosein super-partner reflectionless system H n ′ [5]. Additional restructuring in supercharges andexotic nonlinear superalgebra generated by them can also happen for special values of thephase differences associated with the coinciding pairs of the soliton amplitudes in potentials u n ( x, . . . ) and u n ′ ( x, . . . ) [5]. In all the cases, however, a pair of supercharges are matrixoperators of some even differential order, while another pair of supercharges has an odddifferential order. This is related to the nature of the Lax-Novikov integrals of the subsystems H n and H n ′ , which have odd differential orders 2 n + 1 and 2 n ′ + 1, and that superchargesfrom different pairs effectively provide the factorization of Lax-Novikov integrals into twonon-singular differential operators. For such extended quantum systems H , the phenomenonof transmutation between the exact and partially broken exotic nonlinear supersymmetrieswas observed and interpreted in terms of the soliton scattering in [8]. In the case of theunbroken supersymmetry, the unique ground state of the system H is a zero mode of allthe odd generators Q a and S a and of the even generators H and L of the superalgebra.Coherently with this, the operator L is a central element of the exotic nonlinear N = 4Poincar´e superalgebra. In the phase of the partially broken exotic nonlinear supersymmetry,the even generator L mutually transforms the pairs of the supercharges Q a and S a by meansof the commutator, and the system H has a doubly degenerate lowest energy level, whosecorresponding states are annihilated by a part of the supercharges [5, 8].A similar exotic nonlinear N = 4 Poincar´e supersymmetric structure also describesextended systems H composed from isospectral pairs of the finite-gap quantum systems,and finite-gap systems with soliton defects. In the latter case, the fine structure of the exoticsupersymmetry controls the nature and propagation of soliton defects with energies whichcan be introduced into different prohibited zones of the finite-gap systems [6].The KdV equation has also rational solutions, in which the dynamics of the movingpoles is governed by the Calogero-Moser systems. Such solutions can be obtained via anappropriate limit procedure from multi-soliton solutions by exploiting the Galilean symmetryof the KdV equation. They also can be obtained directly from the free particle by applyingto it singular generalized Darboux transformations based on zero-energy eigenstates ψ =1 and f ψ ( x ) = x of the free particle and Jordan states corresponding to the same zeroenergy [9]. The simplest case of the Darboux-generated in this way system correspondsto the two-particle Calogero model with the omitted center of mass coordinate. Due to asingular nature of the Darboux transformation, the Schr¨odinger symmetry of a free particlereduces and transforms into conformal sl (2 , R ) symmetry of the generated Calogero systemwith a non-degenerate continuos spectrum (0 , ∞ ). Though the Darboux-dressed momentumoperator P = A − p A + in this case commutes with the generated Hamiltonian H n , it is aformal, non-physical integral of motion since acting on non-degenerate eigenstates of H n ittransforms them into non-physical, formal eigenstates of H n which do not satisfy the Dirichletboundary condition at x = 0 [10]. Coherently with a non-physical nature of the formal Lax-Novikov integral, the corresponding extended system H is described by (a non-linear ingeneral case) N = 2 Poincar´e supersymmetry generated by supercharges Q a constructedfrom the operators A − and A + , but the exotic N = 4 Poincar´e supersymmetry is lost.Then the natural question arises whether it is possible to somehow restore the exoticnonlinear N = 4 Poincar´e supersymmetry in the quantum systems associated with rationalsolutions of the KdV equation.In [9] it was recently shown that this indeed can be achieved via the PT -regularization x → x + iα , α ∈ R , α = 0, of Darboux transformations. The key point of such a com-3lex shift is that it allows to recuperate the Schr¨odinger symmetry in a Darboux-generatedsystem, where, however, a higher derivative Lax-Novikov integral expands its algebra andtransforms into a non-linear one. The obtained in such a way systems possess several inter-esting properties. They are not only refectioness, but are perfectly invisible since in themthe transmission amplitude itself, and not only its modulus, is equal to one. Another pecu-liarity is that each of them contains a unique bound state of zero energy at the very edgeof the continuous part of the spectrum, which is described by a quadratically integrablewave function, and in this sense they are zero-gap quantum systems. The paired perfectlyinvisible systems are described by different forms of the nonlinearly extended generalizedsuper-Schr¨odinger symmetry, which can include or not include the superconformal osp (2 , N = 4 Poincar´e supersymmetry in them. The potentials ofsuch perfectly invisible PT -invariant quantum systems can be promoted to the solutionsof the complexified KdV equation (or higher equations of the hierarchy), which exhibit,particularly, a behaviour typical for extreme (rogue) waves.The simplest system with the unbroken exotic N = 4 nonlinear supersymmetry is de-scribed by the Hamiltonian and supercharges H = (cid:18) H α H (cid:19) , Q = (cid:18) D D (cid:19) , S = (cid:18) − iD P i P D (cid:19) , and Q = iσ Q , S = iσ S . Here P = p = − i ddx is the momentum operator of thefree particle H = − d dx , D = ξ ddx ξ − = ddx − ξ − and D = − ξ − ddx ξ = − ddx − ξ − areconstructed on the base of ξ = x + iα which is a non-physical zero-energy eigenstate of H .Operators D and D are the Darboux generators A − and A + for the super-partners H and H α = − d dx + 2 ξ − . The H , Q a and S a generate the non-linear superalgebra [ H , Q a ] =[ H , S a ] = 0, { Q a , Q b } = 2 δ ab H , { S a , S b } = 2 δ ab H , { Q a , S b } = 2 ǫ ab L , where L = (cid:18) P α H P (cid:19) is the bosonic integral of motion being a central charge of this superalgebra. The kernel of theLax-Novikov integral P α = D P D of the PT -regularized two-particle Calogero subsystem H α is ker P α = span { ξ − , ξ, ξ } . Here ξ − is the zero-energy bound state of H α , while ξ and ξ are its Jordan states, H α ξ = 2 ξ − , H α ξ = − ξ . The unique ground state of the system H of zero-energy Ψ = ( D , t = ( − ξ − , t is annihilated by all the supercharges Q a and S a as well as by the even generator L .The set of the even operators H , ( I − Σ), K = diag ( K α , K α ), D = diag ( D α , D α ), andodd operators Q a and λ = − ξσ − tQ , λ = iσ λ generate the osp (2 |
2) superalgebra ofthe matrix system H . Here I = diag (1 ,
1) and Σ = σ ; D α = { G α , P } and K α = ( G α ) are the generators of conformal sl (2 , R ) symmetry of H being its time-dependent, dynamicalintegrals of motion constructed on the base of its generator of Galileo transformations G α = ξ − t P , while D α = { ξ, P } − tH α and K α = ξ − tD α − t H α are the analogous sl (2 , R ) generators for H α . Extension of the set of the generators of superconformal osp (2 | H by the even integral L gives rise to the expansion of the set of4he integrals of motion by the set of the even integralsΣ , P − = (1 − σ ) P , G − = (1 − σ ) G α , G = diag (cid:0) G α , { G α , H } (cid:1) , V = iξ D I − t G − t L , R = ξ I − t V − t G − t L , and by the second order supercharges S a and the odd integrals µ = { ξ, P } σ − i [ ξ, P ] σ − tS , µ = iσ µ , κ = ξ σ − tµ − t S , and κ = iσ κ , where G α = D G α D is theDarboux-dressed free particle integral G α . The resuting nonlinear (quadratic) superalgebrais generated by ten even and ten odd integrals of the system H including a trivial evencentral charge I , and represents a nonlinearly super-extended Schr¨odinger algebra with the osp (2 |
2) sub-superalgebra. The nontrivial bosonic generators ( L , H , G , P − , Σ = σ , D , V , G − , K , R ) are eigenstates of the dilatation generator D , [ D , O ] = is O O , with the eigenvaluesgiven by s O = (3 / , , / , / , , , − / , − / , − , − / S a , Q a , µ a , λ a , κ a ), s O = (1 , / , , − / , − osp (2 |
2) sub-superalgebra with any other generator is linear in generators.A simple example of the system in the phase of the partially broken phase of the exoticnonlinear N = 4 Poincar´e supersymmetry is given by the Hamiltonian H = diag ( H α , H α )composed from two Calogero systems regularized by different complex shifts α > α . Thesubsystems H α and H α can be intertwined by the second order differential operators D α D α and D α D α via the ‘virtual’ free particle system, ( D α D α ) H α = H α ( D α D α ) , ( D α D α ) H α = H α ( D α D α ) . However, there also exists the first order intertwiners, D = ddx + W , D = − ddx + W , where W = ξ − ξ − ξ − ξ , ξ j = x + iα j : DH α = H α D ,D H α = H α D . They satisfy the relations D D = H α − ∆ , DD = H α − ∆ , where∆ = ( α − α ) − . The supercharges and Lax-Novikov integral of this extended system are Q = (cid:18) DD (cid:19) , S = (cid:18) D α D α D α D α (cid:19) , L = (cid:18) P α P α (cid:19) ,Q = σ Q , S = σ s . They satisfy nontrivial superalgebraic relations { Q a , Q b } = 2 δ ab ( H − ∆ ) , { S a , S b } = 2 δ ab H , { Q a , S b } = 2 ( ǫ ab L + iδ ab ∆ H ) . The exotic nonlinear supersymmetry here is in the spontaneously partially broken phase:the doublet of the bound states Ψ ± = ( D α , ± D α t = ( − ξ − , ∓ ξ − ) t of zero energy at thevery edge of the fourfold degenerate continuous spectrum are not annihilated by the firstorder supercharges, Q Ψ ± = ± i ∆Ψ ± .In the case of the system H = diag( H α , H α ), its nonlinear superconformal algebra ismore complicated [11]. The numbers of the even and odd generators are the same as in theprevious example, but no odd fermionic generator has a definite scaling dimension, i.e. isnot an eigenstate of the dilatation operator D . As a consequence, the osp (2 |
2) superalgebrais not contained as a sub-superalgebra in this case.It is interesting to note that the two simplest PT -regularized Calogero models H αℓ = − d dx + ℓ ( ℓ + 1) ξ − with ℓ = 1 , PT -regularized kinks inthe field-theoretical Liouville and SU (3) conformal Toda systems [9].5onsider now the state ψ (1) α,γ = γξ − + ξ , ξ = x + iα , α ∈ R , γ = 12 τ + iνα , ν ∈ (1 , ∞ ), τ ∈ ( −∞ , ∞ ), which is a linear combination of the bound state ξ − of the system H α = − d dx + ξ of zero eigenvalue and of its non-physical partner ξ of the same zeroenergy. Taking it as a seed state for the generalized Darboux transformation, we obtain asuperpotential W (1) α,γ = ddx (cid:16) ln ψ (1) α,γ (cid:17) = − ξ − + 3 ξ ( ξ + γ ) − , and generate the super-partnersystems H ± = − ddx + V ± given in a usual way by the potentials V ± = ( W (1) α,γ ) ± ( W (1) α,γ ) ′ .This yields V + = 2 ξ − , i.e. H + = H α , and H − = H α,γ = − d dx + V − , where V − = − (cid:0) ln W ( ξ, − γ + ξ ) (cid:1) ′′ = 6 ξ − γ ξ + γξ ( ξ + γ ) := V ( x ; α, γ ( τ, ν )) . The first equality with the Wronskian W means here that the system H − can also be produceddirectly from the free particle system by taking as the set of the seed states for the generalizedDarboux transformation the non-physical zero-energy eigenstate ξ of the free particle anda linear combination − γ + ξ of its zero-energy eigenstate − γ and its Jordan state ξ , H ξ = − ξ . As a function of x and τ , the potential V ( x ; α, γ ( τ, ν )) := u ( x, τ ) satisfiesthe complexified KdV equation u τ − uu x + u xxx = 0 being regular function for all valuesof x and τ . In the case α = 0, potential V takes the form of the well known singularrational solution u ( x, τ ) = 6 x x − τ ( x +12 τ ) of the KdV equation. Note also that the potential V ( x ; α, γ ( τ, ν )) as a function of x satisfies simultaneously the higher stationary equation ofthe KdV hierarchy, 30 u u x − u x u xx − uu xxx + u xxxxx = 0. The real and imaginary parts ofthe potential u ( x, τ ) = v ( x, τ ) + iw ( x, τ ) obey the system of the coupled nonlinear equations v τ − v − w ) x + v xxx = 0 and w τ − vw ) x + w xxx = 0, and represent some two-solitonwaves. For appropriately chosen parameters α and ν , they reveal the behaviour typical forextreme (rouge) waves [9], see Figure 1. - -
500 500 1000 x - - - - V - -
500 500 1000 x - - V Figure 1: Evolution of real v ( x, τ ) (on the left) and imaginary w ( x, τ ) (on the right) partsof the potential V + ( x ; α, γ ( τ, ν )) as a complex PT -symmetric solution of the KdV equationat α = 100, ν = 5; dashed lines: τ = − , continuous lines: τ = 0, dotted lines: τ = 10 .In conclusion we note that the rational extensions of the harmonic oscillator, or of theconformal Alfaro, Fubini, Furlan model (AFF) [12, 13, 14] can be constructed from theindicated systems by applying to them dual Darboux transformations with intertwining op-erators to be differential operators of the even and odd orders [15, 16, 4]. The producedin such a way systems reveal a “finite-gap” structure in their discrete spectra, but the dualDarboux schemes generate from the harmonic oscillator or the AFF model the pairs of thesystems described by the Hamiltonians mutually shifted for a nonzero constant. As a con-sequence, instead of the Lax-Novikov type integrals, in this case nontrivial ladder operators6re generated, which allow to connect finite “valence bands” with equidistant infinite partof the spectrum. Using them, one can construct three pairs of the ladder operators whichencode the spectral peculiarities of the system and form a complete spectrum-generating setof the ladder operators. Such rationally extended systems are characterized by nonlinearlydeformed extended conformal (Newton-Hooke) symmetry. They also can be related to thefree particle via the singular Darboux transformations and by the conformal bridge construc-tion described in a recent paper [17]. Both Darboux and conformal bridge transformationssubstantially use zero-energy eigenstates and Jordan states corresponding to zero-energy.Identifying a spatial reflection R as a Z -grading operator, the nonlinear N = 2 Poincar´esupersymmetry can be revealed in many purely bosonic non-extended quantum systems inthe form of the bosonized supersymmetry [18, 19, 20]. In such systems, the Lax-Novikovintegrals P play the role of the local supercharge, while the second supercharge i RP is nonlo-cal due to the nonlocal nature of the reflection operator. Similarly, a hidden superconformalsymmetry is identified in the quantum harmonic oscillator system [21]. Acknowledgements
The work was partially supported by the FONDECYT Project 1190842, the Project USA1899, and by DICYT, USACH.
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