Quantum inverse scattering method and (super)conformal field theory

We consider the possibility of using the quantum inverse scattering method to study the superconformal field theory and its integrable perturbations. The classical limit of the considered constructions is based on the (1|2) super-KdV hierarchy. We introduce the quantum counterpart of the monodromy matrix corresponding to the linear problem associated with the L-operator and use the explicit form of the irreducible representations of _q(1|2) to obtain the fusion relations for the transfer matrices (i.e., the traces of the monodromy matrices in different representations).Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 252–264, February, 2005.     

Fractional monodromy of resonant classical and quantum oscillatorsMonodromie fractionnelle des oscillateurs résonnants classiques et quantiques

We introduce fractional monodromy for a class of integrable fibrations which naturally arise for classical nonlinear oscillator systems with resonance. We show that the same fractional monodromy characterizes the lattice of quantum states in the joint spectrum of the corresponding quantum systems. Results are presented on the example of a two-dimensional oscillator with resonance 1:(?1) and 1:(?2). To cite this article: N.N. Nekhoroshev et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 985–988.     

Super quasiperiodic wave solutions and asymptotic analysis for $$ \mathcal{N} = 1 $$ supersymmetric KdV-type equations

Based on a general multidimensional Riemann theta function and the super Hirota bilinear form, we extend the Hirota method to construct explicit super quasiperiodic (multiperiodic) wave solutions of $ \mathcal{N} = 1 $ \mathcal{N} = 1 supersymmetric KdV-type equations in superspace. We show that the supersymmetric KdV equation does not have an N-periodic wave solution with arbitrary parameters for N ≥ 2. In addition, an interesting influencing band occurs among the super quasiperiodic waves under the presence of a Grassmann variable. We also observe that the super quasiperiodic waves are symmetric about this band but collapse along with it. We present a limit procedure for analyzing the asymptotic properties of the super quasiperiodic waves and rigorously show that the super periodic wave solutions tend to super soliton solutions under some “small amplitude” limits.     

Bilinearization and Soliton Solutions of the Supersymmetric Coupled KdV Equation

Theoretical and Mathematical Physics - We propose an N=1 supersymmetric generalization of the coupled Korteweg-de Vries (KdV) equation and use the Hirota superoperator to obtain a superfield...     

Chaos in M(atrix) theory

We consider the classical and quantum dynamics in M(atrix) theory. Using a simple ansatz we show that a classical trajectory exhibits a chaotic motion. We argue that the holographic feature of M(atrix) theory is related with the repulsive feature of energy eigenvalues in quantum chaotic system. Chaotic dynamics in N = 2 supersymmetric Yang—Mills theory is also discussed. We demonstrate that after the separation of “slow” and “fast” modes there is a singular contribution from the “slow” modes to the Hamiltonian of the “fast” modes.     

Fractional Hamiltonian Monodromy

We introduce fractional monodromy in order to characterize certain non-isolated critical values of the energy–momentum map of integrable Hamiltonian dynamical systems represented by nonlinear resonant two-dimensional oscillators. We give the formal mathematical definition of fractional monodromy, which is a generalization of the definition of monodromy used by other authors before. We prove that the 1:( − 2) resonant oscillator system has monodromy matrix with half-integer coefficients and discuss manifestations of this monodromy in quantum systems. Communicated by Eduard Zehnder Submitted: February 25, 2005; Accepted: November 17, 2005     

Bilinear approach to <Emphasis Type="Italic">N</Emphasis> = 2 supersymmetric KdV equations

The N = 2 supersymmetric KdV equations are studied within the framework of Hirota bilinear method. For two such equations, namely N = 2, a = 4 and N = 2, a = 1 supersymmetric KdV equations, we obtain the corresponding bilinear formulations. Using them, we construct particular solutions for both cases. In particular, a bilinear Bäcklund transformation is given for the N = 2, a = 1 supersymmetric KdV equation.     

Super‐KdV equation: classical solutions and quantization

The supersymmetric extension of the Korteweg‐de Vries equation (super‐KdV) is considered. The direct and inverse scattering problems are studied and the N‐soliton solutions are obtained. The quantization of this dynamical system is introduced and possible application of the quantum inverse scattering method is also dicussed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integrable fermionic extensions of the Garnier system and the anharmonic oscillator

We first derive a fermionic extension of the Garnier system by applying the method of binary nonlinearization of spectral problem to the supersymmetric KdV equation of Kupershmidt and then a fermionic extension of the anharmonic oscillator by a simple reduction. The integrable properties of resulting systems such as Lax representations, corresponding r-matrices and conversed integrals of motion are established.     

Deformations of Euclidean supersymmetries

We consider quantum supergroups that arise in nonanticommutative deformations of the N=(1/2, 1/2) and N=(1, 1) four-dimensional Euclidean supersymmetric theories. Twist operators in the corresponding superspaces and deformed superfield algebras contain left spinor generators. We show that nonanticommutative *-products of superfields transform covariantly under the deformed supersymmetries. This covariance guarantees the invariance of the deformed superfield actions of models involving *-products of superfields. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 270–289, May, 2006.     

On the monodromy of weierstrass points

Summary In this paper we consider the family of curves of genus g=2m with a g ₃ ¹ lying on a particular rational normal scroll S in P^g– 1(C). We define a covering of this family representing the Weierstrass points and we study the monodromy. Applying the techniques of [3] we prove that if g=4 the monodromy is the full symmetric group and for general g=2m it is transitive. We show also that the generic curve of the family has only normal Weierstrass points generalizing a classical result. We work always over the complex numbers.Partially supported by: Ministero della Pubblica Istruzione - Italia; Consiglio Nazionale delle Ricerche — Italia.     

Fractional monodromy: parallel transport of homology cycles

A 2n-dimensional completely integrable system gives rise to a singular fibration whose generic fiber is the n-torus Tⁿ. In the classical setting, it is possible to define a parallel transport of elements of the fundamental group of a fiber along a path, when the path describes a loop around a singular fiber, it defines an automorphism of π₁(Tⁿ) called monodromy transformation [J.J. Duistermaat, On global action-angle coordinates, Communications on Pure and Applied Mathematics 33 (6) (1980) 687–706]. Some systems give rise to a non-classical setting, in which the path can wind around a singular fiber only by crossing a codimension 1 submanifold of special singular fibers (a wall), in this case a non-classical parallel transport can be defined on a subgroup of the fundamental group. This gives rise to what is known as monodromy with fractional coefficients [N. Nekhoroshev, D. Sadovskiì, B. Zhilinskiì, Fractional monodromy of resonant classical and quantum oscillators, Comptes Rendus Mathematique 335 (11) (2002) 985–988]. In this article, we give a precise meaning to the non-classical parallel transport. In particular we show that it is a homologic process and not a homotopic one. We justify this statement by describing the type of singular fibers that generate a wall that can be crossed, by describing the parallel transport in a semi-local neighbourhood of the wall of singularities, and by producing a family of 4-dimensional examples.     

Multiple commutation relations in the models with gl(2|1) symmetry

We consider quantum integrable models with the gl(2|1) symmetry and derive a set of multiple commutation relations between the monodromy matrix elements. These multiple commutation relations allow obtaining different representations for Bethe vectors.     

Exactly solvable two-dimensional complex model with a real spectrum

Using supersymmetric intertwining relations of the second order in derivatives, we construct a two-dimensional quantum model with a complex potential for which all energy levels and the corresponding wave functions are obtained analytically. This model does not admit separation of variables and can be considered a complexified version of the generalized two-dimensional Morse model with an additional sinh ⁻² term. We prove that the energy spectrum of the model is purely real. To our knowledge, this is a rather rare example of a nontrivial exactly solvable model in two dimensions. We explicitly find the symmetry operator, describe the biorthogonal basis, and demonstrate the pseudo-Hermiticity of the Hamiltonian of the model. The obtained wave functions are simultaneously eigenfunctions of the symmetry operator. Dedicated to the 80th birthday of Yuri Victorovich Novozhilov __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 102–111, July, 2006.     

A sequential subspace projection method for extreme Z‐eigenvalues of supersymmetric tensors

Z‐eigenvalues of tensors, especially extreme ones, are quite useful and are related to many problems, such as automatic control, quantum physics, and independent component analysis. For supersymmetric tensors, calculating the smallest/largest Z‐eigenvalue is equivalent to solving a global minimization/maximization problem of a homogenous polynomial over the unit sphere. In this paper, we utilize the sequential subspace projection method (SSPM) to find extreme Z‐eigenvalues and the corresponding Z‐eigenvectors. The main idea of SSPM is to form a 2‐dimensional subspace at the current point and then solve the original optimization problem in the subspace. SSPM benefits from the fact that the 2‐dimensional subproblem can be solved by a direct method. Global convergence and linear convergence are established for supersymmetric tensors under certain assumptions. Preliminary numerical results over several testing problems show that SSPM is very promising. Besides, the globalization strategy of random phase can be easily incorporated into SSPM, which promotes the ability to find extreme Z‐eigenvalues. Copyright © 2014 John Wiley & Sons, Ltd.     

Lax-integrable Laberge–Mathieu hierarchy of supersymmetric nonlinear dynamical systems and its finite-dimensional reduction of Neumann type

A compatibly bi-Hamiltonian Laberge–Mathieu hierarchy of supersymmetric nonlinear dynamical systems is obtained by using a relation for the Casimir functionals of the central extension of a Lie algebra of superconformal even vector fields of two anticommuting variables. Its matrix Lax representation is determined by using the property of the gradient of the supertrace of the monodromy supermatrix for the corresponding matrix spectral problem. For a supersymmetric Laberge–Mathieu hierarchy, we develop a method for reduction to a nonlocal finite-dimensional invariant subspace of the Neumann type. We prove the existence of a canonical even supersymplectic structure on this subspace and the Lax–Liouville integrability of the reduced commuting vector fields generated by the hierarchy.     

Quantum Fourier transform revisited

The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank‐1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix‐2 QFT decomposition to a radix‐d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view.     

Quasiperiodic Wave Solutions of Supersymmetric KdV Equation in Superspace

A direct and unifying scheme for explicitly constructing quasiperiodic wave solutions (multiperiodic wave solutions) of supersymmetric KdV equation in a superspace is proposed. The scheme is based on the concept of super Hirota forms and on the use of super Riemann theta functions. In contrast to ordinary KdV equation with purely bosonic field, some new phenomena on super quasiperiodic waves occur in the supersymmetric KdV equation with the fermionic field. For instance, it is shown that the supersymmetric KdV equation does not possess an N ‐periodic wave solution for N≥ 2 for arbitrary parameters. It is further observed that there is an influencing band occurred among the quasiperiodic waves under the presence of the Grassmann variable. The quasiperiodic waves are symmetric about the band but collapse along with the band. In addition, the relations between the quasiperiodic wave solutions and soliton solutions are rigorously established. It is shown that quasiperiodic wave solution convergence to the soliton solutions under certain conditions and small amplitude limit.