Solutions to the reflection equation and integrable systems for N = 2 SQCD with classical groups

Integrable systems underlying the Seiberg-Witten solutions for the N = 2 SQCD with gauge groups SO(n) and Sp(n) are proposed. They are described by the inhomogeneous XXX spin chain with specific boundary conditions given by reflection matrices. We attribute reflection matrices to orientifold planes in the brane construction and briefly discuss its possible deformations.     

Hyper-symplectic structures on integrable systems

We prove that an integrable system over a symplectic manifold whose symplectic form is covariantly constant carries a natural hyper-symplectic structure. Moreover, a special Kähler structure is induced on the base manifold.     

Geometry of jet spaces and integrable systems

An overview of some recent results on the geometry of partial differential equations in application to integrable systems is given. Lagrangian and Hamiltonian formalism both in the free case (on the space of infinite jets) and with constraints (on a PDE) are discussed. Analogs of tangent and cotangent bundles to a differential equation are introduced and the variational Schouten bracket is defined. General theoretical constructions are illustrated by a series of examples.     

Tau functions and the twistor theory of integrable systems

We first define τ-functions as generalized cross-ratios of four points on a finite- or infinite-dimensional Grassmannian. We show how this definition can be used to construct a natural flat connection on a determinant line bundle associated with two equivariant holomorphic vector bundles over a twistor space, provided that the action of the symmetries on the bundles has the same normal form at the fixed points for the two bundles. The determinant line bundle has a natural meromorphic section of which the logarithmic covariant derivative is the logarithmic derivative of the τ-function. We establish a natural product formula for this τ-function; we show that it vanishes at the jumping lines of one bundle and has poles at the jumping lines of the other. We also show that this definition leads to standard expressions for the τ-functions of the KdV equation, the Ernst equation, and the isomonodromic deformation equations. We describe a new twistor treatment of the isomonodromic deformation equations.     

Open problems in the theory of finite-dimensional integrable systems and related fields

This paper collects a number of open problems in the theory of integrable systems and related fields, their study being suggested by the main lecturers and participants of the Advanced Course on Geometry and Dynamics of Integrable Systems, from September 9th to 14th 2013, as well as the Conference on Integrability, Topological Obstructions to Integrability and Interplay with Geometry, from September 16th to 20th 2013, both held at the Centre de Recerca Mathematica in Barcelona within the Research Programme “Geometry and Dynamics of Integrable Systems”.     

Density-matrix spectra for integrable models

The spectra which occur in numerical density-matrix renormalization group (DMRG) calculations for quantum chains can be obtained analytically for integrable models via corner transfer matrices. This is shown in detail for the transverse Ising chain and the uniaxial XXZ Heisenberg model and explains in particular their exponential character in these cases.     

The affine Weyl group symmetry of Desargues maps and of the non-commutative Hirota-Miwa system

We study recently introduced Desargues maps, which provide simple geometric interpretation of the non-commutative Hirota-Miwa system. We characterize them as maps of the A-type root lattice into a projective space such that images of vertices of any basic regular N-simplex are collinear. Such a characterization is manifestly invariant with respect to the corresponding affine Weyl group action, which leads to related symmetries of the Hirota-Miwa system.     

A construction of N = 2 and centerless N = 4 superconformal fields via affine superalgebras

In this paper we give a new construction of the N = 2 superconformal algebra using currents of the affine superalgebra and free bosonic fields, and also the N = 4 supercomformal algebra without central charge in terms of currents of and free bosonic fields.     

Properties of supersymmetric integrable systems of KP type

The recently proposed supersymmetric extensions of reduced Kadomtsev-Petviashvili (KP) integrable hierarchies in N = 1, 2 superspace are shown to contain in the purely bosonic limit new types of ordinary non-supersymmetric integrable systems. The latter are coupled systems of several multi-component non-linear Schr?dinger-like hierarchies whose basic nonlinear evolution equations contain additional quintic and higher-derivative nonlinear terms. Also, we obtain the N = 2 supersymmetric extension of Toda chain model as Darboux-B?cklund orbit of the simplest reduced N = 2 super-KP hierarchy and find its explicit solution. Received 13 September 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: nissimov@inrne.bas.bg RID="b" ID="b"e-mail: svetlana@inrne.bas.bg     

On a supersymmetric nonlinear integrable equation in (2+1) dimensions

A supersymmetric integrable equation in (2+1) dimensions is constructed by means of the approach of the homogenous space of the super Lie algebra, where the super Lie algebra osp(3/2) is considered. For this (2+1) dimensional integrable equation, we also derive its Bäcklund transformation.     

The moduli space of flat SU (2) and SO (3) connections over surfaces

All the connected components of the moduli space of flat connections on SU (2) and SO (3) (trivial and non-trivial) bundles over closed oriented surfaces are determined. The symplectic structure and volumes of the non-maximal strata of the moduli space are also determined.     

Probing integrable perturbations of conformal theories using singular vectors (II). N = 1 superconformal theories

In this work we pursue the singular-vector analysis of the integrable perturbations of conformal theories that was initiated in our earlier paper [Nucl. Phys. B 475 (1996) 361]. Here we consider the detailed study of the N = 1 superconformal theory and show that all integrable perturbations can be identified from a simple singular-vector argument. We identify these perturbations as theories based on affine Lie superalgebras and show that the results we obtain relating two perturbations can be understood by the extension of affine Toda duality to these theories with fermions. We also discuss how this duality is broken in specific cases.     

Some integrable systems related to affine lie algebras and homogeneous spaces

In an attempt to understand how a result on hamiltonian systems related to hermitian symmetric spaces can be seen as an application of the Adler-Kostant-Symes theorem, it is found that quite substantial extensions to this result can be made.     

Some compatible Poisson structures and integrable bi-Hamiltonian systems on four dimensional and nilpotent six dimensional symplectic real Lie groups

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi’s theorem we obtain new bi-Hamiltonian systems with four dimensional and nilpotent six dimensional symplectic real Lie groups as phase spaces.     

Deviation operator and deviation equations over spaces with affine connections and metrics

The notion of deviation operator over spaces with affine connection (L_n-spaces) is introduced and its applications to deviation equations is considered. On the basis of a deviation identity, by means of sufficient or necessary and sufficient conditions, different deviation equations are obtained and considered. It is shown that the deviation equation for auto-parallel trajectories in L_n-spaces (geodesics in V_n-spaces) allows also other solutions than the well-known solutions for auto-parallel (geodesic) trajectories.     

Contact flows and integrable systems

We consider Hamiltonian systems restricted to the hypersurfaces of contact type and obtain a partial version of the Arnold–Liouville theorem: the system need not be integrable on the whole phase space, while the invariant hypersurface is foliated on an invariant Lagrangian tori. In the second part of the paper we consider contact systems with constraints. As an example, the Reeb flows on Brieskorn manifolds are considered.     

Effect of the modulation waveform on the S/N ratio of laser scanning imaging systems

In a laser scanning imaging system, the modulation of light output is directly achieved by modulating the drive current of a laser diode input signal. So far, what kind of modulation waveform is the best remains undetermined. People usually choose sine modulation for convenience. Through our theoretical analysis and experiment, we have reached the conclusion that for the same receiver, substituting low duty (a duty of is small enough) rectangular modulation for conventional sine modulation will offer at least a 5 dB improvement in S/N ratio with the average incident optical power unchanged.     

Integrable systems on so(n)-valued Lie algebras on higher genus curves

We consider special quasi-graded so(n)-valued Lie algebras on higher genus curves. Using them we find new Lie theoretical interpretation of the integrability of generalized Manakov tops and generalized Clebsch systems and explicitly construct their isomorphism on trajectories.