The six-point families of exceptional representations of the conformal group

Six-point families of topological reducible (indecomposable) representations of the conformal group of space-time M ₄ are studies. The structure of invariant subspaces is analysed and the full set of equivalence relations among the suprepresentations is derived.     

Intertwining operators for solving differential equations,with applications to symmetric spaces

The use of intertwining operators to solve both ordinary and partial differential equations is developed. Classes of intertwining operators are constructed which transform between Laplacians which are self-adjoint with respect to different non-trivial measures. In the two-dimensional case, the intertwining operator transforms a non-separable partial differential operator to a separable one. As an application, the heat kernels on the rank 1 and rank 2 symmetric spaces are constructed.     

Supersymmetry in complex space-time

An interconnection between superluminal transformation and supersymmetric transformations has been investigated in complex C³-space and the evolution of bosonic and fermionic subspaces in such space has been undertaken. Introducing the suitable anticommuting operators to induce grading in Poincare group in C³-space in terms of components of complex angular momentum operator, the supersymmetric algebra connecting bradyonic and tachyonic bosons and fermions has been constructed and it has been demonstrated that the difference between scales of bosonic and fermionic subspaces in C³-space increases quickly in spite of their closeness initially.     

On the group-theoretical approach to extended conformal supersymmetry: Classification of multiplets

Physically important multiplets of indecomposable representations of the conformal superalgebra su(2, 2/N) are constructed and classified adapting results of Kac. Each multiplet exhausts, in particular, all superfields partially equivalent to a given one together with some invariant operators intertwining the superfields. Known examples of the linearized supergauge conformal theories are accommodated in this scheme.On leave of absence from the Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria.     

Hidden algebra of the N-body Calogero problem

A certain generalization of the algebra gl(N, ) of first-order differential operators acting on a space of inhomogeneous polynomials in ^N−1 is constructed. The generators of this (non-) Lie algebra depend on permutation operators. It is shown that the Hamiltonian of the N-body Calogero model can be represented as a second-order polynomial in the generators of this algebra. The representation given implies that the Calogero Hamiltonian possesses infinitely-many finite-dimensional invariant subspaces with explicit bases, which are closely related to the finite-dimensional representations of the above algebra. This representation is an alternative to the standard representation of the Bargmann-Fock type in terms of creation and annihilation operators.     

Renormalization for breakup of invariant tori

We present renormalization group operators for the breakup of invariant tori with winding numbers that are quadratic irrationals. We find the simple fixed points of these operators and interpret the map pairs with critical invariant tori as critical fixed points. Coordinate transformations on the space of maps relate these fixed points, and also induce conjugacies between the corresponding operators.     

Conditional Lie–Bäcklund symmetries and invariant subspaces to nonlinear diffusion equations with source

This paper considers conditional Lie–Bäcklund symmetries of the radially symmetric nonlinear diffusion equations with source. We obtain a complete list of canonical forms for such equations which admit higher-order conditional symmetries. As a consequence, the solutions of the resulting equations are constructed on the invariant subspaces admitted by the corresponding equations.     

Logarithmic Intertwining Operators and Vertex Operators

This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg and lattice vertex algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra M(1) _a , of central charge 1 − 12a ². We classify these operators in terms of depth and provide explicit constructions in all cases. Our intertwining operators resemble puncture operators appearing in quantum Liouville field theory. Furthermore, for a = 0 we focus on the vertex operator subalgebra L(1, 0) of M(1)₀ and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of hidden logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1, 0)-module.     

Two-dimensional Schrödinger Hamiltonians with effective mass in SUSY approach

The general solution of SUSY intertwining relations of first order for two-dimensional Schrödinger operators with position-dependent (effective) mass is built in terms of four arbitrary functions. The procedure of separation of variables for the constructed potentials is demonstrated in general form. The generalization for intertwining of second order is also considered. The general solution for a particular form of intertwining operator is found, its properties—symmetry, irreducibility, and separation of variables—are investigated.     

Polynomial supersymmetry for matrix Hamiltonians

We study intertwining relations for matrix non-Hermitian Hamiltonians by matrix differential operators of arbitrary order. It is established that for any intertwining operator of minimal order there is operator that intertwines the same Hamiltonians in the opposite direction and such that the products of these operators are identical polynomials of the corresponding Hamiltonians. The related polynomial algebra of supersymmetry is constructed. The problems of minimization and reducibility of a matrix intertwining operator are considered and the criteria of minimizability and reducibility are presented. It is shown that there are absolutely irreducible matrix intertwining operators, in contrast to the scalar case.     

Invariant subspaces of clustering operators. II

Clustering operators, when restricted tok-particle invariant subspaces, are shown still to cluster.     

A modular subgroup invariance in a four-dimensional string

A four-dimensional closed supersymmetric string theory is constructed from the left-moving Neveu-Schwarz bosonic string and the right-moving superstring. The gauge group [SU(2)]⁶ arises from an algebraic compactification involving a new affine Kac-Moody construction in terms of Neveu-Schwarz operators. The theory is Lorentz-invariant, tachyon-free and has four-point one-loop amplitudes invariant under a subgroup of the modular group.     

Conformal Operators on Forms and Detour Complexes on Einstein Manifolds

For even dimensional conformal manifolds several new conformally invariant objects were found recently: invariant differential complexes related to, but distinct from, the de Rham complex (these are elliptic in the case of Riemannian signature); the cohomology spaces of these; conformally stable form spaces that we may view as spaces of conformal harmonics; operators that generalise Branson’s Q-curvature; global pairings between differential form bundles that descend to cohomology pairings. Here we show that these operators, spaces, and the theory underlying them, simplify significantly on conformally Einstein manifolds. We give explicit formulae for all the operators concerned. The null spaces for these, the conformal harmonics, and the cohomology spaces are expressed explicitly in terms of direct sums of subspaces of eigenspaces of the form Laplacian. For the case of non-Ricci flat spaces this applies in all signatures and without topological restrictions. In the case of Riemannian signature and compact manifolds, this leads to new results on the global invariant pairings, including for the integral of Q-curvature against the null space of the dimensional order conformal Laplacian of Graham et al.     

The Geometry of Recursion Operators

We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser’s theorem for simultaneous isotopies of two families of symplectic forms. We also consider the geometric structures defined by pairs and triples of symplectic forms for which the squares of the intertwining endomorphisms are plus or minus the identity. For pairs of forms we recover the notions of symplectic pairs and of holomorphic symplectic structures. For triples we recover the notion of a hypersymplectic structure, and we also find three new structures that have not been considered before. One of these is the symplectic formulation of hyper-Kähler geometry, which turns out to be a strict generalization of the usual definition in terms of differential or Kähler geometry.     

Neumann-Bogolyubov-Rosochatius Oscillatory Dynamical Systems and their Integrability Via Dual Moment Maps. Part I

Abstract

The finite-dimensional invariant subspaces of the solutions of intergrable by Lax infinite-dimensional Benney-Kaup dynamical system are presented. These invariant subspaces carry the canonical symplectic structure, with relation to which the Neumann type dynamical systems are Hamiltonian and Liouville intergrable ones. For the Neumann-Bogolyubov and Neumann-Rosochatius dynamical systems, the Lax-type representations via the dual moment maps into some deformed loop algebras as well as the finite hierarchies of conservation laws are constructed.     

Deductive inference of nuclear boson models (IBM, TQM, FQM) based on SU(6) dynamical symmetry

A single deductive inference of Schwinger realization (= interacting boson model—IBM), Holstein-Primakoff realization (= truncated quadrupole phonon model—TQM) and Dyson realization (= finite quadrupole phonon model—FQM) of dynamical SU(6) quadrupole collective algebra (QCA) is presented with a full scope of their isomorphism on the level of representations. Dyson realization of QCA is explicitly constructed by using holomorphically parametrized generalized coherent state and explicit form of root vectors. Utilizing appropriate orthogonalizing operators Holstein-Primakoff realization of QCA has been derived from the Dyson realization. The carrier spaces of Schwinger and Holstein-Primakoff realizations are investigated on the same footing and Marshalek's boson is rigorously derived. The intertwining operator which connects Schwinger and Holstein-Primakoff realizations is constructed and its domain and image are determined. It is shown that the intertwining operator has well-defined inverse in a definite factor space of the IBM basis space which is proved to be isomorphic to the physical subspace of the TQM basis space, meaning equivalence of IBM and TQM on level of representations.     

Vertex Operators, Grassmannians, and Hilbert Schemes

We approximate the infinite Grassmannian by finite-dimensional cutoffs, and define a family of fermionic vertex operators as the limit of geometric correspondences on the equivariant cohomology groups, with respect to a one-dimensional torus action. We prove that in the localization basis, these are the well-known fermionic vertex operators on the infinite wedge representation. Furthermore, the boson-fermion correspondence, locality, and intertwining properties with the Virasoro algebra are the limits of relations on the finite-dimensional cutoff spaces, which are true for geometric reasons. We then show that these operators are also, almost by definition, the vertex operators defined by Okounkov and the author in Carlsson and Okounkov ( [math.AG], 2009), on the equivariant cohomology groups of the Hilbert scheme of points on \mathbb C²{\mathbb C^2} , with respect to a special torus action.     

On kicked quantum systems

The time evolution of a multi-dimensional quantum system which is kicked at random or periodically with a potential is obtained. An interesting aspect of the evolution is that if the operator corresponding to the potential has invariant subspaces (this is characteristic of multi-dimensional problems), the system evolves in these invariant subspaces, i.e., each evolution in the subspaces is independent and there cannot be any mixing between the states of these subspaces.