Nonlinear holomorphic supersymmetry, Dolan-Grady relations and Onsager algebra

Recently, it was noticed by us that the nonlinear holomorphic supersymmetry of order $n\in\N, n>1$, ($n$-HSUSY) has an algebraic origin. We show that the Onsager algebra underlies $n$-HSUSY and investigate the structure of the former in the context of the latter. A new infinite set of mutually commuting charges is found which, unlike those from the Dolan-Grady set, include the terms quadratic in the Onsager algebra generators. This allows us to find the general form of the superalgebra of $n$-HSUSY and fix it explicitly for the cases of $n=2,3,4,5,6$. The similar results are obtained for a new, contracted form of the Onsager algebra generated via the contracted Dolan-Grady relations. As an application, the algebraic structure of the known 1D and 2D systems with $n$-HSUSY is clarified and a generalization of the construction to the case of nonlinear pseudo-supersymmetry is proposed. Such a generalization is discussed in application to some integrable spin models and with its help we obtain a family of quasi-exactly solvable systems appearing in the $PT$-symmetric quantum mechanics.Comment: 18 pages, refs updated; to appear in Nucl. Phys.

Massive Fields of Arbitrary Integer Spin in Symmetrical Einstein Space

We study the propagation of gauge fields with arbitrary integer spins in the symmetrical Einstein space of any dimensionality. We reduce the problem of obtaining a gauge-invariant Lagrangian of integer spin fields in such background to an purely algebraic problem of finding a set of operators with certain features using the representation of high-spin fields in the form of some vectors of pseudo-Hilbert space. We consider such construction in the linear order in the Riemann tensor and scalar curvature and also present an explicit form of interaction Lagrangians and gauge transformations for massive particles with spins 1 and 2 in terms of symmetrical tensor fields.Comment: 15 pages, latex, no figures,minor change

Nonlinear Holomorphic Supersymmetry on Riemann Surfaces

We investigate the nonlinear holomorphic supersymmetry for quantum-mechanical systems on Riemann surfaces subjected to an external magnetic field. The realization is shown to be possible only for Riemann surfaces with constant curvature metrics. The cases of the sphere and Lobachevski plane are elaborated in detail. The partial algebraization of the spectrum of the corresponding Hamiltonians is proved by the reduction to one-dimensional quasi-exactly solvable sl(2,R) families. It is found that these families possess the "duality" transformations, which form a discrete group of symmetries of the corresponding 1D potentials and partially relate the spectra of different 2D systems. The algebraic structure of the systems on the sphere and hyperbolic plane is explored in the context of the Onsager algebra associated with the nonlinear holomorphic supersymmetry. Inspired by this analysis, a general algebraic method for obtaining the covariant form of integrals of motion of the quantum systems in external fields is proposed.Comment: 24 pages, new section and refs added; to appear in Nucl. Phys.

Additional restrictions on quasi-exactly solvable systems

In this paper we discuss constraints on two-dimensional quantum-mechanical systems living in domains with boundaries. The constrains result from the requirement of hermicity of corresponding Hamiltonians. We construct new two-dimensional families of formally exactly solvable systems and applying such constraints show that in real the systems are quasi-exactly solvable at best. Nevertheless in the context of pseudo-Hermitian Hamiltonians some of the constructed families are exactly solvable.Comment: 11 pages, 3 figures, extended version of talk given at the International Workshop on Classical and Quantum Integrable Systems "CQIS-06", Protvino, Russia, January 23-26, 200