165 research outputs found
Nonlinear holomorphic supersymmetry, Dolan-Grady relations and Onsager algebra
Recently, it was noticed by us that the nonlinear holomorphic supersymmetry
of order , (-HSUSY) has an algebraic origin. We show that the
Onsager algebra underlies -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 -HSUSY and fix it explicitly for the cases of
. 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
-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 -symmetric
quantum mechanics.Comment: 18 pages, refs updated; to appear in Nucl. Phys.
Note on antisymmetric spin-tensors
It was known for a long time that in d = 4 dimensions it is impossible to
construct the Lagrangian for antisymmetric second rank spin-tensor that will be
invariant under the gauge transformations with unconstrained spin-vector
parameter. But recently a paper arXiv:0902.1471 appeared where gauge invariant
Lagrangians for antisymmetric spin-tensors of arbitrary rank n in d > 2n were
constructed using powerful BRST approach. To clarify apparent contradiction, in
this note we carry a direct independent analysis of the most general first
order Lagrangian for the massless antisymmetric spin-tensor of second rank. Our
analysis shows that gauge invariant Lagrangian does exist but in d > 4
dimensions only, while in d = 4 this Lagrangian becomes identically zero. As a
byproduct, we obtain a very simple and convenient form of this massless
Lagrangian that makes deformation to AdS space and/or massive case a simple
task as we explicitly show here. Moreover, this simple form admits natural and
straightforward generalization on the case of massive antisymmetric
spin-tensors of rank n for d > 2n.Comment: 7 pages, no figure
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.
Gravitational and higher-derivative interactions of massive spin 5/2 field in (A)dS space
Using on-shell gauge invariant formulation of relativistic dynamics we study
interaction vertices for a massive spin 5/2 Dirac field propagating in (A)dS
space of dimension greater than or equal to four. Gravitational interaction
vertex for the massive spin 5/2 field and all cubic vertices for the massive
spin 5/2 field and massless spin 2 field with two and three derivatives are
obtained. In dimension greater that four, we demonstrate that the gravitational
vertex of the massive spin 5/2 field involves, in addition to the standard
minimal gravitational vertex, contributions with two and three derivatives. We
find that for the massive spin 5/2 and massless spin 2 fields one can build two
higher-derivative vertices with two and three derivatives. Limits of massless
and partial massless spin 5/2 fields in (A)dS space and limits of massive and
massless spin 5/2 fields in flat space are discussed.Comment: 51 pages, LaTeX-2e, v3: Section 1 is divided into Sections 1-6.
Discussion of gravitational and higher-derivative vertices added to Sections
2-6. Tables I, II and Appendices B,C,D,E,F added. Typos correcte
- …