Supersymmetry without Fermions

Annals of Physics, 1996

Within a group-theoretical approach to the description of (2+1)-dimensional anyons, the minimal covariant set of linear differential equations is constructed for the fractional spin fields with the help of the deformed Heisenberg algebra (DHA), [a − , a + ] = 1 + νK, involving the Klein operator K, {K, a ± } = 0, K 2 = 1. The connection of the minimal set of equations with the earlier proposed 'universal' vector set of anyon equations is established. On the basis of this algebra, a bosonization of supersymmetric quantum mechanics is carried out. The construction comprises the cases of exact and spontaneously broken N = 2 supersymmetry allowing us to realize a Bose-Fermi transformation and spin-1/2 representation of SU(2) group in terms of one bosonic oscillator. The construction admits an extension to the case of OSp(2|2) supersymmetry, and, as a consequence, both applications of the DHA turn out to be related. A possibility of 'superimposing' the two applications of the DHA for constructing a supersymmetric (2+1)-dimensional anyon system is discussed. As a consequential result we point out that osp(2|2) superalgebra is realizable as an operator algebra for a quantum mechanical 2-body (nonsupersymmetric) Calogero model.

This paper constitutes a review on N=2 fractional supersymmetric Quantum Mechanics of order k. The presentation is based on the introduction of a generalized Weyl-Heisenberg algebra W_k. It is shown how a general Hamiltonian can be associated with the algebra W_k. This general Hamiltonian covers various supersymmetrical versions of dynamical systems (Morse system, Poschl-Teller system, fractional supersymmetric oscillator of order k, etc.). The case of ordinary supersymmetric Quantum Mechanics corresponds to k=2. A connection between fractional supersymmetric Quantum Mechanics and ordinary supersymmetric Quantum Mechanics is briefly described. A realization of the algebra W_k, of the N=2 supercharges and of the corresponding Hamiltonian is given in terms of deformed-bosons and k-fermions as well as in terms of differential operators.

Modern Physics Letters A, 1996

The minimal bosonization of supersymmetry in terms of one bosonic degree of freedom is considered. A nontrivial relationship of the construction to the Witten supersymmetric quantum mechanics is illustrated with the help of the simplest N=2 SUSY system realized on the basis of the ordinary (undeformed) bosonic oscillator. It is shown that the generalization of such a construction to the case of Vasiliev deformed bosonic oscillator gives a supersymmetric extension of the two-body Calogero model in the phase of exact or spontaneously broken N=2 SUSY. The construction admits an extension to the case of the OSp(2|2) supersymmetry, and, as a consequence, osp(2|2) superalgebra is revealed as a dynamical symmetry algebra for the bosonized supersymmetric Calogero model. Realizing the Klein operator as a parity operator, we construct the bosonized Witten supersymmetric quantum mechanics. Here the general case of the corresponding bosonized N=2 SUSY is given by an odd function being a superpo...

2012

Applying the concept of a momentum map for supersymplectic supervectorspaces to the onedimensional Bose-Fermi oscillator, we show that the largest symmetry group that admits a momentum map is the identity component of the intersection of the orthosymplectic group OSp(2|2) and the group of supersymplectic transformations. This gives a systematic characterization of a certain class of odd supersymmetry transformations that were originally introduced in an ad hoc way.

2004

We demonstrate that two-dimensional N = 8 supersymmetric quantum mechanics which inherits the most interesting properties of N = 2, d = 4 SYM can be constructed if the reduction to one dimension is performed in terms of the basic object, i.e. the N = 2, d = 4 vector multiplet. In such a reduction only complex scalar fields from the N = 2, d = 4 vector multiplet become physical bosons in d = 1, while the rest of the bosonic components are reduced to auxiliary fields, thus giving rise to the (2, 8, 6) supermultiplet in d = 1. We construct the most general action for this supermultiplet with all possible Fayet-Iliopoulos terms included and explicitly demonstrate that the action possesses duality symmetry extended to the fermionic sector of theory. In order to deal with the second-class constraints present in the system, we introduce the Dirac brackets for the canonical variables and find the supercharges and Hamiltonian which form a N = 8 super Poincarè algebra with central charges. Finally, we explicitly present the generalization of two-dimensional N = 8 supersymmetric quantum mechanics to the 2kdimensional case with a special Kähler geometry in the target space.

Central European Journal of Physics, 2010

The N -extended Supersymmetric Quantum Mechanics is deformed via an abelian twist which preserves the super-Hopf algebra structure of its Universal Enveloping Superalgebra. Two constructions are possible. For even N one can identify the 1D N -extended superalgebra with the fermionic Heisenberg algebra. Alternatively, supersymmetry generators can be realized as operators belonging to the Universal Enveloping Superalgebra of one bosonic and several fermionic oscillators.

Physical Review D, 2000

The multidimensional N = 4 supersymmetric quantum mechanics (SUSY QM) is constructed using the superfield approach. As a result, the component form of the classical and quantum Lagrangian and Hamiltonian is obtained. In the considered SUSY QM both classical and quantum N = 4 algebras include central charges, and this opens various possibilities for partial supersymmetry breaking. It is shown that quantum mechanical models with one quarter, one half and three quarters of unbroken(broken) supersymmetries can exist in the framework of the multidimensional N = 4 SUSY QM, while the one-dimensional N = 4 SUSY QM, constructed earlier, admits only one half or total supersymmetry breakdown. We illustrate the constructed general formalism, as well as all possible cases of the partial SUSY breaking taking as an example a direct multidimensional generalization of the one-dimensional N = 4 superconformal quantum mechanical model. Some open questions and possible applications of the constructed multidimensional N = 4 SUSY QM to the known exactly integrable systems and problems of quantum cosmology are briefly discussed. * The supersymmetric quantum mechanics (SUSY QM), being first introduced in [1] - for the N = 2 case, turns out to be a convenient tool for investigating problems of supersymmetric field theories, since it provides the simple and, at the same time, quite adequate understanding of various phenomena arising in relativistic theories.

2010

Supersymmetric quantum mechanics (SUSY QM) is a powerful tool for generating new potentials with known spectra departing from an initial solvable one. In these lecture notes we will present some general formulae concerning SUSY QM of first and second order for one-dimensional arbitrary systems, and we will illustrate the method through the trigonometric Pöschl-Teller potentials. Some intrinsically related subjects, as the algebraic structure inherited by the new Hamiltonians and the corresponding coherent states will be analyzed. The technique will be as well implemented for periodic potentials, for which the corresponding spectrum is composed of allowed bands separated by energy gaps.

Physics Letters B, 1999

A parametrization of the Hamiltonian of the generalized Witten model of the SUSY QM by a single arbitrary function in d = 1 has been obtained for an arbitrary number of the supersymmetries N = 2N. Possible applications of this formalism have been discussed. It has been shown that the N = 1 and 2 conformal SUSY QM is generalized for any N .

Modern Physics Letters A, 2005

We construct a two-parameter deformed SUSY algebra by constructing SUSY generators which are bilinears of n (p,q)-deformed fermions covariant under the quantum group SU p/q(n) and n undeformed bosons. The Fock space representation of the algebra constructed is discussed and the total deformed Hamiltonian for such a system is obtained. Some physical applications of the quantum group covariant two-parameter deformed fermionic oscillator algebra are also considered.